PH Y SICAL RE VIE% A VOLUME 24, NUMBER 1 JULY 1981

Exact spectrmn of the two-dimensional rigid rotator in external fields. II.Zeeman effect

Mark P. SilvermanDepartment ofPhysics, wesleyan University, Middleton, Connecticut 06457

(Received 10 July 1980j

The two-dimensional rigid rotator in a static uniform magnetic field of arbitrary strength is solved exactly. Theeigenvalue spectrum is found to consist of the characteristic values of the Mathieu functions of both even and oddindex and of both n and 2e periodicities. For low field strengths 8 the double degeneracy of the rotatinal states ~mis broken in order 8 . For high field strengths the system exhibits a quasi-Landau spectrum with a regenerated

double degeneracy.

I. INTRODUCTION

In a previous article (designated Paper I) we de-rived the exact eigenvalues and eigenfunctions ofthe two-dimensional rigid rotator in a static uni-form electric field normal to the angular momen-tum. ' The energy spectrum was shown to consistof the characteristic values of the even-indexMathieu functions. In this article we determinethe spectrum of the two-dimensional rigid rotatorin a static uniform magnetic field. Our discussionis again limited to the case of the field orientedperpendicular to the angular momentum. The al-ternative possibility of a field parallel (or anti-parallel) to the angular momentum is triviallysoluble since the solutions remain eigenfunctionsof the angular momentum.

A number of important properties of the Zeemansolutions can be ascertained on the basis of sym-metry considerations alone. In Paper I it wasshown that the Stark solutions span the symmetricand antisymmetric one-dimensional irreduciblerepresentations of a group isomorphic to the sym-metric group S2. An examination of the Hamilton-ian including the rotational kinetic energy anddiamagnetic potential energy (there is no para-magnetic term for the chosen field orientation)shows that it is invariant under a group of opera-tions comprising the identity, reflection acrossthe field axis, reflection across an axis in therotation plane normal to the field, and inversionthrough the origin. This group is isomorphic tothe (Abelian) four group. ' Since the degeneratefield-free basis g, -e" ~ spans a two-dimensionalirreducible representation of the rotation-reflec-tion group, it must split under the magnetic inter-action into nondegenerate states which span theirreducible representations of the four group. Asimple character analysis shows that the statesof even (odd) Im I

span representations of even(odd) inversion parity. The two nondegeneratestates of equal inversion parity derived from adegenerate rotational level of given Im I

differ

in their reflection parities. We therefore expectin the Zeeman-effect stationary states of fourpossible symmetry-types in contrast to the twopossible symmetry-types of the Stark effect. Thehigher symmetry also leads to marked differencesin the energy spectra between the two cases, par-ticularly in the high-field domain.

, II. ZEEMAN EFFECT OF THE TWO-DIMENSIONALRIGID ROTATOR

For convenience we consider the rotator to be apoint mass with charge —Ie I, mass M, and orbitalradius R subjected to a magnetic field B=By nor-mal to the only nonvanishing component of the or-bital angular momentum L,z. The Hamiltonian isthen given by

X =R, +X =I',i2MR'+ 2MuP~R' cos'P,—

where

(la)

82,+ p,

2 —$'cos2$ g p =0, (2a)

where, similar to our usage in Paper I, we haveexpressed the energy eigenvalues as

E = Q p,~/2MR2 (2b)

and defined the dimensionless magnetic interac-tion parameter

$ =M~~R'/K. (2c)

By setting

a= p,*-]'/2,

V= t'/4,

one can reexpress Eq. (2a) in the form

(3a)

(3b)

(&b)

is the Larmor frequency.Use of the above Hamiltonian leads to the Schro-

dinger equation

24

EXACT SPECTRUM OF THE T%0-DIMENSIONAL RIGID. ..

~ +[a-2q cos(24)lk(4}=o, (4)TABLE I. Series expansion of the Zeeman eigen-

values p .~

c(e+») = c(e) (5)

imposed by the physical requirement of continuityand single-valuedness of the wave function is satis-fied by Mathieu functions of both m and 2m perio-dicities. The traditional designation of these func-tions and their characteristic values are as follows(where m=0, 1,2, . . . ).'

(1}ce, (Q, q)—even solutions of period w whichreduce to cos(mQ) as q -0; characteristic valuesare a =a2 ~

(2) ce, „(Q,q}—even solutions of period 2s whichreduce to cos[(2m+1)P] as q —0; characteristicvalues are a a2

(3}se, , (Q, q}—odd solutions of period s whichreduce to sin[(2m+1)g] as q -0; characteristicvalues are a = b2

(4} se, „(Q,q)—odd solutions of period 2w whichreduce to sin[(2m+2}p] as q -0; characteristicvalues are a=b,

The exact Zeeman solutions (to within a normal-ization constant) are therefore

t) „"' = ce (P, $), m =0, 2, 4. . . (6a)

E"' = b [a (() + $ /2]/2MR (6b)

g„"' =se (P, ]), m=2, 4, 6. . . (6c}

E„"' =8 [b ($)+( /2]2MR (6d)

gu" =ce (P, $), m =1, 3, 5. . . (6e)

E„""= b '[a (() + $'/2]/2MR', (6f}

g„""=se (P, $), m=1, 3, 5. . . (Sg)

E"'& =8'[b ($)+(~/2]/2MRI. (Sh)

As discussed in Paper I, the characteristicvalues of the Mathieu functions cannot generallybe expressed in closed form. For sufficientlysmall q or $, however, they can be expanded inan infinite series by solution of a continued frac-tion or of Hill's determinant. Standard degenerateperturbation theory will also yield the same re-sults. In Table I is given the expansion of theeigenvalues p,', for the ground state and first sixexcited states. The eigenvalues are truncated atthe lowest order in $ which splits the zero-fielddouble degeneracy. In contrast to the Stark levelswhich are split by an electric field t" in order 5~~,

which is again recognizable as the canonical formof Mathieu's equation. ' There is an importantdistinction, however, between Eq. (4) above andthe comparable equation (4) in Paper I. Here theangular coordinate P and not 8 = P/2 appears in theequation directly. As a consequence, the boundarycondition

(&)po, = g$2 1 2

(2w) 2 =1+-$3 2f 1+

(2~) 2 =1+&)

(r) 2 =2 +-'f +&2+

(~) 2 22~ & (2 i )4i92

(2~) 2 =3+-5 +—f +—$1 2 i 4 i 6

256 4096

(2&) 2 2 & 2 i 4 i 6"3-=3 + 5 +—5 ——h256 4096

~ Adapted from Ref. 3.

it is seen that the Zeeman levels are split by Bin order B .

For large B the asymptotic expressions for thecharacteristic values of the Mathieu functions'

a2„(q}=b~, (q) =-2q + (8n+ 2)q'~ 2,

a~ ~(q) =b~~2(q) =-2q+(Sn+6)q'~

can be used to derive closed form expressionsfor the energy

(7a)

(7b)

Eu" = (m+"e+E&ai& = (m-"a-E"' =(m+"e+E"' =(m-"m-

2)R(dl,

p)K(dL q

g)R(d ~,

2)R(dl,

m =0, 2, 4. . .

m =2, 4, 6. . .m =1,3, 5. . .

m =1,3, 5. . . .

(Sa)

(8b)

(Sc)

(Sd)

In contrast to the Stark effect, the high-field Zee-man states are again doubly degenerate, includingthe lowest energy level. ' Degenerate pairs com-prise states of opposite periodicity and inversionparity. Thus, for example, the ground states areceo"' and se,"",and the first excited states arece,""and se,"'. Each high-field degenerate pairconsists of states deriving from different irredu-cible representations of the rotation-reflectiongroup. The spectrum given by Eqs. (Sa}-(8d) re-sembles somewhat the Landau spectrum of an elec-tron freely orbiting in a plane perpendicular to themagnetic field. ' The actual behavior of the rota-tor, however, is quite different. The asymptoticMathieu functions of all four symmetry types takeon their maximum values in the vicinity of Q =am/2and drop off rapidly to very small vat.ues at /=0, m

as q- ~.'The high-field rotator executes a librational mo-

tion about these equilibrium points. The similari-ty of the high-field spectrum to the harmonic-os-cillator spectrum can be understood by expanding

MARK P. SILVERMAN 24

the rotator Hamiltonian, Eq. (la), about the points

P =ax/2 to obtain the Hamiltonian of an harmonicoscillator. Similar behavior is exhibited by thehigh-field Stark solutions with the distinction,however, that the classical small-amplitude os-cillations are executed about / =0 (&f&

= w) for adipole moment p &0 (p& 0).

In conclusion it is noted that, as anticipated,the Zeeman solutions, Eqs. (6a), (6c), (6e), and

(6g), span the four irreducible representationsof the four group when one associates the symme-tries ce and se with the eigenvalues + of the re-flection II, and the periodicities n, 2w with theeigenvalues + of the inversion II.

~M. P. Silverman, Phys. Rev. A 24, 339 (1981).E. P. Wigner, @youp Theory (Academic, New York,1959), p. 63.

N. W. McLachlan, Theory and Application of Mathieu

Junctions (Dover, New York, 1964), p. 10.Reference 3, p. 18.

5Reference 3, p. 240.GThe asymptotic relationa„=b„+ &

holds exactly onlyfor infinite q; for finite q the difference b„,&

-a„ is

approximately 2 "' (2/7t) ~ q"+~ exp(-4q /bt).

[See M. Abramowitz and I. A. Stegun, Handbook ofMathematical Eunctions (Dover, New York, 1972).jFor the high values of q at which the asymptotic for-mulas hold, this difference is essentially negligible,and. the levels may be considered degenerate.

L. Landau, Z. Phys. 64, 629 (1930).P. M. Morse and H. Feshbach, Methods of TheoreticalJhysics (McGraw-Hill, New York, 1953), p. 1416.