Research ArticleRegularity and Chaos in the Hydrogen Atom HighlyExcited with a Strong Magnetic Field

M Amdouni1 and H Eleuch2

1 Bizerte Preparatory Engineering Institute University of Carthage 7021 Zarzouna Tunisia2 Department of Physics McGill University Montreal QC Canada H3A 2T8

Correspondence should be addressed to H Eleuch heleuchphysicsmcgillca

Received 19 August 2014 Revised 17 November 2014 Accepted 17 November 2014 Published 3 December 2014

Academic Editor Yan-Jun Liu

Copyright copy 2014 M Amdouni and H EleuchThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The effects of the relativistic corrections on the energy spectra are analyzed Effective simulations based on manipulations ofoperators in the Sturmian basis are developed Discrete and continuous energy spectra of a hydrogen atom with realistic nucleusmass in a strong magnetic field are computedThe transition from regularity to chaos in diamagnetic problem with the effect of thenucleus recoil energy is explored Anticrossing of energy levels is observed for strong magnetic field

1 Introduction

A lot of works have been recently dedicated to the study oftwo- or multilevel systems such as Rydberg atom excitonicsystems and nanoresonators [1ndash13] Its great importance liesin its implication in different branches of physics astro-physics [9 12] atomic physics [14ndash17] molecular physics [1819] spectroscopy [20 21] solid state [22] and plasma physics[23] It leads to interesting quantum features such as solitonpropagation [24ndash30] entanglement [31 32] antibunching[33] squeezing [34] bistability [35] and chaos [36 37]

The Rydberg atom in a static magnetic field is a simplesystem which can be investigated experimentally and theo-retically It turns out a prototype of quantum systems whoseclassical behavior may show chaotical dynamics [38ndash42]Chaotic appearance results in the fact that for certain initialconditions the behavior of the dynamical system becomesunpredictable [43 44] Chaos is identified by high sensitivityto the initial conditions Indeed a system with 119899 degreesof freedom is regular if there are 119899 independent constantsof motion [45] These constants of motion allow remainingunchanged for instance the regularly spaced energy levels[38]

Consequently many researches have been carried outto discover the connections between classical and quantumchaos through Rydberg atoms in strong static fields [46 47]

To a large degree the justification for this concern isattributable to the relevance of chaos in manifold branches ofscience such as physical chemical biological electrical andsocioeconomical systems [38 40 41 48ndash54] In physics thisproblem is related to the atomic magnetism [55] especiallyto explore the impact of the diamagnetic effect [38 56] onthe energy diagrams the overlapping and the anticrossingof energy levels Several techniques [57 58] have beenperformed to explore the complete part of a magnetic field

The hydrogen atom in a strong magnetic field wasexplored [3 12 38 55 59 60] The hydrogenoid and nonhy-drogenoid atoms in the presence of the external fieldwere alsostudied [38 59 61 62] In the regime of strong external fieldDelande has shown that the system behavior is qualitativelysimilar to that of the hydrogen atom [59] All these workswere limited to the nonrelativistic case In our previous paper[63] we have studied the effect of relativistic corrections onthe regularity behavior of the hydrogen atom in a strong staticmagnetic field This paper aims to complete the relativisticstudy by exploring other main effects namely the realisticproton mass and the energy recoil of the nucleus

The paper is structured as follows In Section 2 we intro-duce some fundamental quantum models that are directlyrelated to our subject We describe the numerical methodsin Section 3 We calculate the eigenvalues and eigenvectorsfor different external magnetic fields in the Sturmian basis

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 957652 9 pageshttpdxdoiorg1011552014957652

2 Mathematical Problems in Engineering

In Section 4 we introduce the effect of nucleus recoil energyand we present the results Finally conclusion is given inSection 5

2 Model

21 The Bohr Model A succinct overview of the Bohr modelwhich has connection to the hydrogen atom is useful and isrequested [64 65]Thismodel enables us to introduce funda-mental quantities such us the Bohr radius and the ionizationenergy Moreover the energies determined by Bohr theoryare entirely the eigenvalues 119864

119899of the Hamiltonian This

semiclassical model has been established on the hypothesisthat the electron describes a circular orbit of radius 119903 aroundthe proton satisfying the following equations [55]

119864 =

1

2

120583V2 minus1198902

119903

120583V2

119903

=

1198902

1199032

120583V119903 = 119899ℏ

(1)

The term minus1198902

119903 of the first equation is the correspondingpotential energy of the electrostatic interaction between theproton and the electron In this expression 1198902 is written as

1198902

=

1199022

41205871205760

(2)

where 119902 is the elementary electric charge whereas 120578V2119903 isthe kinetic energy of the relative motion of the proton andthe electron The ratio of 119898

119901to 119898119890is close to 1800 Thus the

reduced mass of this system is very near to 119898119890

120583 =

119898119901sdot 119898119890

119898119901+ 119898119890

≃ 119898119890(1 minus

119898119890

119898119901

) (3)

In this study we substitute the electron mass 119898119890with the

correction term 119898119890119898119901into the expression of the effective

massIn the second equation the term 119890

2

1199032 expresses the

Coulomb force which is applied to the electronThe term V2119903is the acceleration of the electron in uniform circular motion

To explain the existence of the stable discrete energylevels the Bohr quantization rules were established Besidesthe energies 119864

119899given by this model are in agreement with

quantum mechanical theory [66]

119864119899= minus

1198981198901198904

21198992ℏ2 (4)

where 119899 is the principal quantum number and ℏ is Planckrsquosconstant

In atomic units (119890 = 119898119890= ℏ = 1) the energy eigenvalues

equation (4) becomes

119864119899= minus

1

21198992 (5)

From this expression Bohr derived the transmission wave-length between different levels of 119899 using the Rydbergconstant for a nucleus of infinite mass [67]

119877infin

= 10973731568527mminus1 (6)

The dynamic symmetry group has simplified the study ofthe hydrogen atom (with 6 degrees of freedom) This groupis generated by the constants of motion namely the angularmomentum the Runge-Lenz vector (defined by = (and

)119898 minus 119903119903) and the energy 119864For a weak perturbation the group SO4 is appropriated to

explain the degeneracy of the energy levels in nonrelativisticdynamics However for the global investigation of the dis-crete and continuous states the use of the symmetry groupSO(22) is required in particular for strong perturbations InSection 3 more details are given about the group representa-tion of SO(22) for hydrogen atom excited by amagnetic field

22 Relativistic Corrections in an External Strong MagneticField We introduce Diracrsquos equation in order to take intoaccount the relativistic effects The energy levels are nowrelying on two quantum numbers the principal quantumnumber 119899 and the quantum number 119869 associated with thetotal angular momentum of the electron 119869 = + 119878 It canbe written as follows [48]

119864119899119869

= 1198981198882[

[

[

[

1 +

1205722

(radic(119895 + 12)2

minus 1205722+ 119899 minus 119895 minus 12)

2

]

]

]

]

minus12

(7)

where 120572 is the fine structure constant which defines the scaleof the splitting

120572 =

1198902

ℏ119888

(8)

TheDirac equation is generally used as aTaylor expansionin power of 1205722 as follows [48]

119864119899119869

= minus

1198981198882

1205722

21198992

[1 +

1205722

1198992(

119899

119869 + 12

minus

3

4

) + sdot sdot sdot ] (9)

Dependence 11198993 of the fine structure is immediately

apparent in this expression We studied this effect in ourprevious paper [63] Dirac theory takes into account thequantum and the relativistic nature of the electron howeverit does not reflect accurately the finite mass of the nucleus Inthe equations above themass of the nucleus is still consideredinfinite To take into account the finite mass we substitute inthe Dirac equation the mass of the electron by the reducedmass 120583 = 119898119872(119898 + 119872) of the electron-proton system (119872 isthe proton mass) Furthermore we add other corrections tothe energy levels obtainedThe first is a relativistic correctiondue to the recoil of the nucleus It is of the order of 1205722 with

Mathematical Problems in Engineering 3

respect to the ground state energy of the hydrogen atom Itsexpression is quite simple [48]

119864119899(recoil) = minus

1205782

1198882

119872 + 119898

1205724

81198994 (10)

Replacing the electron mass by the reduced mass theexpressions of the energies (4) and (10) are written in atomicunit (119898 = 1 119872 = 1836153996) [48]

119864119903

119899= minus(1 minus

119898

119872

)

1

21198992

119864119899(recoil) = minus

1

119872

(1 minus

3119898

119872

)

1205722

81198994

(11)

Over the last three decades several works have beendedicated to the understanding of Rydberg atoms highlyexcited hydrogen atom in strong static fields These studieshave led to high-resolution numerical spectra as well asimproving the accuracy of the calculations of the magneticfield in many problems [48 68] If the contribution of theelectromagnetic field compared to undisturbed energies isequal or higher then the magnetic field is strong In ourcontext chaos can be realized in the experimentwithRydbergstates (119899 = 30 31)

We will study later this effect on energy diagrams Wewrite the Hamiltonian of hydrogen atom in a uniformmagnetic field as [63]

119867 = 1198671+ 119867119901+ 119867119889 (12)

where 1198671 119867119901 and 119867

119889are defined as follows [66]

1198671= 1198981198901198882

+ 1198670+ 119882119898V + 119882SO + 119882

119863

119867119901=

1198902

2119898119890

sdot ( + 2 119878)

119867119889=

1198902

1198612

8119898119890

1199032sin2 (120579)

(13)

The eigenenergies of the Hamiltonian are given in (9) In thisrelation 119898

1198901198882 is the rest-mass energy of the electron 119867

0=

1199012

2119898119890+ 119881(119903) is the nonrelativistic Hamiltonian and the

following terms

119882119898V =

1199014

81198983

1198901198882

119882SO =

1

21198982

1198901198882

1

119903

119889119881 (119903)

119889119903

sdot 119878

119882119863

=

ℏ2

81198982

1198901198882Δ119881 (119903)

(14)

describe the fine structure and 119878 represent respectively theorbital and spin moments of the atom 119903 is the radius-vectorof the atomic electron and 120579 designs the angle between 119903 and

The paramagnetic term 119867119901is the origin of the linear

Zeeman effect offering the splitting of the atomic levelswhile the diamagnetic energy term 119867

119889is responsible for the

complex nature of the problemSince 119871

119911is a constant of the motion the linear Zeeman

energy 119867119901plays no significant role in the dynamics [68]

For high 119899 levels the diamagnetic term rapidly dominatesthe paramagnetic one and it may even become comparableto or greater than the Coulomb term itself leading to acomplete modification of the structure of atomic spectra Infact the linear Zeeman (paramagnetic) term is independentof 119899 while the diamagnetic term is proportional to 119899

4 Thehydrogen atomHamiltonian becomes nonseparablewhen thediamagnetic term is included [68 69]

In this paper we are studying the effect of the diamagneticterm 119867

119889(in atomic units 119867

119889= (119861

2

8)1199032sin2(120579)) on the

Rydberg atom119867119889is responsible for the chaos manifestation

As an important criterion chaos appears when the valueof the diamagnetic term is comparable to the unperturbedenergies [59 68]

3 Computation Method

All computed spectra presented in this paper are obtained bydigitalization of the Hamiltonian matrix in a suitable basisIt is worth noting that the spherical Sturmian basis has beenapplied for the study of hydrogen atom in a magnetic fieldOur simulations are performed by developing a code basedon MAPLE in which we can add new correction effects It isadvantageous to choose the Sturmian basis well adopted todynamic symmetries and it does not saturate the computermemory

31 Spherical Sturmian Basis Computing the spectra of dia-magnetic hydrogen in a spherical Sturmian basis was firstlyused by Edmonds [70] Clark and Taylor [68] recognized theimportance of being able to shift the length scale of the basisrelying on the region of energy and field Delande applieda complex rotation to the Sturmian basis to compute thespectrum of the diamagnetic hydrogen for positive energybeing aware that this technique has already been extended toother atoms [38]

Efficient calculations are provided by this basis choicewhich needs several considerations of symmetry [71] Allhydrogenoid states that share the same value 119872 are mixedtherefore it is a need to consider a dynamic group of diversestates It is acknowledgeable that the group SO(2 2) appliesthe previous criterion In fact this group is founded on theeigenstates that are themselves the basis of the Sturmianfunctions On one hand this basis enables the global repre-sentation of statesrsquo space (discrete continuum) On the otherhand it results in the loss of orthogonality of the basis vectors

The Sturmian basis is the proper basis describing thesystem of two oscillators equivalent to the Rydberg atom We

4 Mathematical Problems in Engineering

are able to express the Schrodinger equation in function ofthese generators ( 119878

(120572)

119879(120572)) of dynamic group SO(2 2) They

are defined respectively on an adjustable parameter 120572 [38]

119878(120572)

1= minus

120572

4

(1199012

120578+

1198712

119911

1205782) +

1

4120572

1205782

119878(120572)

2= minus

1

2

120578119901120578

119878(120572)

3=

120572

4

(1199012

120578+

1198712

119911

1205782) +

1

4120572

1205782

119879(120572)

1= minus

120572

4

(1199012

] +

1198712

119911

]2) +

1

4120572

]2

119879(120572)

2= minus

1

2

]119901]

119879(120572)

3=

120572

4

(1199012

] +

1198712

119911

]2) +

1

4120572

]2

(15)

where semiparabolic coordinates (120578 ]) are defined as

120578 = radic119903 + 119911

] = radic119903 minus 119911

(16)

while the momentums (119901120578 119901]) are defined as

119901120578= (1205782

+ ]2)119889120578

119889119905

119901] = (1205782

+ ]2)119889]119889119905

(17)

and 119871119911is the 119911-component of the angular momentum

These 6 variables are not independent

1198782

= (119878(120572)

3)

2

minus (119878(120572)

1)

2

minus (119878(120572)

2)

2

=

1198712

119911

4

1198792

= (119879(120572)

3)

2

minus (119879(120572)

1)

2

minus (119879(120572)

2)

2

=

1198712

119911

4

(18)

Thus the dynamic variables ( 119878(120572)

119879(120572)) represent the

classical formof the quantumgenerator of the dynamic groupSO(2 2)

The Hamiltonian ldquooscillatorrdquo obtained coincides with itsexpression in the context of the classic mechanic

(119878(120572)

3+ 119879(120572)

3+

1205742

2 (minus2119864)2(119878(120572)

1+ 119878(120572)

3) (119879(120572)

1+ 119879(120572)

3)

times (119878(120572)

1+ 119878(120572)

3+ 119879(120572)

1+ 119879(120572)

3) minus

1

radicminus2119864

) |Ψ⟩ = 0

(19)

with 120572 = 1radicminus2119864

We can write a more general equation by taking any valueof the adjustable parameter 120572 [57] We find

[119860(120572)

minus 120572 + (minus21198641205722

) 119861(120572)

+

1205742

1205724

2

119862(120572)

] |Ψ⟩ = 0 (20)

with

119860(120572)

=

119878(120572)

3minus 119878(120572)

1

2

+

119879(120572)

3minus 119879(120572)

1

2

119861(120572)

=

119878(120572)

3+ 119878(120572)

1

2

+

119879(120572)

3+ 119879(120572)

1

2

119862(120572)

= (119878(120572)

1+ 119878(120572)

3) (119879(120572)

1+ 119879(120572)

3) (119878(120572)

1+ 119878(120572)

3+ 119879(120572)

1+ 119879(120572)

3)

(21)

In the Sturmian basis the eigenvalues of the hydrogenHamiltonian are obtained by solving the generalized eigen-value problem (20)

The spherical Sturmian basis functions are [72]

Ψ119899119897119898

(119903) = 119878(120572)

119899119897(119903) 119884119897119898

(120579 120593) (22)

where 119878(120572)

119899119897(119903) is the radial Sturmian function

119878(120572)

119899119897(119903) = [

(119899 minus 119897 minus 1)

2 (119899 + 119897)

]

(12)

(

119903

2

)

119897+1

119890minus1199032120572

1198712119897+1

119899minus119897minus1(

119903

120572

)

(23)

and 119871119895

119894is the associated Laguerre polynomial

The full eigenfunctions used can be written as

| 119899119897119898⟩ = 119878(120572)

119899119897119884119897119898

(24)

Then the matrix element of the diamagnetic term(1198612

8)1199032sin2(120579) is given by

⟨1198991015840

1198971015840

119898

100381610038161003816100381610038161003816100381610038161003816

1198612

8

1199032sin2 (120579)

100381610038161003816100381610038161003816100381610038161003816

119899119897119898⟩ =

1

8

1198612

⟨1198991015840

119897101584010038161003816100381610038161003816119903210038161003816100381610038161003816119899119897⟩

times ⟨1198971015840

119898

10038161003816100381610038161003816sin2 (120579)1003816100381610038161003816

1003816119897119898⟩

(25)

Evaluation of the angular matrix elements is then easyThe corresponding matrix elements of the square positionoperator 1199032 are written as [68]

⟨1198991015840

119897101584010038161003816100381610038161003816119903210038161003816100381610038161003816119899119897⟩ = int

infin

0

119878120577

119899119897(119903) 1199032

119878120577

11989910158401198971015840(119903) 119889119903 (26)

Mathematical Problems in Engineering 5

are nonzero only for |119899 minus 1198991015840

| = 0 1 2 3 |119897 minus 1198971015840

| = 0 2 and atthese points take the following forms

⟨1198991015840

119897 + 2

10038161003816100381610038161003816119903210038161003816100381610038161003816119899119897⟩

=

minus

1

2

120577minus3

[(119899 minus 119897 minus 1) (119899 minus 119897 minus 2) (119899 minus 119897 minus 3)

times(119899 minus 119897 minus 4)(119899 minus 119897 minus 5)(119899 + 119897)]12

1198991015840

= 119899 minus 3

120577minus3

[(3119899 + 2119897) (119899 minus 119897 minus 1) (119899 minus 119897 minus 2)

times(119899 minus 119897 minus 3)(119899 minus 119897 minus 4)]12

1198991015840

= 119899 minus 2

minus

5

2

120577minus3

[(3119899 + 119897) (119899 minus 119897 minus 1) (119899 minus 119897 minus 2)

times(119899 minus 119897 minus 3)(119899 + 119897 + 1)]12

1198991015840

= 119899 minus 1

10119899120577minus3

[(119899 + 119897 + 1) (119899 + 119897 + 2)

times(119899 minus 119897 minus 1)(119899 minus 119897 minus 2)]12

1198991015840

= 119899

minus

5

2

120577minus3

(3119899 minus 119897) [(119899 + 119897 + 3) (119899 + 119897 + 2)

times(119899 + 119897 + 1)(119899 minus 119897 minus 1)]12

1198991015840

= 119899 + 1

120577minus3

(3119899 minus 2119897) [(119899 + 119897 + 4) (119899 + 119897 + 3)

times(119899 + 119897 + 2)(119899 + 119897 + 1)]12

1198991015840

= 119899 + 2

minus

1

2

120577minus3

[(119899 + 119897 + 5) (119899 + 119897 + 4) (119899 + 119897 + 3)

times(119899 + 119897 + 2)(119899 + 119897 + 1)(119899 minus 119897)]12

1198991015840

= 119899 + 3

⟨1198991015840

119897

10038161003816100381610038161003816119903210038161003816100381610038161003816119899119897⟩

=

2119899120577minus3

[51198992

minus 3119897 (119897 + 1) + 1] 1198991015840

= 119899

minus

3

2

120577minus3

[5119899 (119899 + 1) minus 119897 (119897 + 1) + 2]

times [(119899 + 119897 + 1) (119899 minus 119897)]12

1198991015840

= 119899 + 1

3120577

minus3

(119899 + 1) [(119899 + 119897 + 1) (119899 minus 119897)

times(119899 + 119897 + 2)(119899 minus 119897 + 1)]12

1198991015840

= 119899 + 2

minus

1

2

120577minus3

[(119899 + 119897 + 1) (119899 + 2 minus 119897) (119899 minus 119897)

times(119899 + 119897 + 3)(119899 + 119897 + 2)(119899 + 1 minus 119897)]12

1198991015840

= 119899 + 3

(27)

The investigation of Clark and Taylor [68] has revealedthat it is usually advisable to select higher values of 120577 Inour simulation we used 120577 = 2119899 in order to be close to theionization limit

Our numerical simulations consist in diagonalizing thematrix representing the operator in a Sturmian basis Thematrix elements of the generators are independent of 120572

32 Numerical Method Equation (20) is equivalent to thegeneral problem for eigenvalues of the form [59]

(119872 minus 120582119873) |Ψ⟩ = 0 (28)

where 119872 and 119873 are fixed matrices and 120582 and |Ψ⟩ are theeigenvalues and the eigenvectors searched

321 Simulation at Fixed Magnetic Field In the case wherewe fix the magnetic field 120574 = 119861119861

119888(120574 denotes the magnetic

field in units of 119861119888≃ 235 times 10

5

119879) after opting for 120572 = 1205720=

119862119905119890 we obtain an equation in the generalized eigenvalueswith

[59]

119872 = 119860(120572)

minus 1205720+

1205742

1205724

0

2

119862(120572)

119873 = 119861(120572)

120582 = minus21198641205722

0

(29)

that determines the energy levels

322 Simulation at Fixed Energy In this simulation wechoose 120572 = 120572

0= 119862119905119890 and we fix the energy So that we get

an equation in the generalized eigenvalues with [59]

119872 = 119860(120572)

minus 1205720+ (minus2119864120572

2

0) 119861(120572)

119873 = 119862(120572)

120582 =

1205742

1205724

0

2

(30)

that establishes the spectrum of magnetic field values relatedto the fixed energy

Both types of the simulations give the same outcomes

323 Diagonalization Algorithm and Convergence CriterionThe matrix components of the generators ( 119878

(120572)

119879(120572)) in a

Sturmian basis are restricted to some selectionsrsquo rules It iswell known that the matrix eigenvalue problems generally tobe solved are real symmetric matrices in strips [38]

Currently stable numerical algorithms have been devel-oped in order to calculate the eigenvalues and eigenvectors[48]

We provide an adoptable parameter 1205720to verify the

convergence for each type of the simulations In fact 1205720is

connected to the natural frequency of the basic oscillatorselected The accuracy of the examined values is not relianton this parameter Exact searched values are not dependenton this parameter

4 Numerical Results and Discussion

41 Energy Diagrams We use the simulation at fixed mag-netic field and we plot for a large number of points diagrams119864119903

119899= 119891(120574

2

) in Figures 1 2 and 3Figure 1 shows the diagram 119864(Hartree) = 119891(120574

2

) forRydberg states 119899 = 30 in the system with rather low field

6 Mathematical Problems in Engineering

minus003

minus004

minus005

minus006

minus007

minus008

minus009

E(H

artre

e)

02 04 06 08 10 12 14 16

1205742(times1010)

Figure 1 Energy levels as a function of the normalized squaremagnetic field 120574

2

(120574 = 119861119861119888 119861119888

= 235 times 105

119879) calculated by thesimulation technique described in Section 321 for weak magneticfield 119861 (119861 lt 3119879) and for the principle quantum number 119899 = 30The rovibration structure is completely removed (inter-119897-mixing)We observe that for a weak magnetic field the classical dynamicsis regular

minus005

minus010

minus015

minus020

E(H

artre

e)

1205742(times1010)

20 30 40 50

Figure 2 Diagram of energy levels of the hydrogen atom instrong magnetic field as function of 120574

2

(120574 = 119861119861119888 119861119888

= 235 times

105

119879) computed with a simulation described in Section 321 Thecalculation is done for a strong magnetic field 119861 (9119879 lt 119861 lt 18119879)

for both levels 119899 = 30 (dotted red line) and 119899 = 31 (solid blue line)Mixing between the energy levels of 119899 = 30 and 119899 = 31 is observed

(119861 lt 3119879) diamagnetism breaks the zero-field 119897-degeneracyof the hydrogen atom

We can explain this by the fact that in very weak field(119897-system intermixing) the degeneracy of the levels (therovibrational structure) proportional to 120574

2 is completelyremoved by the diamagnetic term It is visible that regularsystem disturbed by a weak perturbation remains essentiallyregular for the reason that the diamagnetic interaction is aperturbation of the Coulomb dynamics That is a regularregime there is no sensitivity on initial conditions

At highmagnetic field as shown by Figure 2 the differentenergy levels (119899 = 30 119899 = 31) essentially cross

Indeed for a highly excited state with principal quantumnumber 119899 the diamagnetic interaction scales as 120574

2

1198994 and

1205742(times1010)

20 30 40 50

minus005

minus010

minus015

minus020

E(H

artre

e)

Figure 3 Diagram of the energy levels of the hydrogen atom inthe strong magnetic field for 119899 = 30 computed by the simulationdescribed in Section 321 We note that for a high magnetic field theenergy levels anticrossThis is an indication of the chaos emergence

the zero-field as 11198992 The strong field regime is obtained

when [38]

1205741198993

≃ 1 that is 1198611198993

≃ 119861119888 (31)

This is achieved with 119899 ≃ 30 and 119861 ≃ 9119879 The rotationaland vibrational energy levels really cross By increasing themagnetic field in Figure 3 we observe that more statesanticross The classical smooth transition from regularity tochaos agrees well with this result

42The Effect of the Nucleus Recoil Energy In this section weuse the second simulation simulation at fixed energy wherewe have plotted 120574

2 as a function of the angular momentum 119897We introduced the recoil energy effect given in atomic unitby

119864119899(recoil) = minus

1

119872

(1 minus

3119898

119872

)

1205722

81198994 (32)

which takes into account the relativistic effects we recall thatthe levels shift due to this effect can be written as [68]

Δ119864119897=

1205722

21198993119897 (119897 + 1)

(33)

This recoil energy has been introduced with which wehave realized the simulation with nucleus of finite massThe result of the simulation is shown in Figure 4 wherethe distinction between the nucleus having a finite mass(diamond line) and the one having an infinite mass (circleline) is very clear This difference can be justified by the factthat the term (33) is the amount of three contributions Thefirst contribution is the kinetic energy in which the electronmass and not the reduced mass of the system manifests

119882119898V =

4

81198983

1198901198882 (34)

The second contribution is the Darwin term which takesinto account the electron delocalization but not the nucleus

ℏ2

81198982

119890119888241205871198902

120575 ( 119903) (35)

Mathematical Problems in Engineering 7

01280

01275

01270

01265

01260

01255

10 12 14 16 18 20 22 24 26 28

l

1205742(times1010)

Figure 4 The normalized square magnetic field 1205742

(120574 = 119861119861119888

119861119888

= 235 times 105

119879) for different angular momentum l with aprincipal quantum 119899 = 30 calculated by the simulation describedin Section 322 Two cases are computed first case realistic nucleusmass (finite mass) (red diamond line) second case nucleus infinitemass (blue circle line)

Finally the third contribution is the spin-orbit whichshows the interaction between the electronmagneticmomentand themagnetic field of the nucleus seen in the proper frameof the electron Therefore we should take into considerationthe nucleus speed since the nucleus of finite mass is being inmotion

5 Conclusion

We have studied the diamagnetic behavior of a hydrogenatom highly excited in a strong magnetic field consideringthe realistic finite mass of the nucleus The results revealthat when the magnetic field is weak the diamagnetic termremoves the degeneracy of the energy levels The system istotally regular it is possible to affect quantum numbers tothe various energy levels However with increasing magneticfield any symmetry will be destroyed and the energy levelswill be crossed there is an unruffled transition from regu-larity to chaotical dynamics At higher magnetic field thedynamic is totally chaotic and quantum spectrum includesonly large anticrossing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B Li and H P Liu ldquoAn effective quantum defect theory forthe diamagnetic spectrum of a barium Rydberg atomrdquo ChinesePhysics B vol 22 no 1 Article ID 013203 2013

[2] A E Shabad and V V Usov ldquoModified coulomb law in astrongly magnetized vacuumrdquo Physical Review Letters vol 98no 18 Article ID 180403 2007

[3] L Stevanovic and D Milojevic ldquoCompressed hydrogen atomunder debye screening in strong magnetic fieldrdquo Publications of

the Astronomical Observatory of Belgrade vol 91 pp 327ndash3302012

[4] H Jabri H Eleuch and T Djerad ldquoLifetimes of atomic Rydbergstates by autocorrelation functionrdquo Laser Physics Letters vol 2no 5 pp 253ndash257 2005

[5] H Eleuch and N Rachid ldquoAutocorrelation function ofmicrocavity-emitting field in the non-linear regimerdquoThe Euro-pean Physical Journal D vol 57 no 2 pp 259ndash264 2010

[6] H Eleuch N Ben Nessib and R Bennaceur ldquoQuantum modelof emission in a weakly non ideal plasmardquo The EuropeanPhysical Journal D vol 29 no 3 pp 391ndash395 2004

[7] P K Jha H Eleuch and Y V Rostovtsev ldquoCoherent control ofatomic excitation using off-resonant strong few-cycle pulsesrdquoPhysical Review A vol 82 no 4 Article ID 045805 2010

[8] E A Sete and H Eleuch ldquoStrong squeezing and robustentanglement in cavity electromechanicsrdquo Physical Review Avol 89 no 1 Article ID 013841 2014

[9] M Khatua U Sarkar and P K Chattaraj ldquoReactivity dynamicsof confined atoms in the presence of an external magnetic fieldrdquoThe European Physical Journal D vol 68 no 2 article 22 2014

[10] U Das and B Mukhopadhyay ldquoNew mass limit for whitedwarfs super-Chandrasekhar type Ia supernova as a newstandard candlerdquo Physical Review Letters vol 110 no 7 ArticleID 071102 2013

[11] A D Sargsyan R K Mirzoyan A S Sarkisyan A H Amiryanand D H Sarkisyan ldquoSplitting of N-type optical resonanceformed in Λ-system of 85Rb atoms in a strong transversemagnetic fieldrdquo Journal of Contemporary Physics vol 49 no 1pp 20ndash27 2014

[12] V S Popov and B M Karnakov ldquoHydrogen atom in a strongmagnetic fieldrdquo Physics-Uspekhi vol 57 no 3 pp 257ndash279 2014

[13] E Giacobino J-P Karr G Messin H Eleuch and A BaasldquoQuantum optical effects in semiconductor microcavitiesrdquoComptes Rendus Physique vol 3 no 1 pp 41ndash52 2002

[14] R Juarez-Amaro A Zuniga-Segundo F Soto-Eguibar andH M Moya-Cessa ldquoEquivalence between mirror-field-atomand ion-laser interactionsrdquo Applied Mathematics amp InformationSciences vol 7 no 4 pp 1311ndash1315 2013

[15] H Eleuch S Guerin and H R Jauslin ldquoEffects of an environ-ment on a cavity-quantum-electrodynamics system controlledby bichromatic adiabatic passagerdquoPhysical ReviewA vol 85 no1 Article ID 013830 2012

[16] E A Sete A A Svidzinsky Y V Rostovtsev et al ldquoUsingquantum coherence to generate gain in the xuv and x-raygain-swept superradiance and lasing without inversionrdquo IEEEJournal on Selected Topics in Quantum Electronics vol 18 no 1pp 541ndash553 2012

[17] E A Sete A A Svidzinsky H Eleuch Z Yang R D Nevelsand M O Scully ldquoCorrelated spontaneous emission on theDanuberdquo Journal of Modern Optics vol 57 no 14-15 pp 1311ndash1330 2010

[18] Z BWalters ldquoQuantum coherent dynamics at ambient temper-ature in photosynthetic moleculesrdquo Quantum Physics Lettersvol 1 no 1 pp 21ndash33 2012

[19] H Rabitz ldquoControl in the sciences over vast length and timescalesrdquo Quantum Physics Letters vol 1 no 1 pp 1ndash19 2012

[20] DA Barskiy KVKovtunov I VKoptyug et al ldquoThe feasibilityof formation and kinetics of NMR signal amplification byreversible exchange (SABRE) at high magnetic field (94 T)rdquoJournal of the American Chemical Society vol 136 no 9 pp3322ndash3325 2014

8 Mathematical Problems in Engineering

[21] B Mandal S Adhikari K Choudhary et al ldquoFemtosecondresponse of quantum discrete breathers in SRR based metama-terials and the role of dielectric permittivityrdquo Quantum PhysicsLetters vol 1 no 2 pp 59ndash68 2012

[22] J D Pritchard J A Isaacs M A Beck R McDermottand M Saffman ldquoHybrid atom-photon quantum gate in asuperconducting microwave resonatorrdquo Physical Review A vol89 no 1 Article ID 010301 2014

[23] P Rej and A Ghoshal ldquoRydberg transitions for positron-hydrogen collisions asymptotic cross section and scaling lawrdquoJournal of Physics B Atomic Molecular and Optical Physics vol47 no 1 Article ID 015204 2014

[24] H Eleuch D Elser and R Bennaceur ldquoSoliton propagation inan absorbing three-level atomic systemrdquo Laser Physics Lettersvol 1 no 8 pp 391ndash396 2004

[25] H Eleuch and R Bennaceur ldquoAn optical soliton pair amongabsorbing three-level atomsrdquo Journal of Optics A Pure andApplied Optics vol 5 no 5 pp 528ndash533 2003

[26] E Osman M Khalfallah and H Sapoor ldquoImproved (G1015840G)-expansion method for elliptic-like equation and general trav-eling wave solutions of some class of NLPDEsrdquo InformationScience Letters vol 2 no 2 pp 123ndash127 2013

[27] M Akbari ldquoExact solutions of the coupled Higgs equationand the Maccari system using the modified simplest equationmethodrdquo Information Science Letters vol 2 no 3 pp 155ndash1582013

[28] N Taghizadeh M Mirzazadeh M Akbari and M RahimianldquoExact soliton solutions for generalized equal width equationrdquoMathematical Science Letters vol 2 no 2 pp 99ndash106 2013

[29] E Osman M Khalfallah and H Sapoor ldquoFinding generalsolutions of nonlinear evolution equations by improved (G1015840G)-expansion methodrdquo Mathematical Science Letters vol 3 no 1pp 1ndash8 2014

[30] E M E Zayed ldquoExact traveling wave solutions for a variable-coefficient generalized dispersive water-wave system using thegeneralized (GrsquoG)-expansion methodrdquo Mathematical ScienceLetters vol 3 no 1 pp 9ndash15 2014

[31] E A Sete and H Eleuch ldquoInteraction of a quantum well withsqueezed light quantum-statistical propertiesrdquo Physical ReviewA vol 82 no 4 Article ID 043810 2010

[32] W Yang L Huang and M Tian ldquoTwo novel and secure quan-tum ANDOS schemesrdquo Applied Mathematics and InformationSciences vol 7 no 2 pp 659ndash665 2013

[33] H Eleuch ldquoPhoton statistics of light in semiconductor micro-cavitiesrdquo Journal of Physics B Atomic Molecular and OpticalPhysics vol 41 no 5 Article ID 055502 2008

[34] G Messin J P Karr H Eleuch J M Courty and E GiacobinoldquoSqueezed states and the quantumnoise of light in semiconduc-tor microcavitiesrdquo Journal of Physics Condensed Matter vol 11no 31 pp 6069ndash6078 1999

[35] A Baas J P Karr H Eleuch and E Giacobino ldquoOpticalbistability in semiconductor microcavitiesrdquo Physical Review AAtomic Molecular and Optical Physics vol 69 no 2 2004

[36] J Bonca and S Kruchinin Physical Properties of NanosystemsSpringer 2011

[37] L Tang and J-A Lu ldquoProjective lag synchronization of chaoticsystems with parameter mismatchrdquo Applied Mathematics ampInformation Sciences vol 7 no 1 pp 121ndash125 2013

[38] D Delande ldquoChaos and atoms in strong magnetic fieldsrdquoPhysica Scripta vol 1991 no T34 p 52 1991

[39] H Ruder G Wunner H Herold and F Geyer Atoms in StrongMagnetic Fields Springer Berlin Germany 1994

[40] H Eleuch and A Prasad ldquoChaos and regularity in semiconduc-tormicrocavitiesrdquoPhysics Letters A vol 376 no 26-27 pp 1970ndash1977 2012

[41] A Y Potekhin and A V Turbiner ldquoHydrogen atom in amagnetic field the quadrupole momentrdquo Physical Review AAtomic Molecular and Optical Physics vol 63 no 6 Article ID065402 2001

[42] S N Rasband Chaotic Dynamics of Non linear Systems JohnWiley amp Sons 1990

[43] A J Lichtenberg and M A Lieberman Regular and StochasticMotion Springer New York NY USA 1983

[44] A V Andreev O Agam B D Simons and B L AltshulerldquoQuantum chaos irreversible classical dynamics and randommatrix theoryrdquo Physical Review Letters vol 76 no 21 pp 3947ndash3950 1996

[45] O Bohigas ldquoLesHouches summer school session LIIrdquo inChaosand Quantum Physics M-J Giannoni A Voros and J Zinn-Justin Eds North-Holland Amsterdam The Netherlands1991

[46] T Pohl H R Sadeghpour and P Schmelcher ldquoCold andultracold Rydberg atoms in strong magnetic fieldsrdquo PhysicsReports vol 484 no 6 pp 181ndash229 2009

[47] T F Gallagher Rydberg Atoms Cambridge University PressCambridge UK 1994

[48] M I Eides H Grotch and V A Shelyuto Theory of LightHydrogenic Bound States vol 222 of Springer Tracts in ModernPhysics 2007

[49] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013

[50] C Yang S S Ge and T H Lee ldquoOutput feedback adaptivecontrol of a class of nonlinear discrete-time systems withunknown control directionsrdquoAutomatica vol 45 no 1 pp 270ndash276 2009

[51] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011

[52] C Yang S S Ge C Xiang T Chai and T H Lee ldquoOutputfeedback NN control for two classes of discrete-time systemswith unknown control directions in a unified approachrdquo IEEETransactions on Neural Networks vol 19 no 11 pp 1873ndash18862008

[53] Y-J Liu S-C Tong DWang T-S Li and C L P Chen ldquoAdap-tive neural output feedback controller design with reduced-order observer for a class of uncertain nonlinear SISO SystemsrdquoIEEE Transactions on Neural Networks vol 22 no 8 pp 1328ndash1334 2011

[54] T M Shahwan ldquoA comparison of Bayesian methods andartificial neural networks for forecasting chaotic financial timeseriesrdquo Journal of Statistics Applications amp Probability vol 1 no2 p 89 2012

[55] J W SeriesThe Spectrum of Atomic Hydrogen Oxford Univer-sity Press London UK 1957

[56] T van der Veldt W Vassen and W Hogervorst ldquoHeliumRydberg states in parallel electric and magnetic fieldsrdquo Journalof Physics B Atomic Molecular and Optical Physics vol 26 no13 pp 1945ndash1955 1993

Mathematical Problems in Engineering 9

[57] A Arimondo W D Phillips and F Strumia Laser Manip-ulation of Atoms and Ions North-Holland Amsterdam TheNetherlands 1992

[58] D Weiss K Richter A Menschig et al ldquoQuantized periodicorbits in large antidot arraysrdquo Physical Review Letters vol 70no 26 pp 4118ndash4121 1993

[59] D Delande Atomes de rydberg en champs statiques intenses[These drsquoEtat] Universite Paris 6 Paris France 1988

[60] M Sadhukhan and B M Deb ldquoA dynamical signature ofquantum chaos in hydrogen atom under strong oscillatingmagnetic fieldsrdquo EPL vol 94 no 5 Article ID 50008 2011

[61] A Bruno-Alfonso C Bleasdale G V B de Souza and R ALewis ldquoClosed-orbit dependence on the field direction in theanisotropic diamagnetic Kepler problemrdquo Physical Review AmdashAtomic Molecular and Optical Physics vol 89 no 4 Article ID043425 2014

[62] J-K Xu H-Q Chen and H-P Liu ldquoExact quantum defecttheory approach for lithium in magnetic fieldsrdquo Chinese PhysicsB vol 22 no 1 Article ID 013204 2013

[63] M A Amdouni and H Eleuch ldquoInfluence of relativistic termsin the spectra of hydrogen atom highly excited in an externalstrong magnetic fieldrdquo Applied Mathematics amp InformationSciences vol 5 no 2 pp 307ndash320 2011

[64] J C Slater Quantum Theory of Atomic Structure vol 1-2McGraw-Hill New York NY USA 1996

[65] E U Condon and G H ShortleyTheTheory of Atomic SpectraCambridge University Press Cambridge UK 1953

[66] C C Tannoudji B Diu and F Laloe Quantum MechanicsWiley New York NY USA 1977 Mechanics Wiley

[67] P J Mohr B N Taylor and D B Newell ldquoCODATA recom-mended values of the fundamental physical constants 2006rdquoReviews of Modern Physics vol 80 no 2 pp 633ndash730 2008

[68] C W Clark and K T Taylor ldquoThe quadratic Zeeman effect inhydrogen Rydberg series application of Sturmian functionsrdquoJournal of Physics B Atomic and Molecular Physics vol 15 no8 pp 1175ndash1193 1982

[69] M A Iken F Borondo R M Benito and T Uzer ldquoPeriodic-orbit spectroscopy of the hydrogen atom in parallel electric andmagnetic fieldsrdquo Physical ReviewA vol 49 no 4 pp 2734ndash27471994

[70] A R Edmonds ldquoStudies of the quadratic Zeeman effect IApplication of the sturmian functionsrdquo Journal of Physics BAtomic and Molecular Physics vol 6 no 8 pp 1603ndash1615 1973

[71] K Sacha J Zakrzewski and D Delande ldquoBreaking timereversal symmetry in chaotic driven rydberg atomsrdquo Annals ofPhysics vol 283 no 1 pp 141ndash172 2000

[72] M Courtney Rydberg atoms in strong fields a testing groundfor quantum chaos [PhD thesis] Department of PhysicsMassachusetts Institute of Technology 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

In Section 4 we introduce the effect of nucleus recoil energyand we present the results Finally conclusion is given inSection 5

2 Model

21 The Bohr Model A succinct overview of the Bohr modelwhich has connection to the hydrogen atom is useful and isrequested [64 65]Thismodel enables us to introduce funda-mental quantities such us the Bohr radius and the ionizationenergy Moreover the energies determined by Bohr theoryare entirely the eigenvalues 119864

119899of the Hamiltonian This

semiclassical model has been established on the hypothesisthat the electron describes a circular orbit of radius 119903 aroundthe proton satisfying the following equations [55]

119864 =

1

2

120583V2 minus1198902

119903

120583V2

119903

=

1198902

1199032

120583V119903 = 119899ℏ

(1)

The term minus1198902

119903 of the first equation is the correspondingpotential energy of the electrostatic interaction between theproton and the electron In this expression 1198902 is written as

1198902

=

1199022

41205871205760

(2)

where 119902 is the elementary electric charge whereas 120578V2119903 isthe kinetic energy of the relative motion of the proton andthe electron The ratio of 119898

119901to 119898119890is close to 1800 Thus the

reduced mass of this system is very near to 119898119890

120583 =

119898119901sdot 119898119890

119898119901+ 119898119890

≃ 119898119890(1 minus

119898119890

119898119901

) (3)

In this study we substitute the electron mass 119898119890with the

correction term 119898119890119898119901into the expression of the effective

massIn the second equation the term 119890

2

1199032 expresses the

Coulomb force which is applied to the electronThe term V2119903is the acceleration of the electron in uniform circular motion

To explain the existence of the stable discrete energylevels the Bohr quantization rules were established Besidesthe energies 119864

119899given by this model are in agreement with

quantum mechanical theory [66]

119864119899= minus

1198981198901198904

21198992ℏ2 (4)

where 119899 is the principal quantum number and ℏ is Planckrsquosconstant

In atomic units (119890 = 119898119890= ℏ = 1) the energy eigenvalues

equation (4) becomes

119864119899= minus

1

21198992 (5)

From this expression Bohr derived the transmission wave-length between different levels of 119899 using the Rydbergconstant for a nucleus of infinite mass [67]

119877infin

= 10973731568527mminus1 (6)

The dynamic symmetry group has simplified the study ofthe hydrogen atom (with 6 degrees of freedom) This groupis generated by the constants of motion namely the angularmomentum the Runge-Lenz vector (defined by = (and

)119898 minus 119903119903) and the energy 119864For a weak perturbation the group SO4 is appropriated to

explain the degeneracy of the energy levels in nonrelativisticdynamics However for the global investigation of the dis-crete and continuous states the use of the symmetry groupSO(22) is required in particular for strong perturbations InSection 3 more details are given about the group representa-tion of SO(22) for hydrogen atom excited by amagnetic field

22 Relativistic Corrections in an External Strong MagneticField We introduce Diracrsquos equation in order to take intoaccount the relativistic effects The energy levels are nowrelying on two quantum numbers the principal quantumnumber 119899 and the quantum number 119869 associated with thetotal angular momentum of the electron 119869 = + 119878 It canbe written as follows [48]

119864119899119869

= 1198981198882[

[

[

[

1 +

1205722

(radic(119895 + 12)2

minus 1205722+ 119899 minus 119895 minus 12)

2

]

]

]

]

minus12

(7)

where 120572 is the fine structure constant which defines the scaleof the splitting

120572 =

1198902

ℏ119888

(8)

TheDirac equation is generally used as aTaylor expansionin power of 1205722 as follows [48]

119864119899119869

= minus

1198981198882

1205722

21198992

[1 +

1205722

1198992(

119899

119869 + 12

minus

3

4

) + sdot sdot sdot ] (9)

Dependence 11198993 of the fine structure is immediately

apparent in this expression We studied this effect in ourprevious paper [63] Dirac theory takes into account thequantum and the relativistic nature of the electron howeverit does not reflect accurately the finite mass of the nucleus Inthe equations above themass of the nucleus is still consideredinfinite To take into account the finite mass we substitute inthe Dirac equation the mass of the electron by the reducedmass 120583 = 119898119872(119898 + 119872) of the electron-proton system (119872 isthe proton mass) Furthermore we add other corrections tothe energy levels obtainedThe first is a relativistic correctiondue to the recoil of the nucleus It is of the order of 1205722 with

Mathematical Problems in Engineering 3

respect to the ground state energy of the hydrogen atom Itsexpression is quite simple [48]

119864119899(recoil) = minus

1205782

1198882

119872 + 119898

1205724

81198994 (10)

Replacing the electron mass by the reduced mass theexpressions of the energies (4) and (10) are written in atomicunit (119898 = 1 119872 = 1836153996) [48]

119864119903

119899= minus(1 minus

119898

119872

)

1

21198992

119864119899(recoil) = minus

1

119872

(1 minus

3119898

119872

)

1205722

81198994

(11)

Over the last three decades several works have beendedicated to the understanding of Rydberg atoms highlyexcited hydrogen atom in strong static fields These studieshave led to high-resolution numerical spectra as well asimproving the accuracy of the calculations of the magneticfield in many problems [48 68] If the contribution of theelectromagnetic field compared to undisturbed energies isequal or higher then the magnetic field is strong In ourcontext chaos can be realized in the experimentwithRydbergstates (119899 = 30 31)

We will study later this effect on energy diagrams Wewrite the Hamiltonian of hydrogen atom in a uniformmagnetic field as [63]

119867 = 1198671+ 119867119901+ 119867119889 (12)

where 1198671 119867119901 and 119867

119889are defined as follows [66]

1198671= 1198981198901198882

+ 1198670+ 119882119898V + 119882SO + 119882

119863

119867119901=

1198902

2119898119890

sdot ( + 2 119878)

119867119889=

1198902

1198612

8119898119890

1199032sin2 (120579)

(13)

The eigenenergies of the Hamiltonian are given in (9) In thisrelation 119898

1198901198882 is the rest-mass energy of the electron 119867

0=

1199012

2119898119890+ 119881(119903) is the nonrelativistic Hamiltonian and the

following terms

119882119898V =

1199014

81198983

1198901198882

119882SO =

1

21198982

1198901198882

1

119903

119889119881 (119903)

119889119903

sdot 119878

119882119863

=

ℏ2

81198982

1198901198882Δ119881 (119903)

(14)

describe the fine structure and 119878 represent respectively theorbital and spin moments of the atom 119903 is the radius-vectorof the atomic electron and 120579 designs the angle between 119903 and

The paramagnetic term 119867119901is the origin of the linear

Zeeman effect offering the splitting of the atomic levelswhile the diamagnetic energy term 119867

119889is responsible for the

complex nature of the problemSince 119871

119911is a constant of the motion the linear Zeeman

energy 119867119901plays no significant role in the dynamics [68]

For high 119899 levels the diamagnetic term rapidly dominatesthe paramagnetic one and it may even become comparableto or greater than the Coulomb term itself leading to acomplete modification of the structure of atomic spectra Infact the linear Zeeman (paramagnetic) term is independentof 119899 while the diamagnetic term is proportional to 119899

4 Thehydrogen atomHamiltonian becomes nonseparablewhen thediamagnetic term is included [68 69]

In this paper we are studying the effect of the diamagneticterm 119867

119889(in atomic units 119867

119889= (119861

2

8)1199032sin2(120579)) on the

Rydberg atom119867119889is responsible for the chaos manifestation

As an important criterion chaos appears when the valueof the diamagnetic term is comparable to the unperturbedenergies [59 68]

3 Computation Method

All computed spectra presented in this paper are obtained bydigitalization of the Hamiltonian matrix in a suitable basisIt is worth noting that the spherical Sturmian basis has beenapplied for the study of hydrogen atom in a magnetic fieldOur simulations are performed by developing a code basedon MAPLE in which we can add new correction effects It isadvantageous to choose the Sturmian basis well adopted todynamic symmetries and it does not saturate the computermemory

31 Spherical Sturmian Basis Computing the spectra of dia-magnetic hydrogen in a spherical Sturmian basis was firstlyused by Edmonds [70] Clark and Taylor [68] recognized theimportance of being able to shift the length scale of the basisrelying on the region of energy and field Delande applieda complex rotation to the Sturmian basis to compute thespectrum of the diamagnetic hydrogen for positive energybeing aware that this technique has already been extended toother atoms [38]

Efficient calculations are provided by this basis choicewhich needs several considerations of symmetry [71] Allhydrogenoid states that share the same value 119872 are mixedtherefore it is a need to consider a dynamic group of diversestates It is acknowledgeable that the group SO(2 2) appliesthe previous criterion In fact this group is founded on theeigenstates that are themselves the basis of the Sturmianfunctions On one hand this basis enables the global repre-sentation of statesrsquo space (discrete continuum) On the otherhand it results in the loss of orthogonality of the basis vectors

The Sturmian basis is the proper basis describing thesystem of two oscillators equivalent to the Rydberg atom We

4 Mathematical Problems in Engineering

are able to express the Schrodinger equation in function ofthese generators ( 119878

(120572)

119879(120572)) of dynamic group SO(2 2) They

are defined respectively on an adjustable parameter 120572 [38]

119878(120572)

1= minus

120572

4

(1199012

120578+

1198712

119911

1205782) +

1

4120572

1205782

119878(120572)

2= minus

1

2

120578119901120578

119878(120572)

3=

120572

4

(1199012

120578+

1198712

119911

1205782) +

1

4120572

1205782

119879(120572)

1= minus

120572

4

(1199012

] +

1198712

119911

]2) +

1

4120572

]2

119879(120572)

2= minus

1

2

]119901]

119879(120572)

3=

120572

4

(1199012

] +

1198712

119911

]2) +

1

4120572

]2

(15)

where semiparabolic coordinates (120578 ]) are defined as

120578 = radic119903 + 119911

] = radic119903 minus 119911

(16)

while the momentums (119901120578 119901]) are defined as

119901120578= (1205782

+ ]2)119889120578

119889119905

119901] = (1205782

+ ]2)119889]119889119905

(17)

and 119871119911is the 119911-component of the angular momentum

These 6 variables are not independent

1198782

= (119878(120572)

3)

2

minus (119878(120572)

1)

2

minus (119878(120572)

2)

2

=

1198712

119911

4

1198792

= (119879(120572)

3)

2

minus (119879(120572)

1)

2

minus (119879(120572)

2)

2

=

1198712

119911

4

(18)

Thus the dynamic variables ( 119878(120572)

119879(120572)) represent the

classical formof the quantumgenerator of the dynamic groupSO(2 2)

The Hamiltonian ldquooscillatorrdquo obtained coincides with itsexpression in the context of the classic mechanic

(119878(120572)

3+ 119879(120572)

3+

1205742

2 (minus2119864)2(119878(120572)

1+ 119878(120572)

3) (119879(120572)

1+ 119879(120572)

3)

times (119878(120572)

1+ 119878(120572)

3+ 119879(120572)

1+ 119879(120572)

3) minus

1

radicminus2119864

) |Ψ⟩ = 0

(19)

with 120572 = 1radicminus2119864

We can write a more general equation by taking any valueof the adjustable parameter 120572 [57] We find

[119860(120572)

minus 120572 + (minus21198641205722

) 119861(120572)

+

1205742

1205724

2

119862(120572)

] |Ψ⟩ = 0 (20)

with

119860(120572)

=

119878(120572)

3minus 119878(120572)

1

2

+

119879(120572)

3minus 119879(120572)

1

2

119861(120572)

=

119878(120572)

3+ 119878(120572)

1

2

+

119879(120572)

3+ 119879(120572)

1

2

119862(120572)

= (119878(120572)

1+ 119878(120572)

3) (119879(120572)

1+ 119879(120572)

3) (119878(120572)

1+ 119878(120572)

3+ 119879(120572)

1+ 119879(120572)

3)

(21)

In the Sturmian basis the eigenvalues of the hydrogenHamiltonian are obtained by solving the generalized eigen-value problem (20)

The spherical Sturmian basis functions are [72]

Ψ119899119897119898

(119903) = 119878(120572)

119899119897(119903) 119884119897119898

(120579 120593) (22)

where 119878(120572)

119899119897(119903) is the radial Sturmian function

119878(120572)

119899119897(119903) = [

(119899 minus 119897 minus 1)

2 (119899 + 119897)

]

(12)

(

119903

2

)

119897+1

119890minus1199032120572

1198712119897+1

119899minus119897minus1(

119903

120572

)

(23)

and 119871119895

119894is the associated Laguerre polynomial

The full eigenfunctions used can be written as

| 119899119897119898⟩ = 119878(120572)

119899119897119884119897119898

(24)

Then the matrix element of the diamagnetic term(1198612

8)1199032sin2(120579) is given by

⟨1198991015840

1198971015840

119898

100381610038161003816100381610038161003816100381610038161003816

1198612

8

1199032sin2 (120579)

100381610038161003816100381610038161003816100381610038161003816

119899119897119898⟩ =

1

8

1198612

⟨1198991015840

119897101584010038161003816100381610038161003816119903210038161003816100381610038161003816119899119897⟩

times ⟨1198971015840

119898

10038161003816100381610038161003816sin2 (120579)1003816100381610038161003816

1003816119897119898⟩

(25)

Evaluation of the angular matrix elements is then easyThe corresponding matrix elements of the square positionoperator 1199032 are written as [68]

⟨1198991015840

119897101584010038161003816100381610038161003816119903210038161003816100381610038161003816119899119897⟩ = int

infin

0

119878120577

119899119897(119903) 1199032

119878120577

11989910158401198971015840(119903) 119889119903 (26)

Mathematical Problems in Engineering 5

are nonzero only for |119899 minus 1198991015840

| = 0 1 2 3 |119897 minus 1198971015840

| = 0 2 and atthese points take the following forms

⟨1198991015840

119897 + 2

10038161003816100381610038161003816119903210038161003816100381610038161003816119899119897⟩

=

minus

1

2

120577minus3

[(119899 minus 119897 minus 1) (119899 minus 119897 minus 2) (119899 minus 119897 minus 3)

times(119899 minus 119897 minus 4)(119899 minus 119897 minus 5)(119899 + 119897)]12

1198991015840

= 119899 minus 3

120577minus3

[(3119899 + 2119897) (119899 minus 119897 minus 1) (119899 minus 119897 minus 2)

times(119899 minus 119897 minus 3)(119899 minus 119897 minus 4)]12

1198991015840

= 119899 minus 2

minus

5

2

120577minus3

[(3119899 + 119897) (119899 minus 119897 minus 1) (119899 minus 119897 minus 2)

times(119899 minus 119897 minus 3)(119899 + 119897 + 1)]12

1198991015840

= 119899 minus 1

10119899120577minus3

[(119899 + 119897 + 1) (119899 + 119897 + 2)

times(119899 minus 119897 minus 1)(119899 minus 119897 minus 2)]12

1198991015840

= 119899

minus

5

2

120577minus3

(3119899 minus 119897) [(119899 + 119897 + 3) (119899 + 119897 + 2)

times(119899 + 119897 + 1)(119899 minus 119897 minus 1)]12

1198991015840

= 119899 + 1

120577minus3

(3119899 minus 2119897) [(119899 + 119897 + 4) (119899 + 119897 + 3)

times(119899 + 119897 + 2)(119899 + 119897 + 1)]12

1198991015840

= 119899 + 2

minus

1

2

120577minus3

[(119899 + 119897 + 5) (119899 + 119897 + 4) (119899 + 119897 + 3)

times(119899 + 119897 + 2)(119899 + 119897 + 1)(119899 minus 119897)]12

1198991015840

= 119899 + 3

⟨1198991015840

119897

10038161003816100381610038161003816119903210038161003816100381610038161003816119899119897⟩

=

2119899120577minus3

[51198992

minus 3119897 (119897 + 1) + 1] 1198991015840

= 119899

minus

3

2

120577minus3

[5119899 (119899 + 1) minus 119897 (119897 + 1) + 2]

times [(119899 + 119897 + 1) (119899 minus 119897)]12

1198991015840

= 119899 + 1

3120577

minus3

(119899 + 1) [(119899 + 119897 + 1) (119899 minus 119897)

times(119899 + 119897 + 2)(119899 minus 119897 + 1)]12

1198991015840

= 119899 + 2

minus

1

2

120577minus3

[(119899 + 119897 + 1) (119899 + 2 minus 119897) (119899 minus 119897)

times(119899 + 119897 + 3)(119899 + 119897 + 2)(119899 + 1 minus 119897)]12

1198991015840

= 119899 + 3

(27)

The investigation of Clark and Taylor [68] has revealedthat it is usually advisable to select higher values of 120577 Inour simulation we used 120577 = 2119899 in order to be close to theionization limit

Our numerical simulations consist in diagonalizing thematrix representing the operator in a Sturmian basis Thematrix elements of the generators are independent of 120572

32 Numerical Method Equation (20) is equivalent to thegeneral problem for eigenvalues of the form [59]

(119872 minus 120582119873) |Ψ⟩ = 0 (28)

where 119872 and 119873 are fixed matrices and 120582 and |Ψ⟩ are theeigenvalues and the eigenvectors searched

321 Simulation at Fixed Magnetic Field In the case wherewe fix the magnetic field 120574 = 119861119861

119888(120574 denotes the magnetic

field in units of 119861119888≃ 235 times 10

5

119879) after opting for 120572 = 1205720=

119862119905119890 we obtain an equation in the generalized eigenvalueswith

[59]

119872 = 119860(120572)

minus 1205720+

1205742

1205724

0

2

119862(120572)

119873 = 119861(120572)

120582 = minus21198641205722

0

(29)

that determines the energy levels

322 Simulation at Fixed Energy In this simulation wechoose 120572 = 120572

0= 119862119905119890 and we fix the energy So that we get

an equation in the generalized eigenvalues with [59]

119872 = 119860(120572)

minus 1205720+ (minus2119864120572

2

0) 119861(120572)

119873 = 119862(120572)

120582 =

1205742

1205724

0

2

(30)

that establishes the spectrum of magnetic field values relatedto the fixed energy

Both types of the simulations give the same outcomes

323 Diagonalization Algorithm and Convergence CriterionThe matrix components of the generators ( 119878

(120572)

119879(120572)) in a

Sturmian basis are restricted to some selectionsrsquo rules It iswell known that the matrix eigenvalue problems generally tobe solved are real symmetric matrices in strips [38]

Currently stable numerical algorithms have been devel-oped in order to calculate the eigenvalues and eigenvectors[48]

We provide an adoptable parameter 1205720to verify the

convergence for each type of the simulations In fact 1205720is

connected to the natural frequency of the basic oscillatorselected The accuracy of the examined values is not relianton this parameter Exact searched values are not dependenton this parameter

4 Numerical Results and Discussion

41 Energy Diagrams We use the simulation at fixed mag-netic field and we plot for a large number of points diagrams119864119903

119899= 119891(120574

2

) in Figures 1 2 and 3Figure 1 shows the diagram 119864(Hartree) = 119891(120574

2

) forRydberg states 119899 = 30 in the system with rather low field

6 Mathematical Problems in Engineering

minus003

minus004

minus005

minus006

minus007

minus008

minus009

E(H

artre

e)

02 04 06 08 10 12 14 16

1205742(times1010)

Figure 1 Energy levels as a function of the normalized squaremagnetic field 120574

2

(120574 = 119861119861119888 119861119888

= 235 times 105

119879) calculated by thesimulation technique described in Section 321 for weak magneticfield 119861 (119861 lt 3119879) and for the principle quantum number 119899 = 30The rovibration structure is completely removed (inter-119897-mixing)We observe that for a weak magnetic field the classical dynamicsis regular

minus005

minus010

minus015

minus020

E(H

artre

e)

1205742(times1010)

20 30 40 50

Figure 2 Diagram of energy levels of the hydrogen atom instrong magnetic field as function of 120574

2

(120574 = 119861119861119888 119861119888

= 235 times

105

119879) computed with a simulation described in Section 321 Thecalculation is done for a strong magnetic field 119861 (9119879 lt 119861 lt 18119879)

for both levels 119899 = 30 (dotted red line) and 119899 = 31 (solid blue line)Mixing between the energy levels of 119899 = 30 and 119899 = 31 is observed

(119861 lt 3119879) diamagnetism breaks the zero-field 119897-degeneracyof the hydrogen atom

We can explain this by the fact that in very weak field(119897-system intermixing) the degeneracy of the levels (therovibrational structure) proportional to 120574

2 is completelyremoved by the diamagnetic term It is visible that regularsystem disturbed by a weak perturbation remains essentiallyregular for the reason that the diamagnetic interaction is aperturbation of the Coulomb dynamics That is a regularregime there is no sensitivity on initial conditions

At highmagnetic field as shown by Figure 2 the differentenergy levels (119899 = 30 119899 = 31) essentially cross

Indeed for a highly excited state with principal quantumnumber 119899 the diamagnetic interaction scales as 120574

2

1198994 and

1205742(times1010)

20 30 40 50

minus005

minus010

minus015

minus020

E(H

artre

e)

Figure 3 Diagram of the energy levels of the hydrogen atom inthe strong magnetic field for 119899 = 30 computed by the simulationdescribed in Section 321 We note that for a high magnetic field theenergy levels anticrossThis is an indication of the chaos emergence

the zero-field as 11198992 The strong field regime is obtained

when [38]

1205741198993

≃ 1 that is 1198611198993

≃ 119861119888 (31)

This is achieved with 119899 ≃ 30 and 119861 ≃ 9119879 The rotationaland vibrational energy levels really cross By increasing themagnetic field in Figure 3 we observe that more statesanticross The classical smooth transition from regularity tochaos agrees well with this result

42The Effect of the Nucleus Recoil Energy In this section weuse the second simulation simulation at fixed energy wherewe have plotted 120574

2 as a function of the angular momentum 119897We introduced the recoil energy effect given in atomic unitby

119864119899(recoil) = minus

1

119872

(1 minus

3119898

119872

)

1205722

81198994 (32)

which takes into account the relativistic effects we recall thatthe levels shift due to this effect can be written as [68]

Δ119864119897=

1205722

21198993119897 (119897 + 1)

(33)

This recoil energy has been introduced with which wehave realized the simulation with nucleus of finite massThe result of the simulation is shown in Figure 4 wherethe distinction between the nucleus having a finite mass(diamond line) and the one having an infinite mass (circleline) is very clear This difference can be justified by the factthat the term (33) is the amount of three contributions Thefirst contribution is the kinetic energy in which the electronmass and not the reduced mass of the system manifests

119882119898V =

4

81198983

1198901198882 (34)

The second contribution is the Darwin term which takesinto account the electron delocalization but not the nucleus

ℏ2

81198982

119890119888241205871198902

120575 ( 119903) (35)

Mathematical Problems in Engineering 7

01280

01275

01270

01265

01260

01255

10 12 14 16 18 20 22 24 26 28

l

1205742(times1010)

Figure 4 The normalized square magnetic field 1205742

(120574 = 119861119861119888

119861119888

= 235 times 105

119879) for different angular momentum l with aprincipal quantum 119899 = 30 calculated by the simulation describedin Section 322 Two cases are computed first case realistic nucleusmass (finite mass) (red diamond line) second case nucleus infinitemass (blue circle line)

Finally the third contribution is the spin-orbit whichshows the interaction between the electronmagneticmomentand themagnetic field of the nucleus seen in the proper frameof the electron Therefore we should take into considerationthe nucleus speed since the nucleus of finite mass is being inmotion

5 Conclusion

We have studied the diamagnetic behavior of a hydrogenatom highly excited in a strong magnetic field consideringthe realistic finite mass of the nucleus The results revealthat when the magnetic field is weak the diamagnetic termremoves the degeneracy of the energy levels The system istotally regular it is possible to affect quantum numbers tothe various energy levels However with increasing magneticfield any symmetry will be destroyed and the energy levelswill be crossed there is an unruffled transition from regu-larity to chaotical dynamics At higher magnetic field thedynamic is totally chaotic and quantum spectrum includesonly large anticrossing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B Li and H P Liu ldquoAn effective quantum defect theory forthe diamagnetic spectrum of a barium Rydberg atomrdquo ChinesePhysics B vol 22 no 1 Article ID 013203 2013

[2] A E Shabad and V V Usov ldquoModified coulomb law in astrongly magnetized vacuumrdquo Physical Review Letters vol 98no 18 Article ID 180403 2007

[3] L Stevanovic and D Milojevic ldquoCompressed hydrogen atomunder debye screening in strong magnetic fieldrdquo Publications of

the Astronomical Observatory of Belgrade vol 91 pp 327ndash3302012

[4] H Jabri H Eleuch and T Djerad ldquoLifetimes of atomic Rydbergstates by autocorrelation functionrdquo Laser Physics Letters vol 2no 5 pp 253ndash257 2005

[5] H Eleuch and N Rachid ldquoAutocorrelation function ofmicrocavity-emitting field in the non-linear regimerdquoThe Euro-pean Physical Journal D vol 57 no 2 pp 259ndash264 2010

[6] H Eleuch N Ben Nessib and R Bennaceur ldquoQuantum modelof emission in a weakly non ideal plasmardquo The EuropeanPhysical Journal D vol 29 no 3 pp 391ndash395 2004

[7] P K Jha H Eleuch and Y V Rostovtsev ldquoCoherent control ofatomic excitation using off-resonant strong few-cycle pulsesrdquoPhysical Review A vol 82 no 4 Article ID 045805 2010

[8] E A Sete and H Eleuch ldquoStrong squeezing and robustentanglement in cavity electromechanicsrdquo Physical Review Avol 89 no 1 Article ID 013841 2014

[9] M Khatua U Sarkar and P K Chattaraj ldquoReactivity dynamicsof confined atoms in the presence of an external magnetic fieldrdquoThe European Physical Journal D vol 68 no 2 article 22 2014

[10] U Das and B Mukhopadhyay ldquoNew mass limit for whitedwarfs super-Chandrasekhar type Ia supernova as a newstandard candlerdquo Physical Review Letters vol 110 no 7 ArticleID 071102 2013

[11] A D Sargsyan R K Mirzoyan A S Sarkisyan A H Amiryanand D H Sarkisyan ldquoSplitting of N-type optical resonanceformed in Λ-system of 85Rb atoms in a strong transversemagnetic fieldrdquo Journal of Contemporary Physics vol 49 no 1pp 20ndash27 2014

[12] V S Popov and B M Karnakov ldquoHydrogen atom in a strongmagnetic fieldrdquo Physics-Uspekhi vol 57 no 3 pp 257ndash279 2014

[13] E Giacobino J-P Karr G Messin H Eleuch and A BaasldquoQuantum optical effects in semiconductor microcavitiesrdquoComptes Rendus Physique vol 3 no 1 pp 41ndash52 2002

[14] R Juarez-Amaro A Zuniga-Segundo F Soto-Eguibar andH M Moya-Cessa ldquoEquivalence between mirror-field-atomand ion-laser interactionsrdquo Applied Mathematics amp InformationSciences vol 7 no 4 pp 1311ndash1315 2013

[15] H Eleuch S Guerin and H R Jauslin ldquoEffects of an environ-ment on a cavity-quantum-electrodynamics system controlledby bichromatic adiabatic passagerdquoPhysical ReviewA vol 85 no1 Article ID 013830 2012

[16] E A Sete A A Svidzinsky Y V Rostovtsev et al ldquoUsingquantum coherence to generate gain in the xuv and x-raygain-swept superradiance and lasing without inversionrdquo IEEEJournal on Selected Topics in Quantum Electronics vol 18 no 1pp 541ndash553 2012

[17] E A Sete A A Svidzinsky H Eleuch Z Yang R D Nevelsand M O Scully ldquoCorrelated spontaneous emission on theDanuberdquo Journal of Modern Optics vol 57 no 14-15 pp 1311ndash1330 2010

[18] Z BWalters ldquoQuantum coherent dynamics at ambient temper-ature in photosynthetic moleculesrdquo Quantum Physics Lettersvol 1 no 1 pp 21ndash33 2012

[19] H Rabitz ldquoControl in the sciences over vast length and timescalesrdquo Quantum Physics Letters vol 1 no 1 pp 1ndash19 2012

[20] DA Barskiy KVKovtunov I VKoptyug et al ldquoThe feasibilityof formation and kinetics of NMR signal amplification byreversible exchange (SABRE) at high magnetic field (94 T)rdquoJournal of the American Chemical Society vol 136 no 9 pp3322ndash3325 2014

8 Mathematical Problems in Engineering

[21] B Mandal S Adhikari K Choudhary et al ldquoFemtosecondresponse of quantum discrete breathers in SRR based metama-terials and the role of dielectric permittivityrdquo Quantum PhysicsLetters vol 1 no 2 pp 59ndash68 2012

[22] J D Pritchard J A Isaacs M A Beck R McDermottand M Saffman ldquoHybrid atom-photon quantum gate in asuperconducting microwave resonatorrdquo Physical Review A vol89 no 1 Article ID 010301 2014

[23] P Rej and A Ghoshal ldquoRydberg transitions for positron-hydrogen collisions asymptotic cross section and scaling lawrdquoJournal of Physics B Atomic Molecular and Optical Physics vol47 no 1 Article ID 015204 2014

[24] H Eleuch D Elser and R Bennaceur ldquoSoliton propagation inan absorbing three-level atomic systemrdquo Laser Physics Lettersvol 1 no 8 pp 391ndash396 2004

[25] H Eleuch and R Bennaceur ldquoAn optical soliton pair amongabsorbing three-level atomsrdquo Journal of Optics A Pure andApplied Optics vol 5 no 5 pp 528ndash533 2003

[26] E Osman M Khalfallah and H Sapoor ldquoImproved (G1015840G)-expansion method for elliptic-like equation and general trav-eling wave solutions of some class of NLPDEsrdquo InformationScience Letters vol 2 no 2 pp 123ndash127 2013

[27] M Akbari ldquoExact solutions of the coupled Higgs equationand the Maccari system using the modified simplest equationmethodrdquo Information Science Letters vol 2 no 3 pp 155ndash1582013

[28] N Taghizadeh M Mirzazadeh M Akbari and M RahimianldquoExact soliton solutions for generalized equal width equationrdquoMathematical Science Letters vol 2 no 2 pp 99ndash106 2013

[29] E Osman M Khalfallah and H Sapoor ldquoFinding generalsolutions of nonlinear evolution equations by improved (G1015840G)-expansion methodrdquo Mathematical Science Letters vol 3 no 1pp 1ndash8 2014

[30] E M E Zayed ldquoExact traveling wave solutions for a variable-coefficient generalized dispersive water-wave system using thegeneralized (GrsquoG)-expansion methodrdquo Mathematical ScienceLetters vol 3 no 1 pp 9ndash15 2014

[31] E A Sete and H Eleuch ldquoInteraction of a quantum well withsqueezed light quantum-statistical propertiesrdquo Physical ReviewA vol 82 no 4 Article ID 043810 2010

[32] W Yang L Huang and M Tian ldquoTwo novel and secure quan-tum ANDOS schemesrdquo Applied Mathematics and InformationSciences vol 7 no 2 pp 659ndash665 2013

[33] H Eleuch ldquoPhoton statistics of light in semiconductor micro-cavitiesrdquo Journal of Physics B Atomic Molecular and OpticalPhysics vol 41 no 5 Article ID 055502 2008

[34] G Messin J P Karr H Eleuch J M Courty and E GiacobinoldquoSqueezed states and the quantumnoise of light in semiconduc-tor microcavitiesrdquo Journal of Physics Condensed Matter vol 11no 31 pp 6069ndash6078 1999

[35] A Baas J P Karr H Eleuch and E Giacobino ldquoOpticalbistability in semiconductor microcavitiesrdquo Physical Review AAtomic Molecular and Optical Physics vol 69 no 2 2004

[36] J Bonca and S Kruchinin Physical Properties of NanosystemsSpringer 2011

[37] L Tang and J-A Lu ldquoProjective lag synchronization of chaoticsystems with parameter mismatchrdquo Applied Mathematics ampInformation Sciences vol 7 no 1 pp 121ndash125 2013

[38] D Delande ldquoChaos and atoms in strong magnetic fieldsrdquoPhysica Scripta vol 1991 no T34 p 52 1991

[39] H Ruder G Wunner H Herold and F Geyer Atoms in StrongMagnetic Fields Springer Berlin Germany 1994

[40] H Eleuch and A Prasad ldquoChaos and regularity in semiconduc-tormicrocavitiesrdquoPhysics Letters A vol 376 no 26-27 pp 1970ndash1977 2012

[41] A Y Potekhin and A V Turbiner ldquoHydrogen atom in amagnetic field the quadrupole momentrdquo Physical Review AAtomic Molecular and Optical Physics vol 63 no 6 Article ID065402 2001

[42] S N Rasband Chaotic Dynamics of Non linear Systems JohnWiley amp Sons 1990

[43] A J Lichtenberg and M A Lieberman Regular and StochasticMotion Springer New York NY USA 1983

[44] A V Andreev O Agam B D Simons and B L AltshulerldquoQuantum chaos irreversible classical dynamics and randommatrix theoryrdquo Physical Review Letters vol 76 no 21 pp 3947ndash3950 1996

[45] O Bohigas ldquoLesHouches summer school session LIIrdquo inChaosand Quantum Physics M-J Giannoni A Voros and J Zinn-Justin Eds North-Holland Amsterdam The Netherlands1991

[46] T Pohl H R Sadeghpour and P Schmelcher ldquoCold andultracold Rydberg atoms in strong magnetic fieldsrdquo PhysicsReports vol 484 no 6 pp 181ndash229 2009

[47] T F Gallagher Rydberg Atoms Cambridge University PressCambridge UK 1994

[48] M I Eides H Grotch and V A Shelyuto Theory of LightHydrogenic Bound States vol 222 of Springer Tracts in ModernPhysics 2007

[49] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013

[50] C Yang S S Ge and T H Lee ldquoOutput feedback adaptivecontrol of a class of nonlinear discrete-time systems withunknown control directionsrdquoAutomatica vol 45 no 1 pp 270ndash276 2009

[51] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011

[52] C Yang S S Ge C Xiang T Chai and T H Lee ldquoOutputfeedback NN control for two classes of discrete-time systemswith unknown control directions in a unified approachrdquo IEEETransactions on Neural Networks vol 19 no 11 pp 1873ndash18862008

[53] Y-J Liu S-C Tong DWang T-S Li and C L P Chen ldquoAdap-tive neural output feedback controller design with reduced-order observer for a class of uncertain nonlinear SISO SystemsrdquoIEEE Transactions on Neural Networks vol 22 no 8 pp 1328ndash1334 2011

[54] T M Shahwan ldquoA comparison of Bayesian methods andartificial neural networks for forecasting chaotic financial timeseriesrdquo Journal of Statistics Applications amp Probability vol 1 no2 p 89 2012

[55] J W SeriesThe Spectrum of Atomic Hydrogen Oxford Univer-sity Press London UK 1957

[56] T van der Veldt W Vassen and W Hogervorst ldquoHeliumRydberg states in parallel electric and magnetic fieldsrdquo Journalof Physics B Atomic Molecular and Optical Physics vol 26 no13 pp 1945ndash1955 1993

Mathematical Problems in Engineering 9

[57] A Arimondo W D Phillips and F Strumia Laser Manip-ulation of Atoms and Ions North-Holland Amsterdam TheNetherlands 1992

[58] D Weiss K Richter A Menschig et al ldquoQuantized periodicorbits in large antidot arraysrdquo Physical Review Letters vol 70no 26 pp 4118ndash4121 1993

[59] D Delande Atomes de rydberg en champs statiques intenses[These drsquoEtat] Universite Paris 6 Paris France 1988

[60] M Sadhukhan and B M Deb ldquoA dynamical signature ofquantum chaos in hydrogen atom under strong oscillatingmagnetic fieldsrdquo EPL vol 94 no 5 Article ID 50008 2011

[61] A Bruno-Alfonso C Bleasdale G V B de Souza and R ALewis ldquoClosed-orbit dependence on the field direction in theanisotropic diamagnetic Kepler problemrdquo Physical Review AmdashAtomic Molecular and Optical Physics vol 89 no 4 Article ID043425 2014

[62] J-K Xu H-Q Chen and H-P Liu ldquoExact quantum defecttheory approach for lithium in magnetic fieldsrdquo Chinese PhysicsB vol 22 no 1 Article ID 013204 2013

[63] M A Amdouni and H Eleuch ldquoInfluence of relativistic termsin the spectra of hydrogen atom highly excited in an externalstrong magnetic fieldrdquo Applied Mathematics amp InformationSciences vol 5 no 2 pp 307ndash320 2011

[64] J C Slater Quantum Theory of Atomic Structure vol 1-2McGraw-Hill New York NY USA 1996

[65] E U Condon and G H ShortleyTheTheory of Atomic SpectraCambridge University Press Cambridge UK 1953

[66] C C Tannoudji B Diu and F Laloe Quantum MechanicsWiley New York NY USA 1977 Mechanics Wiley

[67] P J Mohr B N Taylor and D B Newell ldquoCODATA recom-mended values of the fundamental physical constants 2006rdquoReviews of Modern Physics vol 80 no 2 pp 633ndash730 2008

[68] C W Clark and K T Taylor ldquoThe quadratic Zeeman effect inhydrogen Rydberg series application of Sturmian functionsrdquoJournal of Physics B Atomic and Molecular Physics vol 15 no8 pp 1175ndash1193 1982

[69] M A Iken F Borondo R M Benito and T Uzer ldquoPeriodic-orbit spectroscopy of the hydrogen atom in parallel electric andmagnetic fieldsrdquo Physical ReviewA vol 49 no 4 pp 2734ndash27471994

[70] A R Edmonds ldquoStudies of the quadratic Zeeman effect IApplication of the sturmian functionsrdquo Journal of Physics BAtomic and Molecular Physics vol 6 no 8 pp 1603ndash1615 1973

[71] K Sacha J Zakrzewski and D Delande ldquoBreaking timereversal symmetry in chaotic driven rydberg atomsrdquo Annals ofPhysics vol 283 no 1 pp 141ndash172 2000

[72] M Courtney Rydberg atoms in strong fields a testing groundfor quantum chaos [PhD thesis] Department of PhysicsMassachusetts Institute of Technology 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

respect to the ground state energy of the hydrogen atom Itsexpression is quite simple [48]

119864119899(recoil) = minus

1205782

1198882

119872 + 119898

1205724

81198994 (10)

Replacing the electron mass by the reduced mass theexpressions of the energies (4) and (10) are written in atomicunit (119898 = 1 119872 = 1836153996) [48]

119864119903

119899= minus(1 minus

119898

119872

)

1

21198992

119864119899(recoil) = minus

1

119872

(1 minus

3119898

119872

)

1205722

81198994

(11)

Over the last three decades several works have beendedicated to the understanding of Rydberg atoms highlyexcited hydrogen atom in strong static fields These studieshave led to high-resolution numerical spectra as well asimproving the accuracy of the calculations of the magneticfield in many problems [48 68] If the contribution of theelectromagnetic field compared to undisturbed energies isequal or higher then the magnetic field is strong In ourcontext chaos can be realized in the experimentwithRydbergstates (119899 = 30 31)

We will study later this effect on energy diagrams Wewrite the Hamiltonian of hydrogen atom in a uniformmagnetic field as [63]

119867 = 1198671+ 119867119901+ 119867119889 (12)

where 1198671 119867119901 and 119867

119889are defined as follows [66]

1198671= 1198981198901198882

+ 1198670+ 119882119898V + 119882SO + 119882

119863

119867119901=

1198902

2119898119890

sdot ( + 2 119878)

119867119889=

1198902

1198612

8119898119890

1199032sin2 (120579)

(13)

The eigenenergies of the Hamiltonian are given in (9) In thisrelation 119898

1198901198882 is the rest-mass energy of the electron 119867

0=

1199012

2119898119890+ 119881(119903) is the nonrelativistic Hamiltonian and the

following terms

119882119898V =

1199014

81198983

1198901198882

119882SO =

1

21198982

1198901198882

1

119903

119889119881 (119903)

119889119903

sdot 119878

119882119863

=

ℏ2

81198982

1198901198882Δ119881 (119903)

(14)

describe the fine structure and 119878 represent respectively theorbital and spin moments of the atom 119903 is the radius-vectorof the atomic electron and 120579 designs the angle between 119903 and

The paramagnetic term 119867119901is the origin of the linear

Zeeman effect offering the splitting of the atomic levelswhile the diamagnetic energy term 119867

119889is responsible for the

complex nature of the problemSince 119871

119911is a constant of the motion the linear Zeeman

energy 119867119901plays no significant role in the dynamics [68]

For high 119899 levels the diamagnetic term rapidly dominatesthe paramagnetic one and it may even become comparableto or greater than the Coulomb term itself leading to acomplete modification of the structure of atomic spectra Infact the linear Zeeman (paramagnetic) term is independentof 119899 while the diamagnetic term is proportional to 119899

4 Thehydrogen atomHamiltonian becomes nonseparablewhen thediamagnetic term is included [68 69]

In this paper we are studying the effect of the diamagneticterm 119867

119889(in atomic units 119867

119889= (119861

2

8)1199032sin2(120579)) on the

Rydberg atom119867119889is responsible for the chaos manifestation

As an important criterion chaos appears when the valueof the diamagnetic term is comparable to the unperturbedenergies [59 68]

3 Computation Method

All computed spectra presented in this paper are obtained bydigitalization of the Hamiltonian matrix in a suitable basisIt is worth noting that the spherical Sturmian basis has beenapplied for the study of hydrogen atom in a magnetic fieldOur simulations are performed by developing a code basedon MAPLE in which we can add new correction effects It isadvantageous to choose the Sturmian basis well adopted todynamic symmetries and it does not saturate the computermemory

31 Spherical Sturmian Basis Computing the spectra of dia-magnetic hydrogen in a spherical Sturmian basis was firstlyused by Edmonds [70] Clark and Taylor [68] recognized theimportance of being able to shift the length scale of the basisrelying on the region of energy and field Delande applieda complex rotation to the Sturmian basis to compute thespectrum of the diamagnetic hydrogen for positive energybeing aware that this technique has already been extended toother atoms [38]

Efficient calculations are provided by this basis choicewhich needs several considerations of symmetry [71] Allhydrogenoid states that share the same value 119872 are mixedtherefore it is a need to consider a dynamic group of diversestates It is acknowledgeable that the group SO(2 2) appliesthe previous criterion In fact this group is founded on theeigenstates that are themselves the basis of the Sturmianfunctions On one hand this basis enables the global repre-sentation of statesrsquo space (discrete continuum) On the otherhand it results in the loss of orthogonality of the basis vectors

The Sturmian basis is the proper basis describing thesystem of two oscillators equivalent to the Rydberg atom We

4 Mathematical Problems in Engineering

are able to express the Schrodinger equation in function ofthese generators ( 119878

(120572)

119879(120572)) of dynamic group SO(2 2) They

are defined respectively on an adjustable parameter 120572 [38]

119878(120572)

1= minus

120572

4

(1199012

120578+

1198712

119911

1205782) +

1

4120572

1205782

119878(120572)

2= minus

1

2

120578119901120578

119878(120572)

3=

120572

4

(1199012

120578+

1198712

119911

1205782) +

1

4120572

1205782

119879(120572)

1= minus

120572

4

(1199012

] +

1198712

119911

]2) +

1

4120572

]2

119879(120572)

2= minus

1

2

]119901]

119879(120572)

3=

120572

4

(1199012

] +

1198712

119911

]2) +

1

4120572

]2

(15)

where semiparabolic coordinates (120578 ]) are defined as

120578 = radic119903 + 119911

] = radic119903 minus 119911

(16)

while the momentums (119901120578 119901]) are defined as

119901120578= (1205782

+ ]2)119889120578

119889119905

119901] = (1205782

+ ]2)119889]119889119905

(17)

and 119871119911is the 119911-component of the angular momentum

These 6 variables are not independent

1198782

= (119878(120572)

3)

2

minus (119878(120572)

1)

2

minus (119878(120572)

2)

2

=

1198712

119911

4

1198792

= (119879(120572)

3)

2

minus (119879(120572)

1)

2

minus (119879(120572)

2)

2

=

1198712

119911

4

(18)

Thus the dynamic variables ( 119878(120572)

119879(120572)) represent the

classical formof the quantumgenerator of the dynamic groupSO(2 2)

The Hamiltonian ldquooscillatorrdquo obtained coincides with itsexpression in the context of the classic mechanic

(119878(120572)

3+ 119879(120572)

3+

1205742

2 (minus2119864)2(119878(120572)

1+ 119878(120572)

3) (119879(120572)

1+ 119879(120572)

3)

times (119878(120572)

1+ 119878(120572)

3+ 119879(120572)

1+ 119879(120572)

3) minus

1

radicminus2119864

) |Ψ⟩ = 0

(19)

with 120572 = 1radicminus2119864

We can write a more general equation by taking any valueof the adjustable parameter 120572 [57] We find

[119860(120572)

minus 120572 + (minus21198641205722

) 119861(120572)

+

1205742

1205724

2

119862(120572)

] |Ψ⟩ = 0 (20)

with

119860(120572)

=

119878(120572)

3minus 119878(120572)

1

2

+

119879(120572)

3minus 119879(120572)

1

2

119861(120572)

=

119878(120572)

3+ 119878(120572)

1

2

+

119879(120572)

3+ 119879(120572)

1

2

119862(120572)

= (119878(120572)

1+ 119878(120572)

3) (119879(120572)

1+ 119879(120572)

3) (119878(120572)

1+ 119878(120572)

3+ 119879(120572)

1+ 119879(120572)

3)

(21)

In the Sturmian basis the eigenvalues of the hydrogenHamiltonian are obtained by solving the generalized eigen-value problem (20)

The spherical Sturmian basis functions are [72]

Ψ119899119897119898

(119903) = 119878(120572)

119899119897(119903) 119884119897119898

(120579 120593) (22)

where 119878(120572)

119899119897(119903) is the radial Sturmian function

119878(120572)

119899119897(119903) = [

(119899 minus 119897 minus 1)

2 (119899 + 119897)

]

(12)

(

119903

2

)

119897+1

119890minus1199032120572

1198712119897+1

119899minus119897minus1(

119903

120572

)

(23)

and 119871119895

119894is the associated Laguerre polynomial

The full eigenfunctions used can be written as

| 119899119897119898⟩ = 119878(120572)

119899119897119884119897119898

(24)

Then the matrix element of the diamagnetic term(1198612

8)1199032sin2(120579) is given by

⟨1198991015840

1198971015840

119898

100381610038161003816100381610038161003816100381610038161003816

1198612

8

1199032sin2 (120579)

100381610038161003816100381610038161003816100381610038161003816

119899119897119898⟩ =

1

8

1198612

⟨1198991015840

119897101584010038161003816100381610038161003816119903210038161003816100381610038161003816119899119897⟩

times ⟨1198971015840

119898

10038161003816100381610038161003816sin2 (120579)1003816100381610038161003816

1003816119897119898⟩

(25)

Evaluation of the angular matrix elements is then easyThe corresponding matrix elements of the square positionoperator 1199032 are written as [68]

⟨1198991015840

119897101584010038161003816100381610038161003816119903210038161003816100381610038161003816119899119897⟩ = int

infin

0

119878120577

119899119897(119903) 1199032

119878120577

11989910158401198971015840(119903) 119889119903 (26)

Mathematical Problems in Engineering 5

are nonzero only for |119899 minus 1198991015840

| = 0 1 2 3 |119897 minus 1198971015840

| = 0 2 and atthese points take the following forms

⟨1198991015840

119897 + 2

10038161003816100381610038161003816119903210038161003816100381610038161003816119899119897⟩

=

minus

1

2

120577minus3

[(119899 minus 119897 minus 1) (119899 minus 119897 minus 2) (119899 minus 119897 minus 3)

times(119899 minus 119897 minus 4)(119899 minus 119897 minus 5)(119899 + 119897)]12

1198991015840

= 119899 minus 3

120577minus3

[(3119899 + 2119897) (119899 minus 119897 minus 1) (119899 minus 119897 minus 2)

times(119899 minus 119897 minus 3)(119899 minus 119897 minus 4)]12

1198991015840

= 119899 minus 2

minus

5

2

120577minus3

[(3119899 + 119897) (119899 minus 119897 minus 1) (119899 minus 119897 minus 2)

times(119899 minus 119897 minus 3)(119899 + 119897 + 1)]12

1198991015840

= 119899 minus 1

10119899120577minus3

[(119899 + 119897 + 1) (119899 + 119897 + 2)

times(119899 minus 119897 minus 1)(119899 minus 119897 minus 2)]12

1198991015840

= 119899

minus

5

2

120577minus3

(3119899 minus 119897) [(119899 + 119897 + 3) (119899 + 119897 + 2)

times(119899 + 119897 + 1)(119899 minus 119897 minus 1)]12

1198991015840

= 119899 + 1

120577minus3

(3119899 minus 2119897) [(119899 + 119897 + 4) (119899 + 119897 + 3)

times(119899 + 119897 + 2)(119899 + 119897 + 1)]12

1198991015840

= 119899 + 2

minus

1

2

120577minus3

[(119899 + 119897 + 5) (119899 + 119897 + 4) (119899 + 119897 + 3)

times(119899 + 119897 + 2)(119899 + 119897 + 1)(119899 minus 119897)]12

1198991015840

= 119899 + 3

⟨1198991015840

119897

10038161003816100381610038161003816119903210038161003816100381610038161003816119899119897⟩

=

2119899120577minus3

[51198992

minus 3119897 (119897 + 1) + 1] 1198991015840

= 119899

minus

3

2

120577minus3

[5119899 (119899 + 1) minus 119897 (119897 + 1) + 2]

times [(119899 + 119897 + 1) (119899 minus 119897)]12

1198991015840

= 119899 + 1

3120577

minus3

(119899 + 1) [(119899 + 119897 + 1) (119899 minus 119897)

times(119899 + 119897 + 2)(119899 minus 119897 + 1)]12

1198991015840

= 119899 + 2

minus

1

2

120577minus3

[(119899 + 119897 + 1) (119899 + 2 minus 119897) (119899 minus 119897)

times(119899 + 119897 + 3)(119899 + 119897 + 2)(119899 + 1 minus 119897)]12

1198991015840

= 119899 + 3

(27)

The investigation of Clark and Taylor [68] has revealedthat it is usually advisable to select higher values of 120577 Inour simulation we used 120577 = 2119899 in order to be close to theionization limit

Our numerical simulations consist in diagonalizing thematrix representing the operator in a Sturmian basis Thematrix elements of the generators are independent of 120572

32 Numerical Method Equation (20) is equivalent to thegeneral problem for eigenvalues of the form [59]

(119872 minus 120582119873) |Ψ⟩ = 0 (28)

where 119872 and 119873 are fixed matrices and 120582 and |Ψ⟩ are theeigenvalues and the eigenvectors searched

321 Simulation at Fixed Magnetic Field In the case wherewe fix the magnetic field 120574 = 119861119861

119888(120574 denotes the magnetic

field in units of 119861119888≃ 235 times 10

5

119879) after opting for 120572 = 1205720=

119862119905119890 we obtain an equation in the generalized eigenvalueswith

[59]

119872 = 119860(120572)

minus 1205720+

1205742

1205724

0

2

119862(120572)

119873 = 119861(120572)

120582 = minus21198641205722

0

(29)

that determines the energy levels

322 Simulation at Fixed Energy In this simulation wechoose 120572 = 120572

0= 119862119905119890 and we fix the energy So that we get

an equation in the generalized eigenvalues with [59]

119872 = 119860(120572)

minus 1205720+ (minus2119864120572

2

0) 119861(120572)

119873 = 119862(120572)

120582 =

1205742

1205724

0

2

(30)

that establishes the spectrum of magnetic field values relatedto the fixed energy

Both types of the simulations give the same outcomes

323 Diagonalization Algorithm and Convergence CriterionThe matrix components of the generators ( 119878

(120572)

119879(120572)) in a

Sturmian basis are restricted to some selectionsrsquo rules It iswell known that the matrix eigenvalue problems generally tobe solved are real symmetric matrices in strips [38]

Currently stable numerical algorithms have been devel-oped in order to calculate the eigenvalues and eigenvectors[48]

We provide an adoptable parameter 1205720to verify the

convergence for each type of the simulations In fact 1205720is

connected to the natural frequency of the basic oscillatorselected The accuracy of the examined values is not relianton this parameter Exact searched values are not dependenton this parameter

4 Numerical Results and Discussion

41 Energy Diagrams We use the simulation at fixed mag-netic field and we plot for a large number of points diagrams119864119903

119899= 119891(120574

2

) in Figures 1 2 and 3Figure 1 shows the diagram 119864(Hartree) = 119891(120574

2

) forRydberg states 119899 = 30 in the system with rather low field

6 Mathematical Problems in Engineering

minus003

minus004

minus005

minus006

minus007

minus008

minus009

E(H

artre

e)

02 04 06 08 10 12 14 16

1205742(times1010)

Figure 1 Energy levels as a function of the normalized squaremagnetic field 120574

2

(120574 = 119861119861119888 119861119888

= 235 times 105

119879) calculated by thesimulation technique described in Section 321 for weak magneticfield 119861 (119861 lt 3119879) and for the principle quantum number 119899 = 30The rovibration structure is completely removed (inter-119897-mixing)We observe that for a weak magnetic field the classical dynamicsis regular

minus005

minus010

minus015

minus020

E(H

artre

e)

1205742(times1010)

20 30 40 50

Figure 2 Diagram of energy levels of the hydrogen atom instrong magnetic field as function of 120574

2

(120574 = 119861119861119888 119861119888

= 235 times

105

119879) computed with a simulation described in Section 321 Thecalculation is done for a strong magnetic field 119861 (9119879 lt 119861 lt 18119879)

for both levels 119899 = 30 (dotted red line) and 119899 = 31 (solid blue line)Mixing between the energy levels of 119899 = 30 and 119899 = 31 is observed

(119861 lt 3119879) diamagnetism breaks the zero-field 119897-degeneracyof the hydrogen atom

We can explain this by the fact that in very weak field(119897-system intermixing) the degeneracy of the levels (therovibrational structure) proportional to 120574

2 is completelyremoved by the diamagnetic term It is visible that regularsystem disturbed by a weak perturbation remains essentiallyregular for the reason that the diamagnetic interaction is aperturbation of the Coulomb dynamics That is a regularregime there is no sensitivity on initial conditions

At highmagnetic field as shown by Figure 2 the differentenergy levels (119899 = 30 119899 = 31) essentially cross

Indeed for a highly excited state with principal quantumnumber 119899 the diamagnetic interaction scales as 120574

2

1198994 and

1205742(times1010)

20 30 40 50

minus005

minus010

minus015

minus020

E(H

artre

e)

Figure 3 Diagram of the energy levels of the hydrogen atom inthe strong magnetic field for 119899 = 30 computed by the simulationdescribed in Section 321 We note that for a high magnetic field theenergy levels anticrossThis is an indication of the chaos emergence

the zero-field as 11198992 The strong field regime is obtained

when [38]

1205741198993

≃ 1 that is 1198611198993

≃ 119861119888 (31)

This is achieved with 119899 ≃ 30 and 119861 ≃ 9119879 The rotationaland vibrational energy levels really cross By increasing themagnetic field in Figure 3 we observe that more statesanticross The classical smooth transition from regularity tochaos agrees well with this result

42The Effect of the Nucleus Recoil Energy In this section weuse the second simulation simulation at fixed energy wherewe have plotted 120574

2 as a function of the angular momentum 119897We introduced the recoil energy effect given in atomic unitby

119864119899(recoil) = minus

1

119872

(1 minus

3119898

119872

)

1205722

81198994 (32)

which takes into account the relativistic effects we recall thatthe levels shift due to this effect can be written as [68]

Δ119864119897=

1205722

21198993119897 (119897 + 1)

(33)

This recoil energy has been introduced with which wehave realized the simulation with nucleus of finite massThe result of the simulation is shown in Figure 4 wherethe distinction between the nucleus having a finite mass(diamond line) and the one having an infinite mass (circleline) is very clear This difference can be justified by the factthat the term (33) is the amount of three contributions Thefirst contribution is the kinetic energy in which the electronmass and not the reduced mass of the system manifests

119882119898V =

4

81198983

1198901198882 (34)

The second contribution is the Darwin term which takesinto account the electron delocalization but not the nucleus

ℏ2

81198982

119890119888241205871198902

120575 ( 119903) (35)

Mathematical Problems in Engineering 7

01280

01275

01270

01265

01260

01255

10 12 14 16 18 20 22 24 26 28

l

1205742(times1010)

Figure 4 The normalized square magnetic field 1205742

(120574 = 119861119861119888

119861119888

= 235 times 105

119879) for different angular momentum l with aprincipal quantum 119899 = 30 calculated by the simulation describedin Section 322 Two cases are computed first case realistic nucleusmass (finite mass) (red diamond line) second case nucleus infinitemass (blue circle line)

Finally the third contribution is the spin-orbit whichshows the interaction between the electronmagneticmomentand themagnetic field of the nucleus seen in the proper frameof the electron Therefore we should take into considerationthe nucleus speed since the nucleus of finite mass is being inmotion

5 Conclusion

We have studied the diamagnetic behavior of a hydrogenatom highly excited in a strong magnetic field consideringthe realistic finite mass of the nucleus The results revealthat when the magnetic field is weak the diamagnetic termremoves the degeneracy of the energy levels The system istotally regular it is possible to affect quantum numbers tothe various energy levels However with increasing magneticfield any symmetry will be destroyed and the energy levelswill be crossed there is an unruffled transition from regu-larity to chaotical dynamics At higher magnetic field thedynamic is totally chaotic and quantum spectrum includesonly large anticrossing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B Li and H P Liu ldquoAn effective quantum defect theory forthe diamagnetic spectrum of a barium Rydberg atomrdquo ChinesePhysics B vol 22 no 1 Article ID 013203 2013

[2] A E Shabad and V V Usov ldquoModified coulomb law in astrongly magnetized vacuumrdquo Physical Review Letters vol 98no 18 Article ID 180403 2007

[3] L Stevanovic and D Milojevic ldquoCompressed hydrogen atomunder debye screening in strong magnetic fieldrdquo Publications of

the Astronomical Observatory of Belgrade vol 91 pp 327ndash3302012

[4] H Jabri H Eleuch and T Djerad ldquoLifetimes of atomic Rydbergstates by autocorrelation functionrdquo Laser Physics Letters vol 2no 5 pp 253ndash257 2005

[5] H Eleuch and N Rachid ldquoAutocorrelation function ofmicrocavity-emitting field in the non-linear regimerdquoThe Euro-pean Physical Journal D vol 57 no 2 pp 259ndash264 2010

[6] H Eleuch N Ben Nessib and R Bennaceur ldquoQuantum modelof emission in a weakly non ideal plasmardquo The EuropeanPhysical Journal D vol 29 no 3 pp 391ndash395 2004

[7] P K Jha H Eleuch and Y V Rostovtsev ldquoCoherent control ofatomic excitation using off-resonant strong few-cycle pulsesrdquoPhysical Review A vol 82 no 4 Article ID 045805 2010

[8] E A Sete and H Eleuch ldquoStrong squeezing and robustentanglement in cavity electromechanicsrdquo Physical Review Avol 89 no 1 Article ID 013841 2014

[9] M Khatua U Sarkar and P K Chattaraj ldquoReactivity dynamicsof confined atoms in the presence of an external magnetic fieldrdquoThe European Physical Journal D vol 68 no 2 article 22 2014

[10] U Das and B Mukhopadhyay ldquoNew mass limit for whitedwarfs super-Chandrasekhar type Ia supernova as a newstandard candlerdquo Physical Review Letters vol 110 no 7 ArticleID 071102 2013

[11] A D Sargsyan R K Mirzoyan A S Sarkisyan A H Amiryanand D H Sarkisyan ldquoSplitting of N-type optical resonanceformed in Λ-system of 85Rb atoms in a strong transversemagnetic fieldrdquo Journal of Contemporary Physics vol 49 no 1pp 20ndash27 2014

[12] V S Popov and B M Karnakov ldquoHydrogen atom in a strongmagnetic fieldrdquo Physics-Uspekhi vol 57 no 3 pp 257ndash279 2014

[13] E Giacobino J-P Karr G Messin H Eleuch and A BaasldquoQuantum optical effects in semiconductor microcavitiesrdquoComptes Rendus Physique vol 3 no 1 pp 41ndash52 2002

[14] R Juarez-Amaro A Zuniga-Segundo F Soto-Eguibar andH M Moya-Cessa ldquoEquivalence between mirror-field-atomand ion-laser interactionsrdquo Applied Mathematics amp InformationSciences vol 7 no 4 pp 1311ndash1315 2013

[15] H Eleuch S Guerin and H R Jauslin ldquoEffects of an environ-ment on a cavity-quantum-electrodynamics system controlledby bichromatic adiabatic passagerdquoPhysical ReviewA vol 85 no1 Article ID 013830 2012

[16] E A Sete A A Svidzinsky Y V Rostovtsev et al ldquoUsingquantum coherence to generate gain in the xuv and x-raygain-swept superradiance and lasing without inversionrdquo IEEEJournal on Selected Topics in Quantum Electronics vol 18 no 1pp 541ndash553 2012

[17] E A Sete A A Svidzinsky H Eleuch Z Yang R D Nevelsand M O Scully ldquoCorrelated spontaneous emission on theDanuberdquo Journal of Modern Optics vol 57 no 14-15 pp 1311ndash1330 2010

[18] Z BWalters ldquoQuantum coherent dynamics at ambient temper-ature in photosynthetic moleculesrdquo Quantum Physics Lettersvol 1 no 1 pp 21ndash33 2012

[19] H Rabitz ldquoControl in the sciences over vast length and timescalesrdquo Quantum Physics Letters vol 1 no 1 pp 1ndash19 2012

[20] DA Barskiy KVKovtunov I VKoptyug et al ldquoThe feasibilityof formation and kinetics of NMR signal amplification byreversible exchange (SABRE) at high magnetic field (94 T)rdquoJournal of the American Chemical Society vol 136 no 9 pp3322ndash3325 2014

8 Mathematical Problems in Engineering

[21] B Mandal S Adhikari K Choudhary et al ldquoFemtosecondresponse of quantum discrete breathers in SRR based metama-terials and the role of dielectric permittivityrdquo Quantum PhysicsLetters vol 1 no 2 pp 59ndash68 2012

[22] J D Pritchard J A Isaacs M A Beck R McDermottand M Saffman ldquoHybrid atom-photon quantum gate in asuperconducting microwave resonatorrdquo Physical Review A vol89 no 1 Article ID 010301 2014

[23] P Rej and A Ghoshal ldquoRydberg transitions for positron-hydrogen collisions asymptotic cross section and scaling lawrdquoJournal of Physics B Atomic Molecular and Optical Physics vol47 no 1 Article ID 015204 2014

[24] H Eleuch D Elser and R Bennaceur ldquoSoliton propagation inan absorbing three-level atomic systemrdquo Laser Physics Lettersvol 1 no 8 pp 391ndash396 2004

[25] H Eleuch and R Bennaceur ldquoAn optical soliton pair amongabsorbing three-level atomsrdquo Journal of Optics A Pure andApplied Optics vol 5 no 5 pp 528ndash533 2003

[26] E Osman M Khalfallah and H Sapoor ldquoImproved (G1015840G)-expansion method for elliptic-like equation and general trav-eling wave solutions of some class of NLPDEsrdquo InformationScience Letters vol 2 no 2 pp 123ndash127 2013

[27] M Akbari ldquoExact solutions of the coupled Higgs equationand the Maccari system using the modified simplest equationmethodrdquo Information Science Letters vol 2 no 3 pp 155ndash1582013

[28] N Taghizadeh M Mirzazadeh M Akbari and M RahimianldquoExact soliton solutions for generalized equal width equationrdquoMathematical Science Letters vol 2 no 2 pp 99ndash106 2013

[29] E Osman M Khalfallah and H Sapoor ldquoFinding generalsolutions of nonlinear evolution equations by improved (G1015840G)-expansion methodrdquo Mathematical Science Letters vol 3 no 1pp 1ndash8 2014

[30] E M E Zayed ldquoExact traveling wave solutions for a variable-coefficient generalized dispersive water-wave system using thegeneralized (GrsquoG)-expansion methodrdquo Mathematical ScienceLetters vol 3 no 1 pp 9ndash15 2014

[31] E A Sete and H Eleuch ldquoInteraction of a quantum well withsqueezed light quantum-statistical propertiesrdquo Physical ReviewA vol 82 no 4 Article ID 043810 2010

[32] W Yang L Huang and M Tian ldquoTwo novel and secure quan-tum ANDOS schemesrdquo Applied Mathematics and InformationSciences vol 7 no 2 pp 659ndash665 2013

[33] H Eleuch ldquoPhoton statistics of light in semiconductor micro-cavitiesrdquo Journal of Physics B Atomic Molecular and OpticalPhysics vol 41 no 5 Article ID 055502 2008

[34] G Messin J P Karr H Eleuch J M Courty and E GiacobinoldquoSqueezed states and the quantumnoise of light in semiconduc-tor microcavitiesrdquo Journal of Physics Condensed Matter vol 11no 31 pp 6069ndash6078 1999

[35] A Baas J P Karr H Eleuch and E Giacobino ldquoOpticalbistability in semiconductor microcavitiesrdquo Physical Review AAtomic Molecular and Optical Physics vol 69 no 2 2004

[36] J Bonca and S Kruchinin Physical Properties of NanosystemsSpringer 2011

[37] L Tang and J-A Lu ldquoProjective lag synchronization of chaoticsystems with parameter mismatchrdquo Applied Mathematics ampInformation Sciences vol 7 no 1 pp 121ndash125 2013

[38] D Delande ldquoChaos and atoms in strong magnetic fieldsrdquoPhysica Scripta vol 1991 no T34 p 52 1991

[39] H Ruder G Wunner H Herold and F Geyer Atoms in StrongMagnetic Fields Springer Berlin Germany 1994

[40] H Eleuch and A Prasad ldquoChaos and regularity in semiconduc-tormicrocavitiesrdquoPhysics Letters A vol 376 no 26-27 pp 1970ndash1977 2012

[41] A Y Potekhin and A V Turbiner ldquoHydrogen atom in amagnetic field the quadrupole momentrdquo Physical Review AAtomic Molecular and Optical Physics vol 63 no 6 Article ID065402 2001

[42] S N Rasband Chaotic Dynamics of Non linear Systems JohnWiley amp Sons 1990

[43] A J Lichtenberg and M A Lieberman Regular and StochasticMotion Springer New York NY USA 1983

[44] A V Andreev O Agam B D Simons and B L AltshulerldquoQuantum chaos irreversible classical dynamics and randommatrix theoryrdquo Physical Review Letters vol 76 no 21 pp 3947ndash3950 1996

[45] O Bohigas ldquoLesHouches summer school session LIIrdquo inChaosand Quantum Physics M-J Giannoni A Voros and J Zinn-Justin Eds North-Holland Amsterdam The Netherlands1991

[46] T Pohl H R Sadeghpour and P Schmelcher ldquoCold andultracold Rydberg atoms in strong magnetic fieldsrdquo PhysicsReports vol 484 no 6 pp 181ndash229 2009

[47] T F Gallagher Rydberg Atoms Cambridge University PressCambridge UK 1994

[48] M I Eides H Grotch and V A Shelyuto Theory of LightHydrogenic Bound States vol 222 of Springer Tracts in ModernPhysics 2007

[49] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013

[50] C Yang S S Ge and T H Lee ldquoOutput feedback adaptivecontrol of a class of nonlinear discrete-time systems withunknown control directionsrdquoAutomatica vol 45 no 1 pp 270ndash276 2009

[51] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011

[52] C Yang S S Ge C Xiang T Chai and T H Lee ldquoOutputfeedback NN control for two classes of discrete-time systemswith unknown control directions in a unified approachrdquo IEEETransactions on Neural Networks vol 19 no 11 pp 1873ndash18862008

[53] Y-J Liu S-C Tong DWang T-S Li and C L P Chen ldquoAdap-tive neural output feedback controller design with reduced-order observer for a class of uncertain nonlinear SISO SystemsrdquoIEEE Transactions on Neural Networks vol 22 no 8 pp 1328ndash1334 2011

[54] T M Shahwan ldquoA comparison of Bayesian methods andartificial neural networks for forecasting chaotic financial timeseriesrdquo Journal of Statistics Applications amp Probability vol 1 no2 p 89 2012

[55] J W SeriesThe Spectrum of Atomic Hydrogen Oxford Univer-sity Press London UK 1957

[56] T van der Veldt W Vassen and W Hogervorst ldquoHeliumRydberg states in parallel electric and magnetic fieldsrdquo Journalof Physics B Atomic Molecular and Optical Physics vol 26 no13 pp 1945ndash1955 1993

Mathematical Problems in Engineering 9

[57] A Arimondo W D Phillips and F Strumia Laser Manip-ulation of Atoms and Ions North-Holland Amsterdam TheNetherlands 1992

[58] D Weiss K Richter A Menschig et al ldquoQuantized periodicorbits in large antidot arraysrdquo Physical Review Letters vol 70no 26 pp 4118ndash4121 1993

[59] D Delande Atomes de rydberg en champs statiques intenses[These drsquoEtat] Universite Paris 6 Paris France 1988

[60] M Sadhukhan and B M Deb ldquoA dynamical signature ofquantum chaos in hydrogen atom under strong oscillatingmagnetic fieldsrdquo EPL vol 94 no 5 Article ID 50008 2011

[61] A Bruno-Alfonso C Bleasdale G V B de Souza and R ALewis ldquoClosed-orbit dependence on the field direction in theanisotropic diamagnetic Kepler problemrdquo Physical Review AmdashAtomic Molecular and Optical Physics vol 89 no 4 Article ID043425 2014

[62] J-K Xu H-Q Chen and H-P Liu ldquoExact quantum defecttheory approach for lithium in magnetic fieldsrdquo Chinese PhysicsB vol 22 no 1 Article ID 013204 2013

[63] M A Amdouni and H Eleuch ldquoInfluence of relativistic termsin the spectra of hydrogen atom highly excited in an externalstrong magnetic fieldrdquo Applied Mathematics amp InformationSciences vol 5 no 2 pp 307ndash320 2011

[64] J C Slater Quantum Theory of Atomic Structure vol 1-2McGraw-Hill New York NY USA 1996

[65] E U Condon and G H ShortleyTheTheory of Atomic SpectraCambridge University Press Cambridge UK 1953

[66] C C Tannoudji B Diu and F Laloe Quantum MechanicsWiley New York NY USA 1977 Mechanics Wiley

[67] P J Mohr B N Taylor and D B Newell ldquoCODATA recom-mended values of the fundamental physical constants 2006rdquoReviews of Modern Physics vol 80 no 2 pp 633ndash730 2008

[68] C W Clark and K T Taylor ldquoThe quadratic Zeeman effect inhydrogen Rydberg series application of Sturmian functionsrdquoJournal of Physics B Atomic and Molecular Physics vol 15 no8 pp 1175ndash1193 1982

[69] M A Iken F Borondo R M Benito and T Uzer ldquoPeriodic-orbit spectroscopy of the hydrogen atom in parallel electric andmagnetic fieldsrdquo Physical ReviewA vol 49 no 4 pp 2734ndash27471994

[70] A R Edmonds ldquoStudies of the quadratic Zeeman effect IApplication of the sturmian functionsrdquo Journal of Physics BAtomic and Molecular Physics vol 6 no 8 pp 1603ndash1615 1973

[71] K Sacha J Zakrzewski and D Delande ldquoBreaking timereversal symmetry in chaotic driven rydberg atomsrdquo Annals ofPhysics vol 283 no 1 pp 141ndash172 2000

[72] M Courtney Rydberg atoms in strong fields a testing groundfor quantum chaos [PhD thesis] Department of PhysicsMassachusetts Institute of Technology 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

are able to express the Schrodinger equation in function ofthese generators ( 119878

(120572)

119879(120572)) of dynamic group SO(2 2) They

are defined respectively on an adjustable parameter 120572 [38]

119878(120572)

1= minus

120572

4

(1199012

120578+

1198712

119911

1205782) +

1

4120572

1205782

119878(120572)

2= minus

1

2

120578119901120578

119878(120572)

3=

120572

4

(1199012

120578+

1198712

119911

1205782) +

1

4120572

1205782

119879(120572)

1= minus

120572

4

(1199012

] +

1198712

119911

]2) +

1

4120572

]2

119879(120572)

2= minus

1

2

]119901]

119879(120572)

3=

120572

4

(1199012

] +

1198712

119911

]2) +

1

4120572

]2

(15)

where semiparabolic coordinates (120578 ]) are defined as

120578 = radic119903 + 119911

] = radic119903 minus 119911

(16)

while the momentums (119901120578 119901]) are defined as

119901120578= (1205782

+ ]2)119889120578

119889119905

119901] = (1205782

+ ]2)119889]119889119905

(17)

and 119871119911is the 119911-component of the angular momentum

These 6 variables are not independent

1198782

= (119878(120572)

3)

2

minus (119878(120572)

1)

2

minus (119878(120572)

2)

2

=

1198712

119911

4

1198792

= (119879(120572)

3)

2

minus (119879(120572)

1)

2

minus (119879(120572)

2)

2

=

1198712

119911

4

(18)

Thus the dynamic variables ( 119878(120572)

119879(120572)) represent the

classical formof the quantumgenerator of the dynamic groupSO(2 2)

The Hamiltonian ldquooscillatorrdquo obtained coincides with itsexpression in the context of the classic mechanic

(119878(120572)

3+ 119879(120572)

3+

1205742

2 (minus2119864)2(119878(120572)

1+ 119878(120572)

3) (119879(120572)

1+ 119879(120572)

3)

times (119878(120572)

1+ 119878(120572)

3+ 119879(120572)

1+ 119879(120572)

3) minus

1

radicminus2119864

) |Ψ⟩ = 0

(19)

with 120572 = 1radicminus2119864

We can write a more general equation by taking any valueof the adjustable parameter 120572 [57] We find

[119860(120572)

minus 120572 + (minus21198641205722

) 119861(120572)

+

1205742

1205724

2

119862(120572)

] |Ψ⟩ = 0 (20)

with

119860(120572)

=

119878(120572)

3minus 119878(120572)

1

2

+

119879(120572)

3minus 119879(120572)

1

2

119861(120572)

=

119878(120572)

3+ 119878(120572)

1

2

+

119879(120572)

3+ 119879(120572)

1

2

119862(120572)

= (119878(120572)

1+ 119878(120572)

3) (119879(120572)

1+ 119879(120572)

3) (119878(120572)

1+ 119878(120572)

3+ 119879(120572)

1+ 119879(120572)

3)

(21)

In the Sturmian basis the eigenvalues of the hydrogenHamiltonian are obtained by solving the generalized eigen-value problem (20)

The spherical Sturmian basis functions are [72]

Ψ119899119897119898

(119903) = 119878(120572)

119899119897(119903) 119884119897119898

(120579 120593) (22)

where 119878(120572)

119899119897(119903) is the radial Sturmian function

119878(120572)

119899119897(119903) = [

(119899 minus 119897 minus 1)

2 (119899 + 119897)

]

(12)

(

119903

2

)

119897+1

119890minus1199032120572

1198712119897+1

119899minus119897minus1(

119903

120572

)

(23)

and 119871119895

119894is the associated Laguerre polynomial

The full eigenfunctions used can be written as

| 119899119897119898⟩ = 119878(120572)

119899119897119884119897119898

(24)

Then the matrix element of the diamagnetic term(1198612

8)1199032sin2(120579) is given by

⟨1198991015840

1198971015840

119898

100381610038161003816100381610038161003816100381610038161003816

1198612

8

1199032sin2 (120579)

100381610038161003816100381610038161003816100381610038161003816

119899119897119898⟩ =

1

8

1198612

⟨1198991015840

119897101584010038161003816100381610038161003816119903210038161003816100381610038161003816119899119897⟩

times ⟨1198971015840

119898

10038161003816100381610038161003816sin2 (120579)1003816100381610038161003816

1003816119897119898⟩

(25)

Evaluation of the angular matrix elements is then easyThe corresponding matrix elements of the square positionoperator 1199032 are written as [68]

⟨1198991015840

119897101584010038161003816100381610038161003816119903210038161003816100381610038161003816119899119897⟩ = int

infin

0

119878120577

119899119897(119903) 1199032

119878120577

11989910158401198971015840(119903) 119889119903 (26)

Mathematical Problems in Engineering 5

are nonzero only for |119899 minus 1198991015840

| = 0 1 2 3 |119897 minus 1198971015840

| = 0 2 and atthese points take the following forms

⟨1198991015840

119897 + 2

10038161003816100381610038161003816119903210038161003816100381610038161003816119899119897⟩

=

minus

1

2

120577minus3

[(119899 minus 119897 minus 1) (119899 minus 119897 minus 2) (119899 minus 119897 minus 3)

times(119899 minus 119897 minus 4)(119899 minus 119897 minus 5)(119899 + 119897)]12

1198991015840

= 119899 minus 3

120577minus3

[(3119899 + 2119897) (119899 minus 119897 minus 1) (119899 minus 119897 minus 2)

times(119899 minus 119897 minus 3)(119899 minus 119897 minus 4)]12

1198991015840

= 119899 minus 2

minus

5

2

120577minus3

[(3119899 + 119897) (119899 minus 119897 minus 1) (119899 minus 119897 minus 2)

times(119899 minus 119897 minus 3)(119899 + 119897 + 1)]12

1198991015840

= 119899 minus 1

10119899120577minus3

[(119899 + 119897 + 1) (119899 + 119897 + 2)

times(119899 minus 119897 minus 1)(119899 minus 119897 minus 2)]12

1198991015840

= 119899

minus

5

2

120577minus3

(3119899 minus 119897) [(119899 + 119897 + 3) (119899 + 119897 + 2)

times(119899 + 119897 + 1)(119899 minus 119897 minus 1)]12

1198991015840

= 119899 + 1

120577minus3

(3119899 minus 2119897) [(119899 + 119897 + 4) (119899 + 119897 + 3)

times(119899 + 119897 + 2)(119899 + 119897 + 1)]12

1198991015840

= 119899 + 2

minus

1

2

120577minus3

[(119899 + 119897 + 5) (119899 + 119897 + 4) (119899 + 119897 + 3)

times(119899 + 119897 + 2)(119899 + 119897 + 1)(119899 minus 119897)]12

1198991015840

= 119899 + 3

⟨1198991015840

119897

10038161003816100381610038161003816119903210038161003816100381610038161003816119899119897⟩

=

2119899120577minus3

[51198992

minus 3119897 (119897 + 1) + 1] 1198991015840

= 119899

minus

3

2

120577minus3

[5119899 (119899 + 1) minus 119897 (119897 + 1) + 2]

times [(119899 + 119897 + 1) (119899 minus 119897)]12

1198991015840

= 119899 + 1

3120577

minus3

(119899 + 1) [(119899 + 119897 + 1) (119899 minus 119897)

times(119899 + 119897 + 2)(119899 minus 119897 + 1)]12

1198991015840

= 119899 + 2

minus

1

2

120577minus3

[(119899 + 119897 + 1) (119899 + 2 minus 119897) (119899 minus 119897)

times(119899 + 119897 + 3)(119899 + 119897 + 2)(119899 + 1 minus 119897)]12

1198991015840

= 119899 + 3

(27)

The investigation of Clark and Taylor [68] has revealedthat it is usually advisable to select higher values of 120577 Inour simulation we used 120577 = 2119899 in order to be close to theionization limit

Our numerical simulations consist in diagonalizing thematrix representing the operator in a Sturmian basis Thematrix elements of the generators are independent of 120572

32 Numerical Method Equation (20) is equivalent to thegeneral problem for eigenvalues of the form [59]

(119872 minus 120582119873) |Ψ⟩ = 0 (28)

where 119872 and 119873 are fixed matrices and 120582 and |Ψ⟩ are theeigenvalues and the eigenvectors searched

321 Simulation at Fixed Magnetic Field In the case wherewe fix the magnetic field 120574 = 119861119861

119888(120574 denotes the magnetic

field in units of 119861119888≃ 235 times 10

5

119879) after opting for 120572 = 1205720=

119862119905119890 we obtain an equation in the generalized eigenvalueswith

[59]

119872 = 119860(120572)

minus 1205720+

1205742

1205724

0

2

119862(120572)

119873 = 119861(120572)

120582 = minus21198641205722

0

(29)

that determines the energy levels

322 Simulation at Fixed Energy In this simulation wechoose 120572 = 120572

0= 119862119905119890 and we fix the energy So that we get

an equation in the generalized eigenvalues with [59]

119872 = 119860(120572)

minus 1205720+ (minus2119864120572

2

0) 119861(120572)

119873 = 119862(120572)

120582 =

1205742

1205724

0

2

(30)

that establishes the spectrum of magnetic field values relatedto the fixed energy

Both types of the simulations give the same outcomes

323 Diagonalization Algorithm and Convergence CriterionThe matrix components of the generators ( 119878

(120572)

119879(120572)) in a

Sturmian basis are restricted to some selectionsrsquo rules It iswell known that the matrix eigenvalue problems generally tobe solved are real symmetric matrices in strips [38]

Currently stable numerical algorithms have been devel-oped in order to calculate the eigenvalues and eigenvectors[48]

We provide an adoptable parameter 1205720to verify the

convergence for each type of the simulations In fact 1205720is

connected to the natural frequency of the basic oscillatorselected The accuracy of the examined values is not relianton this parameter Exact searched values are not dependenton this parameter

4 Numerical Results and Discussion

41 Energy Diagrams We use the simulation at fixed mag-netic field and we plot for a large number of points diagrams119864119903

119899= 119891(120574

2

) in Figures 1 2 and 3Figure 1 shows the diagram 119864(Hartree) = 119891(120574

2

) forRydberg states 119899 = 30 in the system with rather low field

6 Mathematical Problems in Engineering

minus003

minus004

minus005

minus006

minus007

minus008

minus009

E(H

artre

e)

02 04 06 08 10 12 14 16

1205742(times1010)

Figure 1 Energy levels as a function of the normalized squaremagnetic field 120574

2

(120574 = 119861119861119888 119861119888

= 235 times 105

119879) calculated by thesimulation technique described in Section 321 for weak magneticfield 119861 (119861 lt 3119879) and for the principle quantum number 119899 = 30The rovibration structure is completely removed (inter-119897-mixing)We observe that for a weak magnetic field the classical dynamicsis regular

minus005

minus010

minus015

minus020

E(H

artre

e)

1205742(times1010)

20 30 40 50

Figure 2 Diagram of energy levels of the hydrogen atom instrong magnetic field as function of 120574

2

(120574 = 119861119861119888 119861119888

= 235 times

105

119879) computed with a simulation described in Section 321 Thecalculation is done for a strong magnetic field 119861 (9119879 lt 119861 lt 18119879)

for both levels 119899 = 30 (dotted red line) and 119899 = 31 (solid blue line)Mixing between the energy levels of 119899 = 30 and 119899 = 31 is observed

(119861 lt 3119879) diamagnetism breaks the zero-field 119897-degeneracyof the hydrogen atom

We can explain this by the fact that in very weak field(119897-system intermixing) the degeneracy of the levels (therovibrational structure) proportional to 120574

2 is completelyremoved by the diamagnetic term It is visible that regularsystem disturbed by a weak perturbation remains essentiallyregular for the reason that the diamagnetic interaction is aperturbation of the Coulomb dynamics That is a regularregime there is no sensitivity on initial conditions

At highmagnetic field as shown by Figure 2 the differentenergy levels (119899 = 30 119899 = 31) essentially cross

Indeed for a highly excited state with principal quantumnumber 119899 the diamagnetic interaction scales as 120574

2

1198994 and

1205742(times1010)

20 30 40 50

minus005

minus010

minus015

minus020

E(H

artre

e)

Figure 3 Diagram of the energy levels of the hydrogen atom inthe strong magnetic field for 119899 = 30 computed by the simulationdescribed in Section 321 We note that for a high magnetic field theenergy levels anticrossThis is an indication of the chaos emergence

the zero-field as 11198992 The strong field regime is obtained

when [38]

1205741198993

≃ 1 that is 1198611198993

≃ 119861119888 (31)

This is achieved with 119899 ≃ 30 and 119861 ≃ 9119879 The rotationaland vibrational energy levels really cross By increasing themagnetic field in Figure 3 we observe that more statesanticross The classical smooth transition from regularity tochaos agrees well with this result

42The Effect of the Nucleus Recoil Energy In this section weuse the second simulation simulation at fixed energy wherewe have plotted 120574

2 as a function of the angular momentum 119897We introduced the recoil energy effect given in atomic unitby

119864119899(recoil) = minus

1

119872

(1 minus

3119898

119872

)

1205722

81198994 (32)

which takes into account the relativistic effects we recall thatthe levels shift due to this effect can be written as [68]

Δ119864119897=

1205722

21198993119897 (119897 + 1)

(33)

This recoil energy has been introduced with which wehave realized the simulation with nucleus of finite massThe result of the simulation is shown in Figure 4 wherethe distinction between the nucleus having a finite mass(diamond line) and the one having an infinite mass (circleline) is very clear This difference can be justified by the factthat the term (33) is the amount of three contributions Thefirst contribution is the kinetic energy in which the electronmass and not the reduced mass of the system manifests

119882119898V =

4

81198983

1198901198882 (34)

The second contribution is the Darwin term which takesinto account the electron delocalization but not the nucleus

ℏ2

81198982

119890119888241205871198902

120575 ( 119903) (35)

Mathematical Problems in Engineering 7

01280

01275

01270

01265

01260

01255

10 12 14 16 18 20 22 24 26 28

l

1205742(times1010)

Figure 4 The normalized square magnetic field 1205742

(120574 = 119861119861119888

119861119888

= 235 times 105

119879) for different angular momentum l with aprincipal quantum 119899 = 30 calculated by the simulation describedin Section 322 Two cases are computed first case realistic nucleusmass (finite mass) (red diamond line) second case nucleus infinitemass (blue circle line)

Finally the third contribution is the spin-orbit whichshows the interaction between the electronmagneticmomentand themagnetic field of the nucleus seen in the proper frameof the electron Therefore we should take into considerationthe nucleus speed since the nucleus of finite mass is being inmotion

5 Conclusion

We have studied the diamagnetic behavior of a hydrogenatom highly excited in a strong magnetic field consideringthe realistic finite mass of the nucleus The results revealthat when the magnetic field is weak the diamagnetic termremoves the degeneracy of the energy levels The system istotally regular it is possible to affect quantum numbers tothe various energy levels However with increasing magneticfield any symmetry will be destroyed and the energy levelswill be crossed there is an unruffled transition from regu-larity to chaotical dynamics At higher magnetic field thedynamic is totally chaotic and quantum spectrum includesonly large anticrossing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B Li and H P Liu ldquoAn effective quantum defect theory forthe diamagnetic spectrum of a barium Rydberg atomrdquo ChinesePhysics B vol 22 no 1 Article ID 013203 2013

[2] A E Shabad and V V Usov ldquoModified coulomb law in astrongly magnetized vacuumrdquo Physical Review Letters vol 98no 18 Article ID 180403 2007

[3] L Stevanovic and D Milojevic ldquoCompressed hydrogen atomunder debye screening in strong magnetic fieldrdquo Publications of

the Astronomical Observatory of Belgrade vol 91 pp 327ndash3302012

[4] H Jabri H Eleuch and T Djerad ldquoLifetimes of atomic Rydbergstates by autocorrelation functionrdquo Laser Physics Letters vol 2no 5 pp 253ndash257 2005

[5] H Eleuch and N Rachid ldquoAutocorrelation function ofmicrocavity-emitting field in the non-linear regimerdquoThe Euro-pean Physical Journal D vol 57 no 2 pp 259ndash264 2010

[6] H Eleuch N Ben Nessib and R Bennaceur ldquoQuantum modelof emission in a weakly non ideal plasmardquo The EuropeanPhysical Journal D vol 29 no 3 pp 391ndash395 2004

[7] P K Jha H Eleuch and Y V Rostovtsev ldquoCoherent control ofatomic excitation using off-resonant strong few-cycle pulsesrdquoPhysical Review A vol 82 no 4 Article ID 045805 2010

[8] E A Sete and H Eleuch ldquoStrong squeezing and robustentanglement in cavity electromechanicsrdquo Physical Review Avol 89 no 1 Article ID 013841 2014

[9] M Khatua U Sarkar and P K Chattaraj ldquoReactivity dynamicsof confined atoms in the presence of an external magnetic fieldrdquoThe European Physical Journal D vol 68 no 2 article 22 2014

[10] U Das and B Mukhopadhyay ldquoNew mass limit for whitedwarfs super-Chandrasekhar type Ia supernova as a newstandard candlerdquo Physical Review Letters vol 110 no 7 ArticleID 071102 2013

[11] A D Sargsyan R K Mirzoyan A S Sarkisyan A H Amiryanand D H Sarkisyan ldquoSplitting of N-type optical resonanceformed in Λ-system of 85Rb atoms in a strong transversemagnetic fieldrdquo Journal of Contemporary Physics vol 49 no 1pp 20ndash27 2014

[12] V S Popov and B M Karnakov ldquoHydrogen atom in a strongmagnetic fieldrdquo Physics-Uspekhi vol 57 no 3 pp 257ndash279 2014

[13] E Giacobino J-P Karr G Messin H Eleuch and A BaasldquoQuantum optical effects in semiconductor microcavitiesrdquoComptes Rendus Physique vol 3 no 1 pp 41ndash52 2002

[14] R Juarez-Amaro A Zuniga-Segundo F Soto-Eguibar andH M Moya-Cessa ldquoEquivalence between mirror-field-atomand ion-laser interactionsrdquo Applied Mathematics amp InformationSciences vol 7 no 4 pp 1311ndash1315 2013

[15] H Eleuch S Guerin and H R Jauslin ldquoEffects of an environ-ment on a cavity-quantum-electrodynamics system controlledby bichromatic adiabatic passagerdquoPhysical ReviewA vol 85 no1 Article ID 013830 2012

[16] E A Sete A A Svidzinsky Y V Rostovtsev et al ldquoUsingquantum coherence to generate gain in the xuv and x-raygain-swept superradiance and lasing without inversionrdquo IEEEJournal on Selected Topics in Quantum Electronics vol 18 no 1pp 541ndash553 2012

[17] E A Sete A A Svidzinsky H Eleuch Z Yang R D Nevelsand M O Scully ldquoCorrelated spontaneous emission on theDanuberdquo Journal of Modern Optics vol 57 no 14-15 pp 1311ndash1330 2010

[18] Z BWalters ldquoQuantum coherent dynamics at ambient temper-ature in photosynthetic moleculesrdquo Quantum Physics Lettersvol 1 no 1 pp 21ndash33 2012

[19] H Rabitz ldquoControl in the sciences over vast length and timescalesrdquo Quantum Physics Letters vol 1 no 1 pp 1ndash19 2012

[20] DA Barskiy KVKovtunov I VKoptyug et al ldquoThe feasibilityof formation and kinetics of NMR signal amplification byreversible exchange (SABRE) at high magnetic field (94 T)rdquoJournal of the American Chemical Society vol 136 no 9 pp3322ndash3325 2014

8 Mathematical Problems in Engineering

[21] B Mandal S Adhikari K Choudhary et al ldquoFemtosecondresponse of quantum discrete breathers in SRR based metama-terials and the role of dielectric permittivityrdquo Quantum PhysicsLetters vol 1 no 2 pp 59ndash68 2012

[22] J D Pritchard J A Isaacs M A Beck R McDermottand M Saffman ldquoHybrid atom-photon quantum gate in asuperconducting microwave resonatorrdquo Physical Review A vol89 no 1 Article ID 010301 2014

[23] P Rej and A Ghoshal ldquoRydberg transitions for positron-hydrogen collisions asymptotic cross section and scaling lawrdquoJournal of Physics B Atomic Molecular and Optical Physics vol47 no 1 Article ID 015204 2014

[24] H Eleuch D Elser and R Bennaceur ldquoSoliton propagation inan absorbing three-level atomic systemrdquo Laser Physics Lettersvol 1 no 8 pp 391ndash396 2004

[25] H Eleuch and R Bennaceur ldquoAn optical soliton pair amongabsorbing three-level atomsrdquo Journal of Optics A Pure andApplied Optics vol 5 no 5 pp 528ndash533 2003

[26] E Osman M Khalfallah and H Sapoor ldquoImproved (G1015840G)-expansion method for elliptic-like equation and general trav-eling wave solutions of some class of NLPDEsrdquo InformationScience Letters vol 2 no 2 pp 123ndash127 2013

[27] M Akbari ldquoExact solutions of the coupled Higgs equationand the Maccari system using the modified simplest equationmethodrdquo Information Science Letters vol 2 no 3 pp 155ndash1582013

[28] N Taghizadeh M Mirzazadeh M Akbari and M RahimianldquoExact soliton solutions for generalized equal width equationrdquoMathematical Science Letters vol 2 no 2 pp 99ndash106 2013

[29] E Osman M Khalfallah and H Sapoor ldquoFinding generalsolutions of nonlinear evolution equations by improved (G1015840G)-expansion methodrdquo Mathematical Science Letters vol 3 no 1pp 1ndash8 2014

[30] E M E Zayed ldquoExact traveling wave solutions for a variable-coefficient generalized dispersive water-wave system using thegeneralized (GrsquoG)-expansion methodrdquo Mathematical ScienceLetters vol 3 no 1 pp 9ndash15 2014

[31] E A Sete and H Eleuch ldquoInteraction of a quantum well withsqueezed light quantum-statistical propertiesrdquo Physical ReviewA vol 82 no 4 Article ID 043810 2010

[32] W Yang L Huang and M Tian ldquoTwo novel and secure quan-tum ANDOS schemesrdquo Applied Mathematics and InformationSciences vol 7 no 2 pp 659ndash665 2013

[33] H Eleuch ldquoPhoton statistics of light in semiconductor micro-cavitiesrdquo Journal of Physics B Atomic Molecular and OpticalPhysics vol 41 no 5 Article ID 055502 2008

[34] G Messin J P Karr H Eleuch J M Courty and E GiacobinoldquoSqueezed states and the quantumnoise of light in semiconduc-tor microcavitiesrdquo Journal of Physics Condensed Matter vol 11no 31 pp 6069ndash6078 1999

[35] A Baas J P Karr H Eleuch and E Giacobino ldquoOpticalbistability in semiconductor microcavitiesrdquo Physical Review AAtomic Molecular and Optical Physics vol 69 no 2 2004

[36] J Bonca and S Kruchinin Physical Properties of NanosystemsSpringer 2011

[37] L Tang and J-A Lu ldquoProjective lag synchronization of chaoticsystems with parameter mismatchrdquo Applied Mathematics ampInformation Sciences vol 7 no 1 pp 121ndash125 2013

[38] D Delande ldquoChaos and atoms in strong magnetic fieldsrdquoPhysica Scripta vol 1991 no T34 p 52 1991

[39] H Ruder G Wunner H Herold and F Geyer Atoms in StrongMagnetic Fields Springer Berlin Germany 1994

[40] H Eleuch and A Prasad ldquoChaos and regularity in semiconduc-tormicrocavitiesrdquoPhysics Letters A vol 376 no 26-27 pp 1970ndash1977 2012

[41] A Y Potekhin and A V Turbiner ldquoHydrogen atom in amagnetic field the quadrupole momentrdquo Physical Review AAtomic Molecular and Optical Physics vol 63 no 6 Article ID065402 2001

[42] S N Rasband Chaotic Dynamics of Non linear Systems JohnWiley amp Sons 1990

[43] A J Lichtenberg and M A Lieberman Regular and StochasticMotion Springer New York NY USA 1983

[44] A V Andreev O Agam B D Simons and B L AltshulerldquoQuantum chaos irreversible classical dynamics and randommatrix theoryrdquo Physical Review Letters vol 76 no 21 pp 3947ndash3950 1996

[45] O Bohigas ldquoLesHouches summer school session LIIrdquo inChaosand Quantum Physics M-J Giannoni A Voros and J Zinn-Justin Eds North-Holland Amsterdam The Netherlands1991

[46] T Pohl H R Sadeghpour and P Schmelcher ldquoCold andultracold Rydberg atoms in strong magnetic fieldsrdquo PhysicsReports vol 484 no 6 pp 181ndash229 2009

[47] T F Gallagher Rydberg Atoms Cambridge University PressCambridge UK 1994

[48] M I Eides H Grotch and V A Shelyuto Theory of LightHydrogenic Bound States vol 222 of Springer Tracts in ModernPhysics 2007

[49] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013

[50] C Yang S S Ge and T H Lee ldquoOutput feedback adaptivecontrol of a class of nonlinear discrete-time systems withunknown control directionsrdquoAutomatica vol 45 no 1 pp 270ndash276 2009

[51] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011

[52] C Yang S S Ge C Xiang T Chai and T H Lee ldquoOutputfeedback NN control for two classes of discrete-time systemswith unknown control directions in a unified approachrdquo IEEETransactions on Neural Networks vol 19 no 11 pp 1873ndash18862008

[53] Y-J Liu S-C Tong DWang T-S Li and C L P Chen ldquoAdap-tive neural output feedback controller design with reduced-order observer for a class of uncertain nonlinear SISO SystemsrdquoIEEE Transactions on Neural Networks vol 22 no 8 pp 1328ndash1334 2011

[54] T M Shahwan ldquoA comparison of Bayesian methods andartificial neural networks for forecasting chaotic financial timeseriesrdquo Journal of Statistics Applications amp Probability vol 1 no2 p 89 2012

[55] J W SeriesThe Spectrum of Atomic Hydrogen Oxford Univer-sity Press London UK 1957

[56] T van der Veldt W Vassen and W Hogervorst ldquoHeliumRydberg states in parallel electric and magnetic fieldsrdquo Journalof Physics B Atomic Molecular and Optical Physics vol 26 no13 pp 1945ndash1955 1993

Mathematical Problems in Engineering 9

[57] A Arimondo W D Phillips and F Strumia Laser Manip-ulation of Atoms and Ions North-Holland Amsterdam TheNetherlands 1992

[58] D Weiss K Richter A Menschig et al ldquoQuantized periodicorbits in large antidot arraysrdquo Physical Review Letters vol 70no 26 pp 4118ndash4121 1993

[59] D Delande Atomes de rydberg en champs statiques intenses[These drsquoEtat] Universite Paris 6 Paris France 1988

[60] M Sadhukhan and B M Deb ldquoA dynamical signature ofquantum chaos in hydrogen atom under strong oscillatingmagnetic fieldsrdquo EPL vol 94 no 5 Article ID 50008 2011

[61] A Bruno-Alfonso C Bleasdale G V B de Souza and R ALewis ldquoClosed-orbit dependence on the field direction in theanisotropic diamagnetic Kepler problemrdquo Physical Review AmdashAtomic Molecular and Optical Physics vol 89 no 4 Article ID043425 2014

[62] J-K Xu H-Q Chen and H-P Liu ldquoExact quantum defecttheory approach for lithium in magnetic fieldsrdquo Chinese PhysicsB vol 22 no 1 Article ID 013204 2013

[63] M A Amdouni and H Eleuch ldquoInfluence of relativistic termsin the spectra of hydrogen atom highly excited in an externalstrong magnetic fieldrdquo Applied Mathematics amp InformationSciences vol 5 no 2 pp 307ndash320 2011

[64] J C Slater Quantum Theory of Atomic Structure vol 1-2McGraw-Hill New York NY USA 1996

[65] E U Condon and G H ShortleyTheTheory of Atomic SpectraCambridge University Press Cambridge UK 1953

[66] C C Tannoudji B Diu and F Laloe Quantum MechanicsWiley New York NY USA 1977 Mechanics Wiley

[67] P J Mohr B N Taylor and D B Newell ldquoCODATA recom-mended values of the fundamental physical constants 2006rdquoReviews of Modern Physics vol 80 no 2 pp 633ndash730 2008

[68] C W Clark and K T Taylor ldquoThe quadratic Zeeman effect inhydrogen Rydberg series application of Sturmian functionsrdquoJournal of Physics B Atomic and Molecular Physics vol 15 no8 pp 1175ndash1193 1982

[69] M A Iken F Borondo R M Benito and T Uzer ldquoPeriodic-orbit spectroscopy of the hydrogen atom in parallel electric andmagnetic fieldsrdquo Physical ReviewA vol 49 no 4 pp 2734ndash27471994

[70] A R Edmonds ldquoStudies of the quadratic Zeeman effect IApplication of the sturmian functionsrdquo Journal of Physics BAtomic and Molecular Physics vol 6 no 8 pp 1603ndash1615 1973

[71] K Sacha J Zakrzewski and D Delande ldquoBreaking timereversal symmetry in chaotic driven rydberg atomsrdquo Annals ofPhysics vol 283 no 1 pp 141ndash172 2000

[72] M Courtney Rydberg atoms in strong fields a testing groundfor quantum chaos [PhD thesis] Department of PhysicsMassachusetts Institute of Technology 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

are nonzero only for |119899 minus 1198991015840

| = 0 1 2 3 |119897 minus 1198971015840

| = 0 2 and atthese points take the following forms

⟨1198991015840

119897 + 2

10038161003816100381610038161003816119903210038161003816100381610038161003816119899119897⟩

=

minus

1

2

120577minus3

[(119899 minus 119897 minus 1) (119899 minus 119897 minus 2) (119899 minus 119897 minus 3)

times(119899 minus 119897 minus 4)(119899 minus 119897 minus 5)(119899 + 119897)]12

1198991015840

= 119899 minus 3

120577minus3

[(3119899 + 2119897) (119899 minus 119897 minus 1) (119899 minus 119897 minus 2)

times(119899 minus 119897 minus 3)(119899 minus 119897 minus 4)]12

1198991015840

= 119899 minus 2

minus

5

2

120577minus3

[(3119899 + 119897) (119899 minus 119897 minus 1) (119899 minus 119897 minus 2)

times(119899 minus 119897 minus 3)(119899 + 119897 + 1)]12

1198991015840

= 119899 minus 1

10119899120577minus3

[(119899 + 119897 + 1) (119899 + 119897 + 2)

times(119899 minus 119897 minus 1)(119899 minus 119897 minus 2)]12

1198991015840

= 119899

minus

5

2

120577minus3

(3119899 minus 119897) [(119899 + 119897 + 3) (119899 + 119897 + 2)

times(119899 + 119897 + 1)(119899 minus 119897 minus 1)]12

1198991015840

= 119899 + 1

120577minus3

(3119899 minus 2119897) [(119899 + 119897 + 4) (119899 + 119897 + 3)

times(119899 + 119897 + 2)(119899 + 119897 + 1)]12

1198991015840

= 119899 + 2

minus

1

2

120577minus3

[(119899 + 119897 + 5) (119899 + 119897 + 4) (119899 + 119897 + 3)

times(119899 + 119897 + 2)(119899 + 119897 + 1)(119899 minus 119897)]12

1198991015840

= 119899 + 3

⟨1198991015840

119897

10038161003816100381610038161003816119903210038161003816100381610038161003816119899119897⟩

=

2119899120577minus3

[51198992

minus 3119897 (119897 + 1) + 1] 1198991015840

= 119899

minus

3

2

120577minus3

[5119899 (119899 + 1) minus 119897 (119897 + 1) + 2]

times [(119899 + 119897 + 1) (119899 minus 119897)]12

1198991015840

= 119899 + 1

3120577

minus3

(119899 + 1) [(119899 + 119897 + 1) (119899 minus 119897)

times(119899 + 119897 + 2)(119899 minus 119897 + 1)]12

1198991015840

= 119899 + 2

minus

1

2

120577minus3

[(119899 + 119897 + 1) (119899 + 2 minus 119897) (119899 minus 119897)

times(119899 + 119897 + 3)(119899 + 119897 + 2)(119899 + 1 minus 119897)]12

1198991015840

= 119899 + 3

(27)

The investigation of Clark and Taylor [68] has revealedthat it is usually advisable to select higher values of 120577 Inour simulation we used 120577 = 2119899 in order to be close to theionization limit

Our numerical simulations consist in diagonalizing thematrix representing the operator in a Sturmian basis Thematrix elements of the generators are independent of 120572

32 Numerical Method Equation (20) is equivalent to thegeneral problem for eigenvalues of the form [59]

(119872 minus 120582119873) |Ψ⟩ = 0 (28)

where 119872 and 119873 are fixed matrices and 120582 and |Ψ⟩ are theeigenvalues and the eigenvectors searched

321 Simulation at Fixed Magnetic Field In the case wherewe fix the magnetic field 120574 = 119861119861

119888(120574 denotes the magnetic

field in units of 119861119888≃ 235 times 10

5

119879) after opting for 120572 = 1205720=

119862119905119890 we obtain an equation in the generalized eigenvalueswith

[59]

119872 = 119860(120572)

minus 1205720+

1205742

1205724

0

2

119862(120572)

119873 = 119861(120572)

120582 = minus21198641205722

0

(29)

that determines the energy levels

322 Simulation at Fixed Energy In this simulation wechoose 120572 = 120572

0= 119862119905119890 and we fix the energy So that we get

an equation in the generalized eigenvalues with [59]

119872 = 119860(120572)

minus 1205720+ (minus2119864120572

2

0) 119861(120572)

119873 = 119862(120572)

120582 =

1205742

1205724

0

2

(30)

that establishes the spectrum of magnetic field values relatedto the fixed energy

Both types of the simulations give the same outcomes

323 Diagonalization Algorithm and Convergence CriterionThe matrix components of the generators ( 119878

(120572)

119879(120572)) in a

Sturmian basis are restricted to some selectionsrsquo rules It iswell known that the matrix eigenvalue problems generally tobe solved are real symmetric matrices in strips [38]

Currently stable numerical algorithms have been devel-oped in order to calculate the eigenvalues and eigenvectors[48]

We provide an adoptable parameter 1205720to verify the

convergence for each type of the simulations In fact 1205720is

connected to the natural frequency of the basic oscillatorselected The accuracy of the examined values is not relianton this parameter Exact searched values are not dependenton this parameter

4 Numerical Results and Discussion

41 Energy Diagrams We use the simulation at fixed mag-netic field and we plot for a large number of points diagrams119864119903

119899= 119891(120574

2

) in Figures 1 2 and 3Figure 1 shows the diagram 119864(Hartree) = 119891(120574

2

) forRydberg states 119899 = 30 in the system with rather low field

6 Mathematical Problems in Engineering

minus003

minus004

minus005

minus006

minus007

minus008

minus009

E(H

artre

e)

02 04 06 08 10 12 14 16

1205742(times1010)

Figure 1 Energy levels as a function of the normalized squaremagnetic field 120574

2

(120574 = 119861119861119888 119861119888

= 235 times 105

119879) calculated by thesimulation technique described in Section 321 for weak magneticfield 119861 (119861 lt 3119879) and for the principle quantum number 119899 = 30The rovibration structure is completely removed (inter-119897-mixing)We observe that for a weak magnetic field the classical dynamicsis regular

minus005

minus010

minus015

minus020

E(H

artre

e)

1205742(times1010)

20 30 40 50

Figure 2 Diagram of energy levels of the hydrogen atom instrong magnetic field as function of 120574

2

(120574 = 119861119861119888 119861119888

= 235 times

105

119879) computed with a simulation described in Section 321 Thecalculation is done for a strong magnetic field 119861 (9119879 lt 119861 lt 18119879)

for both levels 119899 = 30 (dotted red line) and 119899 = 31 (solid blue line)Mixing between the energy levels of 119899 = 30 and 119899 = 31 is observed

(119861 lt 3119879) diamagnetism breaks the zero-field 119897-degeneracyof the hydrogen atom

We can explain this by the fact that in very weak field(119897-system intermixing) the degeneracy of the levels (therovibrational structure) proportional to 120574

2 is completelyremoved by the diamagnetic term It is visible that regularsystem disturbed by a weak perturbation remains essentiallyregular for the reason that the diamagnetic interaction is aperturbation of the Coulomb dynamics That is a regularregime there is no sensitivity on initial conditions

At highmagnetic field as shown by Figure 2 the differentenergy levels (119899 = 30 119899 = 31) essentially cross

Indeed for a highly excited state with principal quantumnumber 119899 the diamagnetic interaction scales as 120574

2

1198994 and

1205742(times1010)

20 30 40 50

minus005

minus010

minus015

minus020

E(H

artre

e)

Figure 3 Diagram of the energy levels of the hydrogen atom inthe strong magnetic field for 119899 = 30 computed by the simulationdescribed in Section 321 We note that for a high magnetic field theenergy levels anticrossThis is an indication of the chaos emergence

the zero-field as 11198992 The strong field regime is obtained

when [38]

1205741198993

≃ 1 that is 1198611198993

≃ 119861119888 (31)

This is achieved with 119899 ≃ 30 and 119861 ≃ 9119879 The rotationaland vibrational energy levels really cross By increasing themagnetic field in Figure 3 we observe that more statesanticross The classical smooth transition from regularity tochaos agrees well with this result

42The Effect of the Nucleus Recoil Energy In this section weuse the second simulation simulation at fixed energy wherewe have plotted 120574

2 as a function of the angular momentum 119897We introduced the recoil energy effect given in atomic unitby

119864119899(recoil) = minus

1

119872

(1 minus

3119898

119872

)

1205722

81198994 (32)

which takes into account the relativistic effects we recall thatthe levels shift due to this effect can be written as [68]

Δ119864119897=

1205722

21198993119897 (119897 + 1)

(33)

This recoil energy has been introduced with which wehave realized the simulation with nucleus of finite massThe result of the simulation is shown in Figure 4 wherethe distinction between the nucleus having a finite mass(diamond line) and the one having an infinite mass (circleline) is very clear This difference can be justified by the factthat the term (33) is the amount of three contributions Thefirst contribution is the kinetic energy in which the electronmass and not the reduced mass of the system manifests

119882119898V =

4

81198983

1198901198882 (34)

The second contribution is the Darwin term which takesinto account the electron delocalization but not the nucleus

ℏ2

81198982

119890119888241205871198902

120575 ( 119903) (35)

Mathematical Problems in Engineering 7

01280

01275

01270

01265

01260

01255

10 12 14 16 18 20 22 24 26 28

l

1205742(times1010)

Figure 4 The normalized square magnetic field 1205742

(120574 = 119861119861119888

119861119888

= 235 times 105

119879) for different angular momentum l with aprincipal quantum 119899 = 30 calculated by the simulation describedin Section 322 Two cases are computed first case realistic nucleusmass (finite mass) (red diamond line) second case nucleus infinitemass (blue circle line)

Finally the third contribution is the spin-orbit whichshows the interaction between the electronmagneticmomentand themagnetic field of the nucleus seen in the proper frameof the electron Therefore we should take into considerationthe nucleus speed since the nucleus of finite mass is being inmotion

5 Conclusion

We have studied the diamagnetic behavior of a hydrogenatom highly excited in a strong magnetic field consideringthe realistic finite mass of the nucleus The results revealthat when the magnetic field is weak the diamagnetic termremoves the degeneracy of the energy levels The system istotally regular it is possible to affect quantum numbers tothe various energy levels However with increasing magneticfield any symmetry will be destroyed and the energy levelswill be crossed there is an unruffled transition from regu-larity to chaotical dynamics At higher magnetic field thedynamic is totally chaotic and quantum spectrum includesonly large anticrossing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B Li and H P Liu ldquoAn effective quantum defect theory forthe diamagnetic spectrum of a barium Rydberg atomrdquo ChinesePhysics B vol 22 no 1 Article ID 013203 2013

[2] A E Shabad and V V Usov ldquoModified coulomb law in astrongly magnetized vacuumrdquo Physical Review Letters vol 98no 18 Article ID 180403 2007

[3] L Stevanovic and D Milojevic ldquoCompressed hydrogen atomunder debye screening in strong magnetic fieldrdquo Publications of

the Astronomical Observatory of Belgrade vol 91 pp 327ndash3302012

[4] H Jabri H Eleuch and T Djerad ldquoLifetimes of atomic Rydbergstates by autocorrelation functionrdquo Laser Physics Letters vol 2no 5 pp 253ndash257 2005

[5] H Eleuch and N Rachid ldquoAutocorrelation function ofmicrocavity-emitting field in the non-linear regimerdquoThe Euro-pean Physical Journal D vol 57 no 2 pp 259ndash264 2010

[6] H Eleuch N Ben Nessib and R Bennaceur ldquoQuantum modelof emission in a weakly non ideal plasmardquo The EuropeanPhysical Journal D vol 29 no 3 pp 391ndash395 2004

[7] P K Jha H Eleuch and Y V Rostovtsev ldquoCoherent control ofatomic excitation using off-resonant strong few-cycle pulsesrdquoPhysical Review A vol 82 no 4 Article ID 045805 2010

[8] E A Sete and H Eleuch ldquoStrong squeezing and robustentanglement in cavity electromechanicsrdquo Physical Review Avol 89 no 1 Article ID 013841 2014

[9] M Khatua U Sarkar and P K Chattaraj ldquoReactivity dynamicsof confined atoms in the presence of an external magnetic fieldrdquoThe European Physical Journal D vol 68 no 2 article 22 2014

[10] U Das and B Mukhopadhyay ldquoNew mass limit for whitedwarfs super-Chandrasekhar type Ia supernova as a newstandard candlerdquo Physical Review Letters vol 110 no 7 ArticleID 071102 2013

[11] A D Sargsyan R K Mirzoyan A S Sarkisyan A H Amiryanand D H Sarkisyan ldquoSplitting of N-type optical resonanceformed in Λ-system of 85Rb atoms in a strong transversemagnetic fieldrdquo Journal of Contemporary Physics vol 49 no 1pp 20ndash27 2014

[12] V S Popov and B M Karnakov ldquoHydrogen atom in a strongmagnetic fieldrdquo Physics-Uspekhi vol 57 no 3 pp 257ndash279 2014

[13] E Giacobino J-P Karr G Messin H Eleuch and A BaasldquoQuantum optical effects in semiconductor microcavitiesrdquoComptes Rendus Physique vol 3 no 1 pp 41ndash52 2002

[14] R Juarez-Amaro A Zuniga-Segundo F Soto-Eguibar andH M Moya-Cessa ldquoEquivalence between mirror-field-atomand ion-laser interactionsrdquo Applied Mathematics amp InformationSciences vol 7 no 4 pp 1311ndash1315 2013

[15] H Eleuch S Guerin and H R Jauslin ldquoEffects of an environ-ment on a cavity-quantum-electrodynamics system controlledby bichromatic adiabatic passagerdquoPhysical ReviewA vol 85 no1 Article ID 013830 2012

[16] E A Sete A A Svidzinsky Y V Rostovtsev et al ldquoUsingquantum coherence to generate gain in the xuv and x-raygain-swept superradiance and lasing without inversionrdquo IEEEJournal on Selected Topics in Quantum Electronics vol 18 no 1pp 541ndash553 2012

[17] E A Sete A A Svidzinsky H Eleuch Z Yang R D Nevelsand M O Scully ldquoCorrelated spontaneous emission on theDanuberdquo Journal of Modern Optics vol 57 no 14-15 pp 1311ndash1330 2010

[18] Z BWalters ldquoQuantum coherent dynamics at ambient temper-ature in photosynthetic moleculesrdquo Quantum Physics Lettersvol 1 no 1 pp 21ndash33 2012

[19] H Rabitz ldquoControl in the sciences over vast length and timescalesrdquo Quantum Physics Letters vol 1 no 1 pp 1ndash19 2012

[20] DA Barskiy KVKovtunov I VKoptyug et al ldquoThe feasibilityof formation and kinetics of NMR signal amplification byreversible exchange (SABRE) at high magnetic field (94 T)rdquoJournal of the American Chemical Society vol 136 no 9 pp3322ndash3325 2014

8 Mathematical Problems in Engineering

[21] B Mandal S Adhikari K Choudhary et al ldquoFemtosecondresponse of quantum discrete breathers in SRR based metama-terials and the role of dielectric permittivityrdquo Quantum PhysicsLetters vol 1 no 2 pp 59ndash68 2012

[22] J D Pritchard J A Isaacs M A Beck R McDermottand M Saffman ldquoHybrid atom-photon quantum gate in asuperconducting microwave resonatorrdquo Physical Review A vol89 no 1 Article ID 010301 2014

[23] P Rej and A Ghoshal ldquoRydberg transitions for positron-hydrogen collisions asymptotic cross section and scaling lawrdquoJournal of Physics B Atomic Molecular and Optical Physics vol47 no 1 Article ID 015204 2014

[24] H Eleuch D Elser and R Bennaceur ldquoSoliton propagation inan absorbing three-level atomic systemrdquo Laser Physics Lettersvol 1 no 8 pp 391ndash396 2004

[25] H Eleuch and R Bennaceur ldquoAn optical soliton pair amongabsorbing three-level atomsrdquo Journal of Optics A Pure andApplied Optics vol 5 no 5 pp 528ndash533 2003

[26] E Osman M Khalfallah and H Sapoor ldquoImproved (G1015840G)-expansion method for elliptic-like equation and general trav-eling wave solutions of some class of NLPDEsrdquo InformationScience Letters vol 2 no 2 pp 123ndash127 2013

[27] M Akbari ldquoExact solutions of the coupled Higgs equationand the Maccari system using the modified simplest equationmethodrdquo Information Science Letters vol 2 no 3 pp 155ndash1582013

[28] N Taghizadeh M Mirzazadeh M Akbari and M RahimianldquoExact soliton solutions for generalized equal width equationrdquoMathematical Science Letters vol 2 no 2 pp 99ndash106 2013

[29] E Osman M Khalfallah and H Sapoor ldquoFinding generalsolutions of nonlinear evolution equations by improved (G1015840G)-expansion methodrdquo Mathematical Science Letters vol 3 no 1pp 1ndash8 2014

[30] E M E Zayed ldquoExact traveling wave solutions for a variable-coefficient generalized dispersive water-wave system using thegeneralized (GrsquoG)-expansion methodrdquo Mathematical ScienceLetters vol 3 no 1 pp 9ndash15 2014

[31] E A Sete and H Eleuch ldquoInteraction of a quantum well withsqueezed light quantum-statistical propertiesrdquo Physical ReviewA vol 82 no 4 Article ID 043810 2010

[32] W Yang L Huang and M Tian ldquoTwo novel and secure quan-tum ANDOS schemesrdquo Applied Mathematics and InformationSciences vol 7 no 2 pp 659ndash665 2013

[33] H Eleuch ldquoPhoton statistics of light in semiconductor micro-cavitiesrdquo Journal of Physics B Atomic Molecular and OpticalPhysics vol 41 no 5 Article ID 055502 2008

[34] G Messin J P Karr H Eleuch J M Courty and E GiacobinoldquoSqueezed states and the quantumnoise of light in semiconduc-tor microcavitiesrdquo Journal of Physics Condensed Matter vol 11no 31 pp 6069ndash6078 1999

[35] A Baas J P Karr H Eleuch and E Giacobino ldquoOpticalbistability in semiconductor microcavitiesrdquo Physical Review AAtomic Molecular and Optical Physics vol 69 no 2 2004

[36] J Bonca and S Kruchinin Physical Properties of NanosystemsSpringer 2011

[37] L Tang and J-A Lu ldquoProjective lag synchronization of chaoticsystems with parameter mismatchrdquo Applied Mathematics ampInformation Sciences vol 7 no 1 pp 121ndash125 2013

[38] D Delande ldquoChaos and atoms in strong magnetic fieldsrdquoPhysica Scripta vol 1991 no T34 p 52 1991

[39] H Ruder G Wunner H Herold and F Geyer Atoms in StrongMagnetic Fields Springer Berlin Germany 1994

[40] H Eleuch and A Prasad ldquoChaos and regularity in semiconduc-tormicrocavitiesrdquoPhysics Letters A vol 376 no 26-27 pp 1970ndash1977 2012

[41] A Y Potekhin and A V Turbiner ldquoHydrogen atom in amagnetic field the quadrupole momentrdquo Physical Review AAtomic Molecular and Optical Physics vol 63 no 6 Article ID065402 2001

[42] S N Rasband Chaotic Dynamics of Non linear Systems JohnWiley amp Sons 1990

[43] A J Lichtenberg and M A Lieberman Regular and StochasticMotion Springer New York NY USA 1983

[44] A V Andreev O Agam B D Simons and B L AltshulerldquoQuantum chaos irreversible classical dynamics and randommatrix theoryrdquo Physical Review Letters vol 76 no 21 pp 3947ndash3950 1996

[45] O Bohigas ldquoLesHouches summer school session LIIrdquo inChaosand Quantum Physics M-J Giannoni A Voros and J Zinn-Justin Eds North-Holland Amsterdam The Netherlands1991

[46] T Pohl H R Sadeghpour and P Schmelcher ldquoCold andultracold Rydberg atoms in strong magnetic fieldsrdquo PhysicsReports vol 484 no 6 pp 181ndash229 2009

[47] T F Gallagher Rydberg Atoms Cambridge University PressCambridge UK 1994

[48] M I Eides H Grotch and V A Shelyuto Theory of LightHydrogenic Bound States vol 222 of Springer Tracts in ModernPhysics 2007

[49] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013

[50] C Yang S S Ge and T H Lee ldquoOutput feedback adaptivecontrol of a class of nonlinear discrete-time systems withunknown control directionsrdquoAutomatica vol 45 no 1 pp 270ndash276 2009

[51] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011

[52] C Yang S S Ge C Xiang T Chai and T H Lee ldquoOutputfeedback NN control for two classes of discrete-time systemswith unknown control directions in a unified approachrdquo IEEETransactions on Neural Networks vol 19 no 11 pp 1873ndash18862008

[53] Y-J Liu S-C Tong DWang T-S Li and C L P Chen ldquoAdap-tive neural output feedback controller design with reduced-order observer for a class of uncertain nonlinear SISO SystemsrdquoIEEE Transactions on Neural Networks vol 22 no 8 pp 1328ndash1334 2011

[54] T M Shahwan ldquoA comparison of Bayesian methods andartificial neural networks for forecasting chaotic financial timeseriesrdquo Journal of Statistics Applications amp Probability vol 1 no2 p 89 2012

[55] J W SeriesThe Spectrum of Atomic Hydrogen Oxford Univer-sity Press London UK 1957

[56] T van der Veldt W Vassen and W Hogervorst ldquoHeliumRydberg states in parallel electric and magnetic fieldsrdquo Journalof Physics B Atomic Molecular and Optical Physics vol 26 no13 pp 1945ndash1955 1993

Mathematical Problems in Engineering 9

[57] A Arimondo W D Phillips and F Strumia Laser Manip-ulation of Atoms and Ions North-Holland Amsterdam TheNetherlands 1992

[58] D Weiss K Richter A Menschig et al ldquoQuantized periodicorbits in large antidot arraysrdquo Physical Review Letters vol 70no 26 pp 4118ndash4121 1993

[59] D Delande Atomes de rydberg en champs statiques intenses[These drsquoEtat] Universite Paris 6 Paris France 1988

[60] M Sadhukhan and B M Deb ldquoA dynamical signature ofquantum chaos in hydrogen atom under strong oscillatingmagnetic fieldsrdquo EPL vol 94 no 5 Article ID 50008 2011

[61] A Bruno-Alfonso C Bleasdale G V B de Souza and R ALewis ldquoClosed-orbit dependence on the field direction in theanisotropic diamagnetic Kepler problemrdquo Physical Review AmdashAtomic Molecular and Optical Physics vol 89 no 4 Article ID043425 2014

[62] J-K Xu H-Q Chen and H-P Liu ldquoExact quantum defecttheory approach for lithium in magnetic fieldsrdquo Chinese PhysicsB vol 22 no 1 Article ID 013204 2013

[63] M A Amdouni and H Eleuch ldquoInfluence of relativistic termsin the spectra of hydrogen atom highly excited in an externalstrong magnetic fieldrdquo Applied Mathematics amp InformationSciences vol 5 no 2 pp 307ndash320 2011

[64] J C Slater Quantum Theory of Atomic Structure vol 1-2McGraw-Hill New York NY USA 1996

[65] E U Condon and G H ShortleyTheTheory of Atomic SpectraCambridge University Press Cambridge UK 1953

[66] C C Tannoudji B Diu and F Laloe Quantum MechanicsWiley New York NY USA 1977 Mechanics Wiley

[67] P J Mohr B N Taylor and D B Newell ldquoCODATA recom-mended values of the fundamental physical constants 2006rdquoReviews of Modern Physics vol 80 no 2 pp 633ndash730 2008

[68] C W Clark and K T Taylor ldquoThe quadratic Zeeman effect inhydrogen Rydberg series application of Sturmian functionsrdquoJournal of Physics B Atomic and Molecular Physics vol 15 no8 pp 1175ndash1193 1982

[69] M A Iken F Borondo R M Benito and T Uzer ldquoPeriodic-orbit spectroscopy of the hydrogen atom in parallel electric andmagnetic fieldsrdquo Physical ReviewA vol 49 no 4 pp 2734ndash27471994

[70] A R Edmonds ldquoStudies of the quadratic Zeeman effect IApplication of the sturmian functionsrdquo Journal of Physics BAtomic and Molecular Physics vol 6 no 8 pp 1603ndash1615 1973

[71] K Sacha J Zakrzewski and D Delande ldquoBreaking timereversal symmetry in chaotic driven rydberg atomsrdquo Annals ofPhysics vol 283 no 1 pp 141ndash172 2000

[72] M Courtney Rydberg atoms in strong fields a testing groundfor quantum chaos [PhD thesis] Department of PhysicsMassachusetts Institute of Technology 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

minus003

minus004

minus005

minus006

minus007

minus008

minus009

E(H

artre

e)

02 04 06 08 10 12 14 16

1205742(times1010)

Figure 1 Energy levels as a function of the normalized squaremagnetic field 120574

2

(120574 = 119861119861119888 119861119888

= 235 times 105

119879) calculated by thesimulation technique described in Section 321 for weak magneticfield 119861 (119861 lt 3119879) and for the principle quantum number 119899 = 30The rovibration structure is completely removed (inter-119897-mixing)We observe that for a weak magnetic field the classical dynamicsis regular

minus005

minus010

minus015

minus020

E(H

artre

e)

1205742(times1010)

20 30 40 50

Figure 2 Diagram of energy levels of the hydrogen atom instrong magnetic field as function of 120574

2

(120574 = 119861119861119888 119861119888

= 235 times

105

119879) computed with a simulation described in Section 321 Thecalculation is done for a strong magnetic field 119861 (9119879 lt 119861 lt 18119879)

for both levels 119899 = 30 (dotted red line) and 119899 = 31 (solid blue line)Mixing between the energy levels of 119899 = 30 and 119899 = 31 is observed

(119861 lt 3119879) diamagnetism breaks the zero-field 119897-degeneracyof the hydrogen atom

We can explain this by the fact that in very weak field(119897-system intermixing) the degeneracy of the levels (therovibrational structure) proportional to 120574

2 is completelyremoved by the diamagnetic term It is visible that regularsystem disturbed by a weak perturbation remains essentiallyregular for the reason that the diamagnetic interaction is aperturbation of the Coulomb dynamics That is a regularregime there is no sensitivity on initial conditions

At highmagnetic field as shown by Figure 2 the differentenergy levels (119899 = 30 119899 = 31) essentially cross

Indeed for a highly excited state with principal quantumnumber 119899 the diamagnetic interaction scales as 120574

2

1198994 and

1205742(times1010)

20 30 40 50

minus005

minus010

minus015

minus020

E(H

artre

e)

Figure 3 Diagram of the energy levels of the hydrogen atom inthe strong magnetic field for 119899 = 30 computed by the simulationdescribed in Section 321 We note that for a high magnetic field theenergy levels anticrossThis is an indication of the chaos emergence

the zero-field as 11198992 The strong field regime is obtained

when [38]

1205741198993

≃ 1 that is 1198611198993

≃ 119861119888 (31)

This is achieved with 119899 ≃ 30 and 119861 ≃ 9119879 The rotationaland vibrational energy levels really cross By increasing themagnetic field in Figure 3 we observe that more statesanticross The classical smooth transition from regularity tochaos agrees well with this result

42The Effect of the Nucleus Recoil Energy In this section weuse the second simulation simulation at fixed energy wherewe have plotted 120574

2 as a function of the angular momentum 119897We introduced the recoil energy effect given in atomic unitby

119864119899(recoil) = minus

1

119872

(1 minus

3119898

119872

)

1205722

81198994 (32)

which takes into account the relativistic effects we recall thatthe levels shift due to this effect can be written as [68]

Δ119864119897=

1205722

21198993119897 (119897 + 1)

(33)

This recoil energy has been introduced with which wehave realized the simulation with nucleus of finite massThe result of the simulation is shown in Figure 4 wherethe distinction between the nucleus having a finite mass(diamond line) and the one having an infinite mass (circleline) is very clear This difference can be justified by the factthat the term (33) is the amount of three contributions Thefirst contribution is the kinetic energy in which the electronmass and not the reduced mass of the system manifests

119882119898V =

4

81198983

1198901198882 (34)

The second contribution is the Darwin term which takesinto account the electron delocalization but not the nucleus

ℏ2

81198982

119890119888241205871198902

120575 ( 119903) (35)

Mathematical Problems in Engineering 7

01280

01275

01270

01265

01260

01255

10 12 14 16 18 20 22 24 26 28

l

1205742(times1010)

Figure 4 The normalized square magnetic field 1205742

(120574 = 119861119861119888

119861119888

= 235 times 105

119879) for different angular momentum l with aprincipal quantum 119899 = 30 calculated by the simulation describedin Section 322 Two cases are computed first case realistic nucleusmass (finite mass) (red diamond line) second case nucleus infinitemass (blue circle line)

Finally the third contribution is the spin-orbit whichshows the interaction between the electronmagneticmomentand themagnetic field of the nucleus seen in the proper frameof the electron Therefore we should take into considerationthe nucleus speed since the nucleus of finite mass is being inmotion

5 Conclusion

We have studied the diamagnetic behavior of a hydrogenatom highly excited in a strong magnetic field consideringthe realistic finite mass of the nucleus The results revealthat when the magnetic field is weak the diamagnetic termremoves the degeneracy of the energy levels The system istotally regular it is possible to affect quantum numbers tothe various energy levels However with increasing magneticfield any symmetry will be destroyed and the energy levelswill be crossed there is an unruffled transition from regu-larity to chaotical dynamics At higher magnetic field thedynamic is totally chaotic and quantum spectrum includesonly large anticrossing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B Li and H P Liu ldquoAn effective quantum defect theory forthe diamagnetic spectrum of a barium Rydberg atomrdquo ChinesePhysics B vol 22 no 1 Article ID 013203 2013

[2] A E Shabad and V V Usov ldquoModified coulomb law in astrongly magnetized vacuumrdquo Physical Review Letters vol 98no 18 Article ID 180403 2007

[3] L Stevanovic and D Milojevic ldquoCompressed hydrogen atomunder debye screening in strong magnetic fieldrdquo Publications of

the Astronomical Observatory of Belgrade vol 91 pp 327ndash3302012

[4] H Jabri H Eleuch and T Djerad ldquoLifetimes of atomic Rydbergstates by autocorrelation functionrdquo Laser Physics Letters vol 2no 5 pp 253ndash257 2005

[5] H Eleuch and N Rachid ldquoAutocorrelation function ofmicrocavity-emitting field in the non-linear regimerdquoThe Euro-pean Physical Journal D vol 57 no 2 pp 259ndash264 2010

[6] H Eleuch N Ben Nessib and R Bennaceur ldquoQuantum modelof emission in a weakly non ideal plasmardquo The EuropeanPhysical Journal D vol 29 no 3 pp 391ndash395 2004

[7] P K Jha H Eleuch and Y V Rostovtsev ldquoCoherent control ofatomic excitation using off-resonant strong few-cycle pulsesrdquoPhysical Review A vol 82 no 4 Article ID 045805 2010

[8] E A Sete and H Eleuch ldquoStrong squeezing and robustentanglement in cavity electromechanicsrdquo Physical Review Avol 89 no 1 Article ID 013841 2014

[9] M Khatua U Sarkar and P K Chattaraj ldquoReactivity dynamicsof confined atoms in the presence of an external magnetic fieldrdquoThe European Physical Journal D vol 68 no 2 article 22 2014

[10] U Das and B Mukhopadhyay ldquoNew mass limit for whitedwarfs super-Chandrasekhar type Ia supernova as a newstandard candlerdquo Physical Review Letters vol 110 no 7 ArticleID 071102 2013

[11] A D Sargsyan R K Mirzoyan A S Sarkisyan A H Amiryanand D H Sarkisyan ldquoSplitting of N-type optical resonanceformed in Λ-system of 85Rb atoms in a strong transversemagnetic fieldrdquo Journal of Contemporary Physics vol 49 no 1pp 20ndash27 2014

[12] V S Popov and B M Karnakov ldquoHydrogen atom in a strongmagnetic fieldrdquo Physics-Uspekhi vol 57 no 3 pp 257ndash279 2014

[13] E Giacobino J-P Karr G Messin H Eleuch and A BaasldquoQuantum optical effects in semiconductor microcavitiesrdquoComptes Rendus Physique vol 3 no 1 pp 41ndash52 2002

[14] R Juarez-Amaro A Zuniga-Segundo F Soto-Eguibar andH M Moya-Cessa ldquoEquivalence between mirror-field-atomand ion-laser interactionsrdquo Applied Mathematics amp InformationSciences vol 7 no 4 pp 1311ndash1315 2013

[15] H Eleuch S Guerin and H R Jauslin ldquoEffects of an environ-ment on a cavity-quantum-electrodynamics system controlledby bichromatic adiabatic passagerdquoPhysical ReviewA vol 85 no1 Article ID 013830 2012

[16] E A Sete A A Svidzinsky Y V Rostovtsev et al ldquoUsingquantum coherence to generate gain in the xuv and x-raygain-swept superradiance and lasing without inversionrdquo IEEEJournal on Selected Topics in Quantum Electronics vol 18 no 1pp 541ndash553 2012

[17] E A Sete A A Svidzinsky H Eleuch Z Yang R D Nevelsand M O Scully ldquoCorrelated spontaneous emission on theDanuberdquo Journal of Modern Optics vol 57 no 14-15 pp 1311ndash1330 2010

[18] Z BWalters ldquoQuantum coherent dynamics at ambient temper-ature in photosynthetic moleculesrdquo Quantum Physics Lettersvol 1 no 1 pp 21ndash33 2012

[19] H Rabitz ldquoControl in the sciences over vast length and timescalesrdquo Quantum Physics Letters vol 1 no 1 pp 1ndash19 2012

[20] DA Barskiy KVKovtunov I VKoptyug et al ldquoThe feasibilityof formation and kinetics of NMR signal amplification byreversible exchange (SABRE) at high magnetic field (94 T)rdquoJournal of the American Chemical Society vol 136 no 9 pp3322ndash3325 2014

8 Mathematical Problems in Engineering

[21] B Mandal S Adhikari K Choudhary et al ldquoFemtosecondresponse of quantum discrete breathers in SRR based metama-terials and the role of dielectric permittivityrdquo Quantum PhysicsLetters vol 1 no 2 pp 59ndash68 2012

[22] J D Pritchard J A Isaacs M A Beck R McDermottand M Saffman ldquoHybrid atom-photon quantum gate in asuperconducting microwave resonatorrdquo Physical Review A vol89 no 1 Article ID 010301 2014

[23] P Rej and A Ghoshal ldquoRydberg transitions for positron-hydrogen collisions asymptotic cross section and scaling lawrdquoJournal of Physics B Atomic Molecular and Optical Physics vol47 no 1 Article ID 015204 2014

[24] H Eleuch D Elser and R Bennaceur ldquoSoliton propagation inan absorbing three-level atomic systemrdquo Laser Physics Lettersvol 1 no 8 pp 391ndash396 2004

[25] H Eleuch and R Bennaceur ldquoAn optical soliton pair amongabsorbing three-level atomsrdquo Journal of Optics A Pure andApplied Optics vol 5 no 5 pp 528ndash533 2003

[26] E Osman M Khalfallah and H Sapoor ldquoImproved (G1015840G)-expansion method for elliptic-like equation and general trav-eling wave solutions of some class of NLPDEsrdquo InformationScience Letters vol 2 no 2 pp 123ndash127 2013

[27] M Akbari ldquoExact solutions of the coupled Higgs equationand the Maccari system using the modified simplest equationmethodrdquo Information Science Letters vol 2 no 3 pp 155ndash1582013

[28] N Taghizadeh M Mirzazadeh M Akbari and M RahimianldquoExact soliton solutions for generalized equal width equationrdquoMathematical Science Letters vol 2 no 2 pp 99ndash106 2013

[29] E Osman M Khalfallah and H Sapoor ldquoFinding generalsolutions of nonlinear evolution equations by improved (G1015840G)-expansion methodrdquo Mathematical Science Letters vol 3 no 1pp 1ndash8 2014

[30] E M E Zayed ldquoExact traveling wave solutions for a variable-coefficient generalized dispersive water-wave system using thegeneralized (GrsquoG)-expansion methodrdquo Mathematical ScienceLetters vol 3 no 1 pp 9ndash15 2014

[31] E A Sete and H Eleuch ldquoInteraction of a quantum well withsqueezed light quantum-statistical propertiesrdquo Physical ReviewA vol 82 no 4 Article ID 043810 2010

[32] W Yang L Huang and M Tian ldquoTwo novel and secure quan-tum ANDOS schemesrdquo Applied Mathematics and InformationSciences vol 7 no 2 pp 659ndash665 2013

[33] H Eleuch ldquoPhoton statistics of light in semiconductor micro-cavitiesrdquo Journal of Physics B Atomic Molecular and OpticalPhysics vol 41 no 5 Article ID 055502 2008

[34] G Messin J P Karr H Eleuch J M Courty and E GiacobinoldquoSqueezed states and the quantumnoise of light in semiconduc-tor microcavitiesrdquo Journal of Physics Condensed Matter vol 11no 31 pp 6069ndash6078 1999

[35] A Baas J P Karr H Eleuch and E Giacobino ldquoOpticalbistability in semiconductor microcavitiesrdquo Physical Review AAtomic Molecular and Optical Physics vol 69 no 2 2004

[36] J Bonca and S Kruchinin Physical Properties of NanosystemsSpringer 2011

[37] L Tang and J-A Lu ldquoProjective lag synchronization of chaoticsystems with parameter mismatchrdquo Applied Mathematics ampInformation Sciences vol 7 no 1 pp 121ndash125 2013

[38] D Delande ldquoChaos and atoms in strong magnetic fieldsrdquoPhysica Scripta vol 1991 no T34 p 52 1991

[39] H Ruder G Wunner H Herold and F Geyer Atoms in StrongMagnetic Fields Springer Berlin Germany 1994

[40] H Eleuch and A Prasad ldquoChaos and regularity in semiconduc-tormicrocavitiesrdquoPhysics Letters A vol 376 no 26-27 pp 1970ndash1977 2012

[41] A Y Potekhin and A V Turbiner ldquoHydrogen atom in amagnetic field the quadrupole momentrdquo Physical Review AAtomic Molecular and Optical Physics vol 63 no 6 Article ID065402 2001

[42] S N Rasband Chaotic Dynamics of Non linear Systems JohnWiley amp Sons 1990

[43] A J Lichtenberg and M A Lieberman Regular and StochasticMotion Springer New York NY USA 1983

[44] A V Andreev O Agam B D Simons and B L AltshulerldquoQuantum chaos irreversible classical dynamics and randommatrix theoryrdquo Physical Review Letters vol 76 no 21 pp 3947ndash3950 1996

[45] O Bohigas ldquoLesHouches summer school session LIIrdquo inChaosand Quantum Physics M-J Giannoni A Voros and J Zinn-Justin Eds North-Holland Amsterdam The Netherlands1991

[46] T Pohl H R Sadeghpour and P Schmelcher ldquoCold andultracold Rydberg atoms in strong magnetic fieldsrdquo PhysicsReports vol 484 no 6 pp 181ndash229 2009

[47] T F Gallagher Rydberg Atoms Cambridge University PressCambridge UK 1994

[48] M I Eides H Grotch and V A Shelyuto Theory of LightHydrogenic Bound States vol 222 of Springer Tracts in ModernPhysics 2007

[49] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013

[50] C Yang S S Ge and T H Lee ldquoOutput feedback adaptivecontrol of a class of nonlinear discrete-time systems withunknown control directionsrdquoAutomatica vol 45 no 1 pp 270ndash276 2009

[51] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011

[52] C Yang S S Ge C Xiang T Chai and T H Lee ldquoOutputfeedback NN control for two classes of discrete-time systemswith unknown control directions in a unified approachrdquo IEEETransactions on Neural Networks vol 19 no 11 pp 1873ndash18862008

[53] Y-J Liu S-C Tong DWang T-S Li and C L P Chen ldquoAdap-tive neural output feedback controller design with reduced-order observer for a class of uncertain nonlinear SISO SystemsrdquoIEEE Transactions on Neural Networks vol 22 no 8 pp 1328ndash1334 2011

[54] T M Shahwan ldquoA comparison of Bayesian methods andartificial neural networks for forecasting chaotic financial timeseriesrdquo Journal of Statistics Applications amp Probability vol 1 no2 p 89 2012

[55] J W SeriesThe Spectrum of Atomic Hydrogen Oxford Univer-sity Press London UK 1957

[56] T van der Veldt W Vassen and W Hogervorst ldquoHeliumRydberg states in parallel electric and magnetic fieldsrdquo Journalof Physics B Atomic Molecular and Optical Physics vol 26 no13 pp 1945ndash1955 1993

Mathematical Problems in Engineering 9

[57] A Arimondo W D Phillips and F Strumia Laser Manip-ulation of Atoms and Ions North-Holland Amsterdam TheNetherlands 1992

[58] D Weiss K Richter A Menschig et al ldquoQuantized periodicorbits in large antidot arraysrdquo Physical Review Letters vol 70no 26 pp 4118ndash4121 1993

[59] D Delande Atomes de rydberg en champs statiques intenses[These drsquoEtat] Universite Paris 6 Paris France 1988

[60] M Sadhukhan and B M Deb ldquoA dynamical signature ofquantum chaos in hydrogen atom under strong oscillatingmagnetic fieldsrdquo EPL vol 94 no 5 Article ID 50008 2011

[61] A Bruno-Alfonso C Bleasdale G V B de Souza and R ALewis ldquoClosed-orbit dependence on the field direction in theanisotropic diamagnetic Kepler problemrdquo Physical Review AmdashAtomic Molecular and Optical Physics vol 89 no 4 Article ID043425 2014

[62] J-K Xu H-Q Chen and H-P Liu ldquoExact quantum defecttheory approach for lithium in magnetic fieldsrdquo Chinese PhysicsB vol 22 no 1 Article ID 013204 2013

[63] M A Amdouni and H Eleuch ldquoInfluence of relativistic termsin the spectra of hydrogen atom highly excited in an externalstrong magnetic fieldrdquo Applied Mathematics amp InformationSciences vol 5 no 2 pp 307ndash320 2011

[64] J C Slater Quantum Theory of Atomic Structure vol 1-2McGraw-Hill New York NY USA 1996

[65] E U Condon and G H ShortleyTheTheory of Atomic SpectraCambridge University Press Cambridge UK 1953

[66] C C Tannoudji B Diu and F Laloe Quantum MechanicsWiley New York NY USA 1977 Mechanics Wiley

[67] P J Mohr B N Taylor and D B Newell ldquoCODATA recom-mended values of the fundamental physical constants 2006rdquoReviews of Modern Physics vol 80 no 2 pp 633ndash730 2008

[68] C W Clark and K T Taylor ldquoThe quadratic Zeeman effect inhydrogen Rydberg series application of Sturmian functionsrdquoJournal of Physics B Atomic and Molecular Physics vol 15 no8 pp 1175ndash1193 1982

[69] M A Iken F Borondo R M Benito and T Uzer ldquoPeriodic-orbit spectroscopy of the hydrogen atom in parallel electric andmagnetic fieldsrdquo Physical ReviewA vol 49 no 4 pp 2734ndash27471994

[70] A R Edmonds ldquoStudies of the quadratic Zeeman effect IApplication of the sturmian functionsrdquo Journal of Physics BAtomic and Molecular Physics vol 6 no 8 pp 1603ndash1615 1973

[71] K Sacha J Zakrzewski and D Delande ldquoBreaking timereversal symmetry in chaotic driven rydberg atomsrdquo Annals ofPhysics vol 283 no 1 pp 141ndash172 2000

[72] M Courtney Rydberg atoms in strong fields a testing groundfor quantum chaos [PhD thesis] Department of PhysicsMassachusetts Institute of Technology 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

01280

01275

01270

01265

01260

01255

10 12 14 16 18 20 22 24 26 28

l

1205742(times1010)

Figure 4 The normalized square magnetic field 1205742

(120574 = 119861119861119888

119861119888

= 235 times 105

119879) for different angular momentum l with aprincipal quantum 119899 = 30 calculated by the simulation describedin Section 322 Two cases are computed first case realistic nucleusmass (finite mass) (red diamond line) second case nucleus infinitemass (blue circle line)

Finally the third contribution is the spin-orbit whichshows the interaction between the electronmagneticmomentand themagnetic field of the nucleus seen in the proper frameof the electron Therefore we should take into considerationthe nucleus speed since the nucleus of finite mass is being inmotion

5 Conclusion

We have studied the diamagnetic behavior of a hydrogenatom highly excited in a strong magnetic field consideringthe realistic finite mass of the nucleus The results revealthat when the magnetic field is weak the diamagnetic termremoves the degeneracy of the energy levels The system istotally regular it is possible to affect quantum numbers tothe various energy levels However with increasing magneticfield any symmetry will be destroyed and the energy levelswill be crossed there is an unruffled transition from regu-larity to chaotical dynamics At higher magnetic field thedynamic is totally chaotic and quantum spectrum includesonly large anticrossing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B Li and H P Liu ldquoAn effective quantum defect theory forthe diamagnetic spectrum of a barium Rydberg atomrdquo ChinesePhysics B vol 22 no 1 Article ID 013203 2013

[2] A E Shabad and V V Usov ldquoModified coulomb law in astrongly magnetized vacuumrdquo Physical Review Letters vol 98no 18 Article ID 180403 2007

[3] L Stevanovic and D Milojevic ldquoCompressed hydrogen atomunder debye screening in strong magnetic fieldrdquo Publications of

the Astronomical Observatory of Belgrade vol 91 pp 327ndash3302012

[4] H Jabri H Eleuch and T Djerad ldquoLifetimes of atomic Rydbergstates by autocorrelation functionrdquo Laser Physics Letters vol 2no 5 pp 253ndash257 2005

[5] H Eleuch and N Rachid ldquoAutocorrelation function ofmicrocavity-emitting field in the non-linear regimerdquoThe Euro-pean Physical Journal D vol 57 no 2 pp 259ndash264 2010

[6] H Eleuch N Ben Nessib and R Bennaceur ldquoQuantum modelof emission in a weakly non ideal plasmardquo The EuropeanPhysical Journal D vol 29 no 3 pp 391ndash395 2004

[7] P K Jha H Eleuch and Y V Rostovtsev ldquoCoherent control ofatomic excitation using off-resonant strong few-cycle pulsesrdquoPhysical Review A vol 82 no 4 Article ID 045805 2010

[8] E A Sete and H Eleuch ldquoStrong squeezing and robustentanglement in cavity electromechanicsrdquo Physical Review Avol 89 no 1 Article ID 013841 2014

[9] M Khatua U Sarkar and P K Chattaraj ldquoReactivity dynamicsof confined atoms in the presence of an external magnetic fieldrdquoThe European Physical Journal D vol 68 no 2 article 22 2014

[10] U Das and B Mukhopadhyay ldquoNew mass limit for whitedwarfs super-Chandrasekhar type Ia supernova as a newstandard candlerdquo Physical Review Letters vol 110 no 7 ArticleID 071102 2013

[11] A D Sargsyan R K Mirzoyan A S Sarkisyan A H Amiryanand D H Sarkisyan ldquoSplitting of N-type optical resonanceformed in Λ-system of 85Rb atoms in a strong transversemagnetic fieldrdquo Journal of Contemporary Physics vol 49 no 1pp 20ndash27 2014

[12] V S Popov and B M Karnakov ldquoHydrogen atom in a strongmagnetic fieldrdquo Physics-Uspekhi vol 57 no 3 pp 257ndash279 2014

[13] E Giacobino J-P Karr G Messin H Eleuch and A BaasldquoQuantum optical effects in semiconductor microcavitiesrdquoComptes Rendus Physique vol 3 no 1 pp 41ndash52 2002

[14] R Juarez-Amaro A Zuniga-Segundo F Soto-Eguibar andH M Moya-Cessa ldquoEquivalence between mirror-field-atomand ion-laser interactionsrdquo Applied Mathematics amp InformationSciences vol 7 no 4 pp 1311ndash1315 2013

[15] H Eleuch S Guerin and H R Jauslin ldquoEffects of an environ-ment on a cavity-quantum-electrodynamics system controlledby bichromatic adiabatic passagerdquoPhysical ReviewA vol 85 no1 Article ID 013830 2012

[16] E A Sete A A Svidzinsky Y V Rostovtsev et al ldquoUsingquantum coherence to generate gain in the xuv and x-raygain-swept superradiance and lasing without inversionrdquo IEEEJournal on Selected Topics in Quantum Electronics vol 18 no 1pp 541ndash553 2012

[17] E A Sete A A Svidzinsky H Eleuch Z Yang R D Nevelsand M O Scully ldquoCorrelated spontaneous emission on theDanuberdquo Journal of Modern Optics vol 57 no 14-15 pp 1311ndash1330 2010

[18] Z BWalters ldquoQuantum coherent dynamics at ambient temper-ature in photosynthetic moleculesrdquo Quantum Physics Lettersvol 1 no 1 pp 21ndash33 2012

[19] H Rabitz ldquoControl in the sciences over vast length and timescalesrdquo Quantum Physics Letters vol 1 no 1 pp 1ndash19 2012

[20] DA Barskiy KVKovtunov I VKoptyug et al ldquoThe feasibilityof formation and kinetics of NMR signal amplification byreversible exchange (SABRE) at high magnetic field (94 T)rdquoJournal of the American Chemical Society vol 136 no 9 pp3322ndash3325 2014

8 Mathematical Problems in Engineering

[21] B Mandal S Adhikari K Choudhary et al ldquoFemtosecondresponse of quantum discrete breathers in SRR based metama-terials and the role of dielectric permittivityrdquo Quantum PhysicsLetters vol 1 no 2 pp 59ndash68 2012

[22] J D Pritchard J A Isaacs M A Beck R McDermottand M Saffman ldquoHybrid atom-photon quantum gate in asuperconducting microwave resonatorrdquo Physical Review A vol89 no 1 Article ID 010301 2014

[23] P Rej and A Ghoshal ldquoRydberg transitions for positron-hydrogen collisions asymptotic cross section and scaling lawrdquoJournal of Physics B Atomic Molecular and Optical Physics vol47 no 1 Article ID 015204 2014

[24] H Eleuch D Elser and R Bennaceur ldquoSoliton propagation inan absorbing three-level atomic systemrdquo Laser Physics Lettersvol 1 no 8 pp 391ndash396 2004

[25] H Eleuch and R Bennaceur ldquoAn optical soliton pair amongabsorbing three-level atomsrdquo Journal of Optics A Pure andApplied Optics vol 5 no 5 pp 528ndash533 2003

[26] E Osman M Khalfallah and H Sapoor ldquoImproved (G1015840G)-expansion method for elliptic-like equation and general trav-eling wave solutions of some class of NLPDEsrdquo InformationScience Letters vol 2 no 2 pp 123ndash127 2013

[27] M Akbari ldquoExact solutions of the coupled Higgs equationand the Maccari system using the modified simplest equationmethodrdquo Information Science Letters vol 2 no 3 pp 155ndash1582013

[28] N Taghizadeh M Mirzazadeh M Akbari and M RahimianldquoExact soliton solutions for generalized equal width equationrdquoMathematical Science Letters vol 2 no 2 pp 99ndash106 2013

[29] E Osman M Khalfallah and H Sapoor ldquoFinding generalsolutions of nonlinear evolution equations by improved (G1015840G)-expansion methodrdquo Mathematical Science Letters vol 3 no 1pp 1ndash8 2014

[30] E M E Zayed ldquoExact traveling wave solutions for a variable-coefficient generalized dispersive water-wave system using thegeneralized (GrsquoG)-expansion methodrdquo Mathematical ScienceLetters vol 3 no 1 pp 9ndash15 2014

[31] E A Sete and H Eleuch ldquoInteraction of a quantum well withsqueezed light quantum-statistical propertiesrdquo Physical ReviewA vol 82 no 4 Article ID 043810 2010

[32] W Yang L Huang and M Tian ldquoTwo novel and secure quan-tum ANDOS schemesrdquo Applied Mathematics and InformationSciences vol 7 no 2 pp 659ndash665 2013

[33] H Eleuch ldquoPhoton statistics of light in semiconductor micro-cavitiesrdquo Journal of Physics B Atomic Molecular and OpticalPhysics vol 41 no 5 Article ID 055502 2008

[34] G Messin J P Karr H Eleuch J M Courty and E GiacobinoldquoSqueezed states and the quantumnoise of light in semiconduc-tor microcavitiesrdquo Journal of Physics Condensed Matter vol 11no 31 pp 6069ndash6078 1999

[35] A Baas J P Karr H Eleuch and E Giacobino ldquoOpticalbistability in semiconductor microcavitiesrdquo Physical Review AAtomic Molecular and Optical Physics vol 69 no 2 2004

[36] J Bonca and S Kruchinin Physical Properties of NanosystemsSpringer 2011

[37] L Tang and J-A Lu ldquoProjective lag synchronization of chaoticsystems with parameter mismatchrdquo Applied Mathematics ampInformation Sciences vol 7 no 1 pp 121ndash125 2013

[38] D Delande ldquoChaos and atoms in strong magnetic fieldsrdquoPhysica Scripta vol 1991 no T34 p 52 1991

[39] H Ruder G Wunner H Herold and F Geyer Atoms in StrongMagnetic Fields Springer Berlin Germany 1994

[40] H Eleuch and A Prasad ldquoChaos and regularity in semiconduc-tormicrocavitiesrdquoPhysics Letters A vol 376 no 26-27 pp 1970ndash1977 2012

[41] A Y Potekhin and A V Turbiner ldquoHydrogen atom in amagnetic field the quadrupole momentrdquo Physical Review AAtomic Molecular and Optical Physics vol 63 no 6 Article ID065402 2001

[42] S N Rasband Chaotic Dynamics of Non linear Systems JohnWiley amp Sons 1990

[43] A J Lichtenberg and M A Lieberman Regular and StochasticMotion Springer New York NY USA 1983

[44] A V Andreev O Agam B D Simons and B L AltshulerldquoQuantum chaos irreversible classical dynamics and randommatrix theoryrdquo Physical Review Letters vol 76 no 21 pp 3947ndash3950 1996

[45] O Bohigas ldquoLesHouches summer school session LIIrdquo inChaosand Quantum Physics M-J Giannoni A Voros and J Zinn-Justin Eds North-Holland Amsterdam The Netherlands1991

[46] T Pohl H R Sadeghpour and P Schmelcher ldquoCold andultracold Rydberg atoms in strong magnetic fieldsrdquo PhysicsReports vol 484 no 6 pp 181ndash229 2009

[47] T F Gallagher Rydberg Atoms Cambridge University PressCambridge UK 1994

[48] M I Eides H Grotch and V A Shelyuto Theory of LightHydrogenic Bound States vol 222 of Springer Tracts in ModernPhysics 2007

[49] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013

[50] C Yang S S Ge and T H Lee ldquoOutput feedback adaptivecontrol of a class of nonlinear discrete-time systems withunknown control directionsrdquoAutomatica vol 45 no 1 pp 270ndash276 2009

[51] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011

[52] C Yang S S Ge C Xiang T Chai and T H Lee ldquoOutputfeedback NN control for two classes of discrete-time systemswith unknown control directions in a unified approachrdquo IEEETransactions on Neural Networks vol 19 no 11 pp 1873ndash18862008

[53] Y-J Liu S-C Tong DWang T-S Li and C L P Chen ldquoAdap-tive neural output feedback controller design with reduced-order observer for a class of uncertain nonlinear SISO SystemsrdquoIEEE Transactions on Neural Networks vol 22 no 8 pp 1328ndash1334 2011

[54] T M Shahwan ldquoA comparison of Bayesian methods andartificial neural networks for forecasting chaotic financial timeseriesrdquo Journal of Statistics Applications amp Probability vol 1 no2 p 89 2012

[55] J W SeriesThe Spectrum of Atomic Hydrogen Oxford Univer-sity Press London UK 1957

[56] T van der Veldt W Vassen and W Hogervorst ldquoHeliumRydberg states in parallel electric and magnetic fieldsrdquo Journalof Physics B Atomic Molecular and Optical Physics vol 26 no13 pp 1945ndash1955 1993

Mathematical Problems in Engineering 9

[57] A Arimondo W D Phillips and F Strumia Laser Manip-ulation of Atoms and Ions North-Holland Amsterdam TheNetherlands 1992

[58] D Weiss K Richter A Menschig et al ldquoQuantized periodicorbits in large antidot arraysrdquo Physical Review Letters vol 70no 26 pp 4118ndash4121 1993

[59] D Delande Atomes de rydberg en champs statiques intenses[These drsquoEtat] Universite Paris 6 Paris France 1988

[60] M Sadhukhan and B M Deb ldquoA dynamical signature ofquantum chaos in hydrogen atom under strong oscillatingmagnetic fieldsrdquo EPL vol 94 no 5 Article ID 50008 2011

[61] A Bruno-Alfonso C Bleasdale G V B de Souza and R ALewis ldquoClosed-orbit dependence on the field direction in theanisotropic diamagnetic Kepler problemrdquo Physical Review AmdashAtomic Molecular and Optical Physics vol 89 no 4 Article ID043425 2014

[62] J-K Xu H-Q Chen and H-P Liu ldquoExact quantum defecttheory approach for lithium in magnetic fieldsrdquo Chinese PhysicsB vol 22 no 1 Article ID 013204 2013

[63] M A Amdouni and H Eleuch ldquoInfluence of relativistic termsin the spectra of hydrogen atom highly excited in an externalstrong magnetic fieldrdquo Applied Mathematics amp InformationSciences vol 5 no 2 pp 307ndash320 2011

[64] J C Slater Quantum Theory of Atomic Structure vol 1-2McGraw-Hill New York NY USA 1996

[65] E U Condon and G H ShortleyTheTheory of Atomic SpectraCambridge University Press Cambridge UK 1953

[66] C C Tannoudji B Diu and F Laloe Quantum MechanicsWiley New York NY USA 1977 Mechanics Wiley

[67] P J Mohr B N Taylor and D B Newell ldquoCODATA recom-mended values of the fundamental physical constants 2006rdquoReviews of Modern Physics vol 80 no 2 pp 633ndash730 2008

[68] C W Clark and K T Taylor ldquoThe quadratic Zeeman effect inhydrogen Rydberg series application of Sturmian functionsrdquoJournal of Physics B Atomic and Molecular Physics vol 15 no8 pp 1175ndash1193 1982

[69] M A Iken F Borondo R M Benito and T Uzer ldquoPeriodic-orbit spectroscopy of the hydrogen atom in parallel electric andmagnetic fieldsrdquo Physical ReviewA vol 49 no 4 pp 2734ndash27471994

[70] A R Edmonds ldquoStudies of the quadratic Zeeman effect IApplication of the sturmian functionsrdquo Journal of Physics BAtomic and Molecular Physics vol 6 no 8 pp 1603ndash1615 1973

[71] K Sacha J Zakrzewski and D Delande ldquoBreaking timereversal symmetry in chaotic driven rydberg atomsrdquo Annals ofPhysics vol 283 no 1 pp 141ndash172 2000

[72] M Courtney Rydberg atoms in strong fields a testing groundfor quantum chaos [PhD thesis] Department of PhysicsMassachusetts Institute of Technology 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

[21] B Mandal S Adhikari K Choudhary et al ldquoFemtosecondresponse of quantum discrete breathers in SRR based metama-terials and the role of dielectric permittivityrdquo Quantum PhysicsLetters vol 1 no 2 pp 59ndash68 2012

[22] J D Pritchard J A Isaacs M A Beck R McDermottand M Saffman ldquoHybrid atom-photon quantum gate in asuperconducting microwave resonatorrdquo Physical Review A vol89 no 1 Article ID 010301 2014

[23] P Rej and A Ghoshal ldquoRydberg transitions for positron-hydrogen collisions asymptotic cross section and scaling lawrdquoJournal of Physics B Atomic Molecular and Optical Physics vol47 no 1 Article ID 015204 2014

[24] H Eleuch D Elser and R Bennaceur ldquoSoliton propagation inan absorbing three-level atomic systemrdquo Laser Physics Lettersvol 1 no 8 pp 391ndash396 2004

[25] H Eleuch and R Bennaceur ldquoAn optical soliton pair amongabsorbing three-level atomsrdquo Journal of Optics A Pure andApplied Optics vol 5 no 5 pp 528ndash533 2003

[26] E Osman M Khalfallah and H Sapoor ldquoImproved (G1015840G)-expansion method for elliptic-like equation and general trav-eling wave solutions of some class of NLPDEsrdquo InformationScience Letters vol 2 no 2 pp 123ndash127 2013

[27] M Akbari ldquoExact solutions of the coupled Higgs equationand the Maccari system using the modified simplest equationmethodrdquo Information Science Letters vol 2 no 3 pp 155ndash1582013

[28] N Taghizadeh M Mirzazadeh M Akbari and M RahimianldquoExact soliton solutions for generalized equal width equationrdquoMathematical Science Letters vol 2 no 2 pp 99ndash106 2013

[29] E Osman M Khalfallah and H Sapoor ldquoFinding generalsolutions of nonlinear evolution equations by improved (G1015840G)-expansion methodrdquo Mathematical Science Letters vol 3 no 1pp 1ndash8 2014

[30] E M E Zayed ldquoExact traveling wave solutions for a variable-coefficient generalized dispersive water-wave system using thegeneralized (GrsquoG)-expansion methodrdquo Mathematical ScienceLetters vol 3 no 1 pp 9ndash15 2014

[31] E A Sete and H Eleuch ldquoInteraction of a quantum well withsqueezed light quantum-statistical propertiesrdquo Physical ReviewA vol 82 no 4 Article ID 043810 2010

[32] W Yang L Huang and M Tian ldquoTwo novel and secure quan-tum ANDOS schemesrdquo Applied Mathematics and InformationSciences vol 7 no 2 pp 659ndash665 2013

[33] H Eleuch ldquoPhoton statistics of light in semiconductor micro-cavitiesrdquo Journal of Physics B Atomic Molecular and OpticalPhysics vol 41 no 5 Article ID 055502 2008

[34] G Messin J P Karr H Eleuch J M Courty and E GiacobinoldquoSqueezed states and the quantumnoise of light in semiconduc-tor microcavitiesrdquo Journal of Physics Condensed Matter vol 11no 31 pp 6069ndash6078 1999

[35] A Baas J P Karr H Eleuch and E Giacobino ldquoOpticalbistability in semiconductor microcavitiesrdquo Physical Review AAtomic Molecular and Optical Physics vol 69 no 2 2004

[36] J Bonca and S Kruchinin Physical Properties of NanosystemsSpringer 2011

[37] L Tang and J-A Lu ldquoProjective lag synchronization of chaoticsystems with parameter mismatchrdquo Applied Mathematics ampInformation Sciences vol 7 no 1 pp 121ndash125 2013

[38] D Delande ldquoChaos and atoms in strong magnetic fieldsrdquoPhysica Scripta vol 1991 no T34 p 52 1991

[39] H Ruder G Wunner H Herold and F Geyer Atoms in StrongMagnetic Fields Springer Berlin Germany 1994

[40] H Eleuch and A Prasad ldquoChaos and regularity in semiconduc-tormicrocavitiesrdquoPhysics Letters A vol 376 no 26-27 pp 1970ndash1977 2012

[41] A Y Potekhin and A V Turbiner ldquoHydrogen atom in amagnetic field the quadrupole momentrdquo Physical Review AAtomic Molecular and Optical Physics vol 63 no 6 Article ID065402 2001

[42] S N Rasband Chaotic Dynamics of Non linear Systems JohnWiley amp Sons 1990

[43] A J Lichtenberg and M A Lieberman Regular and StochasticMotion Springer New York NY USA 1983

[44] A V Andreev O Agam B D Simons and B L AltshulerldquoQuantum chaos irreversible classical dynamics and randommatrix theoryrdquo Physical Review Letters vol 76 no 21 pp 3947ndash3950 1996

[45] O Bohigas ldquoLesHouches summer school session LIIrdquo inChaosand Quantum Physics M-J Giannoni A Voros and J Zinn-Justin Eds North-Holland Amsterdam The Netherlands1991

[46] T Pohl H R Sadeghpour and P Schmelcher ldquoCold andultracold Rydberg atoms in strong magnetic fieldsrdquo PhysicsReports vol 484 no 6 pp 181ndash229 2009

[47] T F Gallagher Rydberg Atoms Cambridge University PressCambridge UK 1994

[48] M I Eides H Grotch and V A Shelyuto Theory of LightHydrogenic Bound States vol 222 of Springer Tracts in ModernPhysics 2007

[49] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013

[50] C Yang S S Ge and T H Lee ldquoOutput feedback adaptivecontrol of a class of nonlinear discrete-time systems withunknown control directionsrdquoAutomatica vol 45 no 1 pp 270ndash276 2009

[51] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011

[52] C Yang S S Ge C Xiang T Chai and T H Lee ldquoOutputfeedback NN control for two classes of discrete-time systemswith unknown control directions in a unified approachrdquo IEEETransactions on Neural Networks vol 19 no 11 pp 1873ndash18862008

[53] Y-J Liu S-C Tong DWang T-S Li and C L P Chen ldquoAdap-tive neural output feedback controller design with reduced-order observer for a class of uncertain nonlinear SISO SystemsrdquoIEEE Transactions on Neural Networks vol 22 no 8 pp 1328ndash1334 2011

[54] T M Shahwan ldquoA comparison of Bayesian methods andartificial neural networks for forecasting chaotic financial timeseriesrdquo Journal of Statistics Applications amp Probability vol 1 no2 p 89 2012

[55] J W SeriesThe Spectrum of Atomic Hydrogen Oxford Univer-sity Press London UK 1957

[56] T van der Veldt W Vassen and W Hogervorst ldquoHeliumRydberg states in parallel electric and magnetic fieldsrdquo Journalof Physics B Atomic Molecular and Optical Physics vol 26 no13 pp 1945ndash1955 1993

Mathematical Problems in Engineering 9

[57] A Arimondo W D Phillips and F Strumia Laser Manip-ulation of Atoms and Ions North-Holland Amsterdam TheNetherlands 1992

[58] D Weiss K Richter A Menschig et al ldquoQuantized periodicorbits in large antidot arraysrdquo Physical Review Letters vol 70no 26 pp 4118ndash4121 1993

[59] D Delande Atomes de rydberg en champs statiques intenses[These drsquoEtat] Universite Paris 6 Paris France 1988

[60] M Sadhukhan and B M Deb ldquoA dynamical signature ofquantum chaos in hydrogen atom under strong oscillatingmagnetic fieldsrdquo EPL vol 94 no 5 Article ID 50008 2011

[61] A Bruno-Alfonso C Bleasdale G V B de Souza and R ALewis ldquoClosed-orbit dependence on the field direction in theanisotropic diamagnetic Kepler problemrdquo Physical Review AmdashAtomic Molecular and Optical Physics vol 89 no 4 Article ID043425 2014

[62] J-K Xu H-Q Chen and H-P Liu ldquoExact quantum defecttheory approach for lithium in magnetic fieldsrdquo Chinese PhysicsB vol 22 no 1 Article ID 013204 2013

[63] M A Amdouni and H Eleuch ldquoInfluence of relativistic termsin the spectra of hydrogen atom highly excited in an externalstrong magnetic fieldrdquo Applied Mathematics amp InformationSciences vol 5 no 2 pp 307ndash320 2011

[64] J C Slater Quantum Theory of Atomic Structure vol 1-2McGraw-Hill New York NY USA 1996

[65] E U Condon and G H ShortleyTheTheory of Atomic SpectraCambridge University Press Cambridge UK 1953

[66] C C Tannoudji B Diu and F Laloe Quantum MechanicsWiley New York NY USA 1977 Mechanics Wiley

[67] P J Mohr B N Taylor and D B Newell ldquoCODATA recom-mended values of the fundamental physical constants 2006rdquoReviews of Modern Physics vol 80 no 2 pp 633ndash730 2008

[68] C W Clark and K T Taylor ldquoThe quadratic Zeeman effect inhydrogen Rydberg series application of Sturmian functionsrdquoJournal of Physics B Atomic and Molecular Physics vol 15 no8 pp 1175ndash1193 1982

[69] M A Iken F Borondo R M Benito and T Uzer ldquoPeriodic-orbit spectroscopy of the hydrogen atom in parallel electric andmagnetic fieldsrdquo Physical ReviewA vol 49 no 4 pp 2734ndash27471994

[70] A R Edmonds ldquoStudies of the quadratic Zeeman effect IApplication of the sturmian functionsrdquo Journal of Physics BAtomic and Molecular Physics vol 6 no 8 pp 1603ndash1615 1973

[71] K Sacha J Zakrzewski and D Delande ldquoBreaking timereversal symmetry in chaotic driven rydberg atomsrdquo Annals ofPhysics vol 283 no 1 pp 141ndash172 2000

[72] M Courtney Rydberg atoms in strong fields a testing groundfor quantum chaos [PhD thesis] Department of PhysicsMassachusetts Institute of Technology 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

[57] A Arimondo W D Phillips and F Strumia Laser Manip-ulation of Atoms and Ions North-Holland Amsterdam TheNetherlands 1992

[58] D Weiss K Richter A Menschig et al ldquoQuantized periodicorbits in large antidot arraysrdquo Physical Review Letters vol 70no 26 pp 4118ndash4121 1993

[59] D Delande Atomes de rydberg en champs statiques intenses[These drsquoEtat] Universite Paris 6 Paris France 1988

[60] M Sadhukhan and B M Deb ldquoA dynamical signature ofquantum chaos in hydrogen atom under strong oscillatingmagnetic fieldsrdquo EPL vol 94 no 5 Article ID 50008 2011

[61] A Bruno-Alfonso C Bleasdale G V B de Souza and R ALewis ldquoClosed-orbit dependence on the field direction in theanisotropic diamagnetic Kepler problemrdquo Physical Review AmdashAtomic Molecular and Optical Physics vol 89 no 4 Article ID043425 2014

[62] J-K Xu H-Q Chen and H-P Liu ldquoExact quantum defecttheory approach for lithium in magnetic fieldsrdquo Chinese PhysicsB vol 22 no 1 Article ID 013204 2013

[63] M A Amdouni and H Eleuch ldquoInfluence of relativistic termsin the spectra of hydrogen atom highly excited in an externalstrong magnetic fieldrdquo Applied Mathematics amp InformationSciences vol 5 no 2 pp 307ndash320 2011

[64] J C Slater Quantum Theory of Atomic Structure vol 1-2McGraw-Hill New York NY USA 1996

[65] E U Condon and G H ShortleyTheTheory of Atomic SpectraCambridge University Press Cambridge UK 1953

[66] C C Tannoudji B Diu and F Laloe Quantum MechanicsWiley New York NY USA 1977 Mechanics Wiley

[67] P J Mohr B N Taylor and D B Newell ldquoCODATA recom-mended values of the fundamental physical constants 2006rdquoReviews of Modern Physics vol 80 no 2 pp 633ndash730 2008

[68] C W Clark and K T Taylor ldquoThe quadratic Zeeman effect inhydrogen Rydberg series application of Sturmian functionsrdquoJournal of Physics B Atomic and Molecular Physics vol 15 no8 pp 1175ndash1193 1982

[69] M A Iken F Borondo R M Benito and T Uzer ldquoPeriodic-orbit spectroscopy of the hydrogen atom in parallel electric andmagnetic fieldsrdquo Physical ReviewA vol 49 no 4 pp 2734ndash27471994

[70] A R Edmonds ldquoStudies of the quadratic Zeeman effect IApplication of the sturmian functionsrdquo Journal of Physics BAtomic and Molecular Physics vol 6 no 8 pp 1603ndash1615 1973

[71] K Sacha J Zakrzewski and D Delande ldquoBreaking timereversal symmetry in chaotic driven rydberg atomsrdquo Annals ofPhysics vol 283 no 1 pp 141ndash172 2000

[72] M Courtney Rydberg atoms in strong fields a testing groundfor quantum chaos [PhD thesis] Department of PhysicsMassachusetts Institute of Technology 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of