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Matrix mechanics


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Quantum mechanics

Uncertainty principle

IntroductionMathematical

formulations

Background

Classical mechanics

Old quantum theory

InterferenceBra-ket notationHamiltonian

Fundamental concepts

Quantum stateWave function

SuperpositionEntanglement

ComplementarityDuality

Uncertainty

MeasurementExclusion

DecoherenceEhrenfest

theoremTunnelling

Experiments

Double-slit experiment

DavissonGermer experimen

SternGerlach experiment

Bell's inequality experiment

Popper's experiment

Schrdinger's cat

ElitzurVaidman bomb-tester

Quantum eraser

Formulations

Schrdinger picture

Heisenberg picture

Interaction picture

Matrix mechanics

Sum over histories

Wikipedia

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Matrix mechanics is a formulation ofquantum mechanics created by WernerHeisenberg, Max Born, and Pascual Jordan in 1925.

Matrix mechanics was the first complete and correct definition of quantummechanics. It extended the Bohr Model by describing how the quantum jumpsoccur. It did so by interpreting the physical properties of particles as matricesthat evolve in time. It is equivalent to the Schrdinger wave formulation ofquantum mechanics, and is the basis ofDirac's bra-ket notation for the wavefunction.

Contents

1 Development of matrix mechanics1.1 Epiphany at Heligoland1.2 The Three Papers1.3 Heisenberg's reasoning

2 Further discussion3 Nobel Prize4 Mathematical development

4.1 Harmonic oscillator4.2 Conservation of energy4.3 Differentiation trick canonical commutation relations4.4 State vector modern quantum mechanics

5 Further results5.1 Wave mechanics5.2 Transformation theory5.3 Selection rules5.4 Sum rules

6 The three formulating papers7 See also8 Bibliography9 Footnotes

10 External links

Development of matrix mechanics

In 1925, Werner Heisenberg, Max Born, and Pascual Jordan formulated thematrix mechanics representation of quantum mechanics.


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Equations

Schrdinger equation

Pauli equation

KleinGordon equation

Dirac equation

Rydberg formula

Interpretations

de BroglieBohmCCC

Consistent histories

CopenhagenEnsembleHidde

variablesMany-worlds

Objective collapse

PondicherryQuantum logic

RelationalStochastic

Transactional

Advanced topics

Quantum information science

Scattering theory

Quantum field theory

Quantum chaos

Scientists

Bell Bohm Bohr Born

Bose de Broglie Dirac

Ehrenfest Everett Feynman

Heisenberg Jordan Kramers

von Neumann Pauli Planck

Schrdinger Sommerfeld

Wien Wigner Salam

vde

Epiphany at Heligoland

In 1925 Werner Heisenberg was working in Gttingen on the problem ofcalculating the spectral lines ofhydrogen. By May 1925 he began trying todescribe atomic systems by observables only. On June 7, to escape the effects ofa bad attack ofhay fever, Heisenberg left for the pollen free North Sea island ofHeligoland. While there, in between climbing and learning by heart poems from

Goethe's West-stlicher Diwan, he continued to ponder the spectral issue andeventually realised that adopting non-commuting observables might solve the

problem, and he later wrote[1]

"It was about three o' clock at night when the final result of the calculationlay before me. At first I was deeply shaken. I was so excited that I couldnot think of sleep. So I left the house and awaited the sunrise on the top ofa rock."

The Three Papers

After Heisenberg returned to Gttingen, he showed Wolfgang Pauli his

calculations, commenting at one point:[2]

"Everything is still vague and unclear to me, but it seems as if theelectrons will no more move on orbits."

On July 9 Heisenberg gave the same paper of his calculations to Max Born,saying, "...he had written a crazy paper and did not dare to send it in forpublication, and that Born should read it and advise him on it..." prior topublication. Heisenberg then departed for a while, leaving Born to analyse the

paper.[3]

In the paper, Heisenberg formulated quantum theory without sharp electronorbits. Hendrik Kramers had earlier calculated the relative intensities of spectrallines in the Sommerfeld model by interpreting the Fourier coefficients of theorbits as intensities. But his answer, like all other calculations in the old quantumtheory, was only correct for large orbits.

Heisenberg, after a collaboration with Kramers[4], began to understand that the transition probabilities were noquite classical quantities, because the only frequencies that appear in the Fourier series should be the ones thaare observed in quantum jumps, not the fictional ones that come from Fourier-analyzing sharp classical orbits

He replaced the classical Fourier series with a matrix of coefficients, a fuzzed-out quantum analog of theFourier series. Classically, the Fourier coefficients give the intensity of the emitted radiation, so in quantummechanics the magnitude of the matrix elements of the position operator were the intensity of radiation in thebright-line spectrum.

The quantities in Heisenberg's formulation were the classical position and momentum, but now they were nolonger sharply defined. Each quantity was represented by a collection ofFourier coefficients with two indices

corresponding to the initial and final states.[5] When Born read the paper, he recognized the formulation as on


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which could be transcribed and extended to the systematic language of matrices,[6] which he had learned from

his study under Jakob Rosanes[7] at Breslau University. Born, with the help of his assistant and former studenPascual Jordan, began immediately to make the transcription and extension, and they submitted their results fo

publication; the paper was received for publication just 60 days after Heisenbergs paper.[8] A follow-on pape

was submitted for publication before the end of the year by all three authors.[9] (A brief review of Borns rolethe development of the matrix mechanics formulation of quantum mechanics along with a discussion of the ke

formula involving the non-commutivity of the probability amplitudes can be found in an article by JeremyBernstein.[10] A detailed historical and technical account can be found in Mehra and Rechenbergs book TheHistorical Development of Quantum Theory. Volume 3. The Formulation of Matrix Mechanics and Its

Modifications 19251926.[11])

Up until this time, matrices were seldom used by physicists, they were considered to belong to the realm of pumathematics. Gustav Mie had used them in a paper on electrodynamics in 1912 and Born had used them in hiswork on the lattices theory of crystals in 1921. While matrices were used in these cases, the algebra of matricewith their multiplication did not enter the picture as they did in the matrix formulation of quantum

mechanics.[12] Born, however, had learned matrix algebra from Rosanes, as already noted, but Born had alsolearned Hilberts theory of integral equations and quadratic forms for an infinite number of variables as was

apparent from a citation by Born of Hilberts work Grundzge einer allgemeinen Theorie der LinearenIntegralgleichungen published in 1912.[13][14] Jordan, too was well equipped for the task. For a number ofyears, he had been an assistant to Richard Courant at Gttingen in the preparation of Courant and David

Hilberts bookMethoden der mathematischen Physik I, which was published in 1924.[15] This book,fortuitously, contained a great many of the mathematical tools necessary for the continued development ofquantum mechanics. In 1926, John von Neumann became assistant to David Hilbert, and he would coin the teHilbert space to describe the algebra and analysis which were used in the development of quantum

mechanics.[16][17]

Heisenberg's reasoning

Before matrix mechanics, the old quantum theory described the motion of a particle by a classical orbit, withwell defined position and momentumX(t),P(t), with the restriction that the time integral over one period T ofthe momentum times the velocity must be a positive integer multiple ofPlanck's constant

While this restriction correctly selects orbits with more or less the right energy valuesEn, the old quantum

mechanical formalism did not describe time dependent processes, such as the emission or absorption of

radiation.

When a classical particle is weakly coupled to a radiation field, so that the radiative damping can be neglectedit will emit radiation in a pattern which repeats itself every orbital period. The frequencies which make up theoutgoing wave are then integer multiples of the orbital frequency, and this is a reflection of the fact that X(t) iperiodic, so that its Fourier representation has frequencies 2n / Tonly.


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The coefficientsXn are complex numbers. The ones with negative frequencies must be the complex conjugate

of the ones with positive frequencies, so that X(t) will always be real,

.

A quantum mechanical particle, on the other hand, can't emit radiation continuously, it can only emit photons.Assuming that the quantum particle started in orbit number n, emitted a photon, then ended up in orbit numberm, the energy of the photon isEnEm, which means that its frequency is (EnEm) / h.

For large n and m, but with nm relatively small, these are the classical frequencies by Bohr's correspondencprinciple

In the formula above, Tis the classical period of either orbit n or orbit m, since the difference between them ishigher order in h. But for n and m small, or ifnm is large, the frequencies are not integer multiples of anysingle frequency.

Since the frequencies which the particle emits are the same as the frequencies in the fourier description of itsmotion, this suggests that something in the time-dependent description of the particle is oscillating withfrequency (EnEm) / h . Heisenberg called this quantityXnm, and demanded that it should reduce to the

classical Fourier coefficients in the classical limit. For large values ofn, m but with nm relatively small,Xnis the (nm)th fourier coefficient of the classical motion at orbit n. SinceXnm has opposite frequency toXmn,

the condition that X is real becomes:

.

By definition,Xnm only has the frequency (EnEm) / h, so its time evolution is simple:

.

This is the original form of Heisenberg's equation of motion.

Given two arraysXnm and Pnm describing two physical quantities, Heisenberg could form a new array of the

same type by combining the termsXnkPkm, which also oscillate with the right frequency. Since the Fourier

coefficients of the product of two quantities is the convolution of the Fourier coefficients of each one separatethe correspondence with Fourier series allowed Heisenberg to deduce the rule by which the arrays should bemultiplied:

Born pointed out that this is the law of matrix multiplication, so that the position, the momentum, the energy,


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the observable quantities in the theory, are interpreted as matrices. Because of the multiplication rule, theproduct depends on the order:XP is different from PX.

The X matrix is a complete description of the motion of a quantum mechanical particle. Because the frequencin the quantum motion are not multiples of a common frequency, the matrix elements cannot be interpreted asthe Fourier coefficients of a sharp classical trajectory. Nevertheless, as matrices,X(t) and P(t) satisfy theclassical equations of motion.

Further discussion

When it was introduced by Werner Heisenberg, Max Born and Pascual Jordan in 1925, matrix mechanics wasnot immediately accepted and was a source of great controversy. Schrdinger's later introduction ofwavemechanics was favored.

Part of the reason was that Heisenberg's formulation was in a strange new mathematical language, whileSchrdinger's formulation was based on familiar wave equations. But there was also a deeper sociologicalreason. Quantum mechanics had been developing by two paths, one under the direction of Einstein and theother under the direction of Bohr. Einstein emphasized wave-particle duality, while Bohr emphasized thediscrete energy states and quantum jumps. DeBroglie had shown how to reproduce the discrete energy states iEinstein's framework--- the quantum condition is the standing wave condition, and this gave hope to those inthe Einstein school that all the discrete aspects of quantum mechanics would be subsumed into a continuouswave mechanics.

Matrix mechanics, on the other hand, came from the Bohr school, which was concerned with discrete energystates and quantum jumps. Bohr's followers did not appreciate physical models which pictured electrons aswaves, or as anything at all. They preferred to focus on the quantities which were directly connected toexperiments.

In atomic physics, spectroscopy gave observational data on atomic transitions arising from the interactions of

atoms with light quanta. The Bohr school required that only those quantities which were in principle measurabby spectroscopy should appear in the theory. These quantities include the energy levels and their intensities buthey do not include the exact location of a particle in its Bohr orbit. It is very hard to imagine an experimentwhich could determine whether an electron in the ground state of a hydrogen atom is to the right or to the left the nucleus. It was a deep conviction that such questions did not have an answer.

The matrix formulation was built on the premise that all physical observables are represented by matrices whoelements are indexed by two different energy levels. The set ofeigenvalues of the matrix were eventuallyunderstood to be the set of all possible values that the observable can have. Since Heisenberg's matrices areHermitian, the eigenvalues are real.

If an observable is measured and the result is a certain eigenvalue, the corresponding eigenvector is the state othe system immediately after the measurement. The act of measurement in matrix mechanics 'collapses' the staof the system. If you measure two observables simultaneously, the state of the system should collapse to acommon eigenvector of the two observables. Since most matrices don't have any eigenvectors in common, moobservables can never be measured precisely at the same time. This is the uncertainty principle.

If two matrices share their eigenvectors, they can be simultaneously diagonalized. In the basis where they areboth diagonal, it is clear that their product does not depend on their order because multiplication of diagonal


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matrices is just multiplication of numbers. The Uncertainty Principle then is a consequence of the fact that twomatrices A and B do not always commute,i.e,that A B - B A does not necessarily equal 0. The famouscommutation relation of matrix mechanics:

shows that there are no states which simultaneously have a definite position and momentum. But the principleof uncertainty (also called complementarity by Bohr) holds for most other pairs of observables too. Forexample, the energy does not commute with the position either, so it is impossible to precisely determine theposition and energy of an electron in an atom.

Nobel Prize

In 1928, Albert Einstein nominated Heisenberg, Born, and Jordan for the Nobel Prize in Physics,[18] but it wa

not to be. The announcement of the Nobel Prize in Physics for 1932 was delayed until November 1933.[19] Itwas at that time that it was announced Heisenberg had won the Prize for 1932 for the creation of quantum

mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen[2

and Erwin Schrdinger and Paul Adrien Maurice Dirac shared the 1933 Prize "for the discovery of new

productive forms of atomic theory".[20] One can rightly ask why Born was not awarded the Prize in 1932 alonwith Heisenberg, and Bernstein gives some speculations on this matter. One of them is related to Jordan joinin

the Nazi Party on May 1, 1933 and becoming a Storm Trooper.[21] Hence, Jordans Party affiliations andJordans links to Born may have affected Borns chance at the Prize at that time. Bernstein also notes that wheBorn won the Prize in 1954, Jordan was still alive, and the Prize was awarded for the statistical interpretation

quantum mechanics, attributable alone to Born.[22]

Heisenbergs reactions to Born for Heisenberg receiving the Prize for 1932 and for Born receiving the Prize in1954 are also instructive in evaluating whether Born should have shared the Prize with Heisenberg. OnNovember 25, 1933 Born received a letter from Heisenberg in which he said he had been delayed in writing duto a bad conscience that he alone had received the Prize for work done in Gttingen in collaboration youJordan and I. Heisenberg went on to say that Born and Jordans contribution to quantum mechanics cannot b

changed by a wrong decision from the outside.[23] In 1954, Heisenberg wrote an article honoring Max Planfor his insight in 1900. In the article, Heisenberg credited Born and Jordan for the final mathematicalformulation of matrix mechanics and Heisenberg went on to stress how great their contributions were to

quantum mechanics, which were not adequately acknowledged in the public eye. [24]

Mathematical developmentOnce Heisenberg introduced the matrices for X and P, he could find their matrix elements in special cases byguesswork, guided by the correspondence principle. Since the matrix elements are the quantum mechanicalanalogs ofFourier coefficients of the classical orbits, the simplest case is the harmonic oscillator, where X(t)and P(t) are sinusoidal.

Harmonic oscillator


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In units where the mass and frequency of the oscillator are equal to one, the energy of the oscillator is

The level sets of H are the orbits, and they are nested circles. The classical orbit with energy E is:

The old quantum condition says that the integral of P dX over an orbit, which is the area of the circle in phasespace, must be an integer multiple ofPlanck's constant. The area of the circle of radius is 2ESo

or, in units of length where is one, the energy is an integer.

The Fourier components ofX(t) and P(t) are very simple, even more so if they are combined into the quantitie

both and have only a single frequency, andXand P can be recovered from their sum and difference.

SinceA(t) has a classical fourier series with only the lowest frequency, and the matrix element Amn is the (m-

n)th fourier coefficient of the classical orbit, the matrix for is nonzero only on the line just above thediagonal, where it is equal to . The matrix for is likewise only nonzero on the line below the diagonalwith the same elements. Reconstructing X and P from and :

which, up to the choice of units, are the Heisenberg matrices for the harmonic oscillator. Notice that bothmatrices are hermitian, since they are constructed from the Fourier coefficients of real quantities. To findX(t)


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and P(t) is simple, since they are quantum Fourier coefficients so they evolve simply with time.

The matrix product ofXand P is not hermitian, but has a real and imaginary part. The real part is one half thesymmetric expression (XP + PX), while the imaginary part is proportional to the commutator [X,P] = (XPPX). It is easy to verify explicitly that (XPPX) in the case of the harmonic oscillator, is ih/2, multiplied by

the identity. It is also easy to verify that the matrix

is a diagonal matrix, with eigenvaluesEi.

Conservation of energy

The harmonic oscillator is too special. It is too easy to find the matrices exactly, and it is too hard to discovergeneral conditions from these special forms. For this reason, Heisenberg investigated the anharmonic oscillatowith Hamiltonian

In this case, theXand P matrices are no longer simple off diagonal matrices, since the corresponding classicaorbits are slightly squashed and displaced, so that they have Fourier coefficients at every classical frequency. Tdetermine the matrix elements, Heisenberg required that the classical equations of motion be obeyed as matrixequations:

He noticed that if this could be done then H considered as a matrix function of X and P, will have zero timederivative.

WhereA *B is the symmetric product.

.

Given that all the off diagonal elements have a nonzero frequency; H being constant implies that H is diagonaIt was clear to Heisenberg that in this system, the energy could be exactly conserved in an arbitrary quantumsystem, a very encouraging sign.

The process of emission and absorption of photons seemed to demand that the conservation of energy will hol


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at best on average. If a wave containing exactly one photon passes over some atoms, and one of them absorbs that atom needs to tell the others that they can't absorb the photon anymore. But if the atoms are far apart, anysignal cannot reach the other atoms in time, and they might end up absorbing the same photon anyway anddissipating the energy to the environment. When the signal reached them, the other atoms would have tosomehow recall that energy. This paradox led Bohr, Kramers and Slater to abandon exact conservation ofenergy. Heisenberg's formalism, when extended to include the electromagnetic field, was obviously going tosidestep this problem, a hint that the interpretation of the theory will involve wavefunction collapse.

Differentiation trick canonical commutation relations

Demanding that the classical equations of motion are preserved is not a strong enough condition to determinethe matrix elements. Planck's constant does not appear in the classical equations, so that the matrices could beconstructed for many different values of and still satisfy the equations of motion, but with different energylevels.

So in order to implement his program, Heisenberg needed to use the old quantum condition to fix the energylevels, then fill in the matrices with Fourier coefficients of the classical equations, then alter the matrixcoefficients and the energy levels slightly to make sure the classical equations are satisfied. This is clearly not

satisfactory. The old quantum conditions refer to the area enclosed by the sharp classical orbits, which do notexist in the new formalism.

The most important thing that Heisenberg discovered is how to translate the old quantum condition into asimple statement in matrix mechanics. To do this, he investigated the action integral as a matrix quantity:

There are several problems with this integral, all stemming from the incompatibility of the matrix formalismwith the old picture of orbits. Which period T should you use? Semiclassically, it should be either m or n, butthe difference is order h and we want an answer to order h. The quantum condition tells us thatJmn is 2n on

the diagonal, then the fact that J is classically constant tells us that the off diagonal elements are zero.

His crucial insight was to differentiate the quantum condition with respect to n. This idea only makes completsense in the classical limit, where n is not an integer but the continuous action variableJ, but Heisenbergperformed analogous manipulations with matrices, where the intermediate expressions are sometimes discretedifferences and sometimes derivatives. In the following discussion, for the sake of clarity, the differentiationwill be performed on the classical variables, and the transition to matrix mechanics will be done afterwardsusing the correspondence principle.

In the classical setting, the derivative is the derivative with respect to J of the integral which defines J, so it istautologically equal to 1.


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Where the derivatives dp/dJ dx/dJ should be interpreted as differences with respect to J at corresponding timeson nearby orbits, exactly what you would get if you differentiated the Fourier coefficients of the orbital motioThese derivatives are symplectically orthogonal in phase space to the time derivatives dP/dt dX/dt. The finalexpression is clarified by introducing the variable canonically conjugate to J, which is called the angle variabl. The derivative with respect to time is a derivative with respect to , up to a factor of 2/ T.

So the quantum condition integral is the average value over one cycle of the Poisson bracket of X and P. Ananalogous differentiation of the Fourier series of P dX demonstrates that the off diagonal elements of thePoisson bracket are all zero. The Poisson bracket of two canonically conjugate variables, such as X and P, is tconstant value 1, so this integral really is the average value of 1, so it is 1, as we knew all along, because it isdJ/dJ after all. But Heisenberg, Born and Jordan weren't familiar with the theory of Poisson brackets, so forthem, the differentiation effectively evaluated {X,P} in J coordinates.

The Poisson Bracket, unlike the action integral, has a simple translation to matrix mechanics--- it is theimaginary part of the product of two variables, the commutator. To see this, examine the product of two

matrices A and B in the correspondence limit, where the matrix elements are slowly varying functions of theindex, keeping in mind that the answer is zero classically.

In the correspondence limit, when indices mn are large and nearby, while k,r are small, the rate of change of tmatrix elements in the diagonal direction is the matrix element of theJderivative of the corresponding classicquantity. So we can shift any matrix element diagonally using the formula:

Where the right hand side is really only the (m-n)'th Fourier component of (dA/dJ) at orbit near m to this

semiclassical order, not a full well defined matrix.

The semiclassical time derivative of a matrix element is obtained up to a factor of i by multiplying by thedistance from the diagonal,

Since the coefficientAm(m + k) is semiclassically the k'th Fourier coefficient of the m-th classical orbit.

The imaginary part of the product of A and B can be evaluated by shifting the matrix elements around so as to

reproduce the classical answer, which is zero. The leading nonzero residual is then given entirely by theshifting. Since all the matrix elements are at indices which have a small distance from the large index position(m,m), it helps to introduce two temporary notations:A[r,k] =A(m + r)(m + k), for the matrices, and (dA / dJ)[r]

for the r'th Fourier components of classical quantities.


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Flipping the summation variable in the first sum from r to r'=k-r, the matrix element becomes:

and it is clear that the main part cancels. The leading quantum part, neglecting the higher order product ofderivatives, is

which can be identified as i times the k-th classical Fourier component of the Poisson bracket. Heisenberg'soriginal differentiation trick of was eventually extended to a full semiclassical derivation of the quantumcondition in collaboration with Born and Jordan.

Once they were able to establish that:

this condition replaced and extended the old quantization rule, allowing the matrix elements of P and X for an

arbitrary system to be determined simply from the form of the Hamiltonian. The new quantization rule wasassumed to be universally true, even though the derivation from the old quantum theory required semiclassicareasoning.

State vector modern quantum mechanics

To make the transition to modern quantum mechanics, the most important further addition was the quantum

state vector, now written , which is the vector that the matrices act on. Without the state vector, it is not clewhich particular motion the Heisenberg matrices are describing, since they include all the motions somewhere

The interpretation of the state vector, whose components are written m

, was given by Born. The interpretatio

is statistical: the result of a measurement of the physical quantity corresponding to the matrix A is random, wian average value equal to

Alternatively and equivalently, the state vector gives the probability amplitudei for the quantum system to b

in the energy state i. Once the state vector was introduced, matrix mechanics could be rotated to any basis,


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where the H matrix was no longer diagonal. The Heisenberg equation of motion in its original form states thatAmn evolves in time like a Fourier component

which can be recast in differential form

and it can be restated so that it is true in an arbitrary basis by noting that the H matrix is diagonal with diagonavaluesEm:

This is now a matrix equation, so it holds in any basis. This is the modern form of the Heisenberg equation ofmotion. The formal solution is:

All the forms of the equation of motion above say the same thing, that A(t) is equal to A(0) up to a basis

rotation by the unitary matrix eiHt. By rotating the basis for the state vector at each time by eiHt, you can undothe time dependence in the matrices. The matrices are now time independent, but the state vector rotates:

This is the Schroedinger equation for the state vector, and the time dependent change of basis is thetransformation to the Schroedinger picture.

In quantum mechanics in the Heisenberg picture the state vector, does not change with time, and anobservableA satisfies

The extra term is for operators likeA = (X+ t2P) which have an explicit time dependence in addition to the tim

dependence from unitary evolution. The Heisenberg picture does not distinguish time from space, so it is nicefor relativistic theories.

Moreover, the similarity to classical physics is more obvious: the Hamiltonian equations of motion for classicmechanics are recovered by replacing the commutator above by the Poisson bracket.

By the Stone-von Neumann theorem, the Heisenberg picture and the Schrdinger picture are unitarilyequivalent.


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See also Schrdinger picture.

Further results

Matrix mechanics rapidly developed into modern quantum mechanics, and gave interesting physical results onthe spectra of atoms.

Wave mechanics

Jordan noted that the commutation relations ensure thatp acts as a differential operator, and came very close tformulating the Schrdinger equation. The identity

allows the evaluation of the commutator ofp with any power ofx, and it implies that

which, together with linearity, implies that ap commutator differentiates any analytic matrix function ofx.Assuming limits are defined sensibly, this will extend to arbitrary functions, but the extension does not need tobe made explicit until a certain degree of mathematical rigor is required.

Sincex is a Hermitian matrix, it should be diagonalizable, and it will be clear from the eventual form ofp thatevery real number can be an eigenvalue. This makes some of the mathematics subtle, since there is a separateeigenvector for every point in space. In the basis wherex is diagonal, an arbitrary state can be written as asuperposition of states with eigenvaluesx:

and the operatorx multiplies each eigenvector byx.

Define a linear operator D which differentiates :

and note that:

so that the operator - iD obeys the same commutation relation as p. The difference between p and - iD must


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commute with x.

so it may be simultaneously diagonalized with x: its value acting on any eigenstate of x is some function f of teigenvalue x. This function must be real, because both p and -iD are Hermitian:

rotating each state |x> by a phase f(x), that is, redefining the phase of the wavefunction:

the operator iD is redefined by an amount:

which means that in the rotated basis, p is equal to -iD. So there is always a basis for the eigenvalues of x whe

the action of p on any wavefunction is known:

and the Hamiltonian in this basis is a linear differential operator on the state vector components:

So that the equation of motion for the state vector is the differential equation:

Since D is a differential operator, in order for it to be sensibly defined, there must be eigenvalues of x whichneighbor every given value. This suggests that the only possibility is that the space of all eigenvalues of x is areal numbers, and that p is iD up to the phase rotation. To make this rigorous requires a sensible discussion ofthe limiting space of functions, and in this space this is the Stone-von Neumann theorem--- any operators x anp which obey the commutation relations can be made to act on a space of wavefunctions, with p a derivativeoperator. This implies that a Schrdinger picture is always available.

Unlike the Schrdinger approach, matrix mechanics could be extended to many degrees of freedom in anobvious way. Each degree of freedom has a separate x operator and a separate differential operator p, and thewavefunction is a function of all the possible eigenvalues of the independent commuting x variables.


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In particular, this means that a system of N interacting particles in 3 dimensions are described by one vectorwhose components in a basis where all the X are diagonal is a mathematical function of 3N dimensional spacewhich describes all their possible positions, which is a much bigger collection of values than N threedimensional wavefunctions in physical space. Schrdinger came to the same conclusion independently, andeventually proved the equivalence of the his own formalism to Heisenberg's.

Since the wavefunction is a property of the whole system, not of any one part, the description in quantum

mechanics is not entirely local. The description of several particles can be quantumly correlated, or entangledThis entanglement leads to strange correlations between distant particles which violate the classical Bell'sinequality.

Even if the particles can only be in two positions, the wavefunction for N particles requires 2Ncomplexnumbers, one for each configuration of positions. This is exponentially many numbers in N, so simulatingquantum mechanics on a computer requires exponential resources. This suggests that it might be possible to

find quantum systems of size N which physically compute the answers to problems which classically require 2bits to solve, which is the motivation for quantum computing.

Transformation theory

Main article: Transformation theory (quantum mechanics)

In classical mechanics, a canonical transformation of phase space coordinates is one which preserves thestructure of the Poisson brackets. The new variables x',p' have the same Poisson brackets with each other as thoriginal variables x,p. Time evolution is a canonical transformation, since the phase space at any time is just agood a choice of variables as the phase space at any other time.

The Hamiltonian flow is then the canonicalcanonical transformation:

Since the Hamiltonian can be an arbitrary function of x and p, there are infinitesimal canonical transformationcorresponding to every classical quantity G, where G is used as the Hamiltonian to generate a flow of points iphase space for an increment of time s.

For a general function A(x,p) on phase space, the infinitesimal change at every step ds under the map is:


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The quantity G is called the infinitesimal generator of the canonical transformation.

In quantum mechanics, G is a Hermitian matrix, and the equations of motion are commutators:

The infinitesimal canonial motions can be formally integrated, just as the Heisenberg equation of motion wereintegrated:

where U= eiGs and s is an arbitrary parameter. The definition of a canonical transformation is an arbitraryunitary change of basis on the space of all state vectors. U is an arbitrary unitary matrix, a complex rotation inphase space.

these transformations leave the sum of the absolute square of the wavefunction components invariant, and takstates which are multiples of each other (including states which are imaginary multiples of each other ) to statewhich are the same multiple of each other.

The interpretation of the matrices is that they act as generators of motions on the space of states. The motiongenerated by P can be found by solving the Heisenberg equation of motion using P as the Hamiltonian:

They are translations of the matrix X which add a multiple of the identity: . This is also the

interpretation of the derivative operator D eiPs = eD, the exponential of a derivative operator is a translation. T

X operator likewise generates translations in P. The Hamiltonian generates translations in time, the angularmomentum generates rotations in physical space, and the operatorX2 + P2 generates rotations in phase space.

When a transformation, like a rotation in physical space, commutes with the Hamiltonian, the transformation called a symmetry. The Hamiltonian expressed in terms of rotated coordinates is the same as the originalHamiltonian. This means that the change in the Hamiltonian under the infinitesimal generator L is zero:

It follows that the change in the generator under time translation is also zero:

So that the matrix L is constant in time. The one-to-one association of infinitesimal symmetry generators andconservation laws was first discovered by Emmy Noether for classical mechanics, where the commutators arePoisson brackets but the argument is identical.


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In quantum mechanics, any unitary symmetry transformation gives a conservation law, since if the matrix U hthe property that

it follows that UH=HUand that the time derivative of U is zero.

The eigenvalues of unitary matrices are pure phases, so that the value of a unitary conserved quantity is acomplex number of unit magnitude, not a real number. Another way of saying this is that a unitary matrix is thexponential of i times a Hermitian matrix, so that the additive conserved real quantity, the phase, is only welldefined only up to an integer multiple of 2\pi. Only when the unitary symmetry matrix is part of a family thatcomes arbitrarily close to the identity are the conserved real quantities single-valued, and then the demand thathey are conserved become a much more exacting constraint.

Symmetries which can be continuously connected to the identity are called continuous, and translations,rotations, and boosts are examples. Symmetries which cannot be continuously connected to the identity arediscrete, and the operation of space-inversion, or parity, and charge conjugation are examples.

The interpretation of the matrices as generators of canonical transformations is due to Paul Dirac[25]. Thecorrespondence between symmetries and matrices was shown by Eugene Wigner to be complete, ifantiunitarmatrices which describe symmetries which include time-reversal are included.

Selection rules

It was physically clear to Heisenberg that the absolute squares of the matrix elements of X, which are the fourcoefficients of the oscillation, would be the rate of emission of electromagnetic radiation.

In the classical limit of large orbits, if a charge with position X(t) and charge q is oscillating next to an equaland opposite charge at position 0, the instantaneous dipole moment is qX(t), and the time variation of the

moment translates directly into the space-time variation of the vector potential, which produces nested outgoinspherical waves. For atoms the wavelength of the emitted light is about 10,000 times the atomic radius, thedipole moment is the only contribution to the radiative field and all other details of the atomic chargedistribution can be ignored.

Ignoring back-reaction, the power radiated in each outgoing mode is a sum of separate contributions from thesquare of each independent time Fourier mode of d:

And in Heisenberg's representation, the Fourier coefficients of the dipole moment are the matrix elements of XThe correspondence allowed Heisenberg to provide the rule for the transition intensities, the fraction of the timthat, starting from an initial state i, a photon is emitted and the atom jumps to a final state j:

This allowed the magnitude of the matrix elements to be interpreted statistically--- they give the intensity of th


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spectral lines, the probability for quantum jumps from the emission of dipole radiation.

Since the transition rates are given by the matrix elements of X, whereverXijis zero, the corresponding

transition should be absent. These were called the selection rules, and they were a puzzle before matrixmechanics.

An arbitrary state of the Hydrogen atom, ignoring spin, is labelled by |n;l,m>, where the value of l is a measur

of the total orbital angular momentum and m is its z-component, which defines the orbit orientation.

The components of the angular momentum pseudovector are:

and the products in this expression are independent of order and real, because different components of x and pcommute.

The commutation relations of L with x (or with any vector) are easy to find:

This verifies that L generates rotations between the components of the vector X.

From this, the commutator of L_z and the coordinate matrices x,y,z can be read off,

Which means that the quantities x+iy,x-iy have a simple commutation rule:

Just like the matrix elements of x+ip and x-ip for the harmonic oscillator hamiltonian, this commutation lawimplies that these operators only have certain off diagonal matrix elements in states of definite m.

meaning that the matrix (x+iy) takes an eigenvector ofLz with eigenvalue m to an eigenvector with eigenvalu

m+1. Similarly, (x-iy) decrease m by one unit, and z does not change the value of m.

So in a basis of |l,m> states whereL2 andLz have definite values, the matrix elements of any of the threecomponents of the position are zero except when m is the same or changes by one unit.

This places a constraint on the change in total angular momentum. Any state can be rotated so that its angularmomentum is in the z-direction as much as possible, where m=l. The matrix element of the position acting on|l,m> can only produce values of m which are bigger by one unit, so that if the coordinates are rotated so thatthe final state is |l',l'>, the value of l' can be at most one bigger than the biggest value of l that occurs in theinitial state. So l' is at most l+1. The matrix elements vanish for l'>l+1, and the reverse matrix element is


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determined by Hermiticity, so these vanish also when l'


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Bibliography

Jeremy BernsteinMax Born and the Quantum Theory,Am. J. Phys.73 (11) 999-1008 (2005). Departmeof Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030. Received 14 April 2005;accepted 29 July 2005.

Max Born The statistical interpretation of quantum mechanics. Nobel Lecture December 11, 1954.

Nancy Thorndike Greenspan, "The End of the Certain World: The Life and Science of Max Born" (BasBooks, 2005) ISBN 0-7382-0693-8. Also published in Germany:Max Born - Baumeister derQuantenweld. Eine Biographie (Spektrum Akademischer Verlag, 2005), ISBN 3-8274-1640-X.

Max Jammer The Conceptual Development of Quantum Mechanics (McGraw-Hill, 1966)

Jagdish Mehra and Helmut Rechenberg The Historical Development of Quantum Theory. Volume 3. ThFormulation of Matrix Mechanics and Its Modifications 19251926. (Springer, 2001) ISBN 0-387-95177-6

B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-48661881-1

Ian J. R. Aitchisona, David A. MacManus, Thomas M. Snyder, "Understanding Heisenbergs magicalpaper of July 1925: A new look at the calculational details"

Footnotes

1. W. Heisenberg, "Der Teil und das Ganze", Piper, Munich, (1969)The Birth of Quantum Mechanics.2. The Birth of Quantum Mechanics

3. W. Heisenberg, ber quantentheoretische Umdeutung kinematischer und mechanischer BeziehungenZeitschrift fr Physik, 33, 879-893, 1925 (received July 29, 1925). [English translation in: B. L. van derWaerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1(English title: Quantum-Theoretical Re-interpretation of Kinematic and Mechanical Relations).]

4. H. A. Kramers und W. Heisenberg, ber die Streuung von Strahlung durch Atome, Zeitschrift frPhysik 31, 681-708 (1925).

5. Emilio Segr, From X-Rays to Quarks: Modern Physicists and their Discoveries (W. H. Freeman andCompany, 1980) ISBN 0-7167-1147-8, pp 153 - 157.

6. Abraham Pais,Niels Bohrs Times in Physics, Philosophy, and Polity (Clarendon Press, 1991) ISBN

19-852049-2, pp 275 - 279.7. Max Born Nobel Lecture (1954)

8. M. Born and P. Jordan,Zur Quantenmechanik,Zeitschrift fr Physik, 34, 858-888, 1925 (receivedSeptember 27, 1925). [English translation in: B. L. van der Waerden, editor, Sources of QuantumMechanics (Dover Publications, 1968) ISBN 0-486-61881-1]

9. M. Born, W. Heisenberg, and P. Jordan,Zur Quantenmechanik II,Zeitschrift fr Physik, 35, 557-6151925 (received November 16, 1925). [English translation in: B. L. van der Waerden, editor, Sources ofQuantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1]


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10. Jeremy BernsteinMax Born and the Quantum Theory,Am. J. Phys.73 (11) 999-1008 (2005)11. Mehra, Volume 3 (Springer, 2001)

12. Jammer, 1966, pp. 206-207.

13. van der Waerden, 1968, p. 51.

14. The citation by Born was in Born and Jordan's paper, the second paper in the trilogy which launched

the matrix mechanics formulation. See van der Waerden, 1968, p. 351.15. Constance Ried Courant (Springer, 1996) p. 93.

16. John von NeumannAllgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren,MathematischAnnalen102 49131 (1929)

17. When von Neumann left Gttingen in 1932, his book on the mathematical foundations of quantum

mechanics, based on Hilberts mathematics, was published under the titleMathematische Grundlagen dQuantenmechanik. See: Norman Macrae,John von Neumann: The Scientific Genius Who Pioneered theModern Computer, Game Theory, Nuclear Deterrence, and Much More (Reprinted by the AmericanMathematical Society, 1999) and Constance Reid,Hilbert(Springer-Verlag, 1996) ISBN 0-387-94674-

18. Bernstein, 2004, p. 1004.

19. Greenspan, 2005, p. 190.

20. 20.020.1Nobel Prize in Physics and 1933 Nobel Prize Presentation Speech.

21. Bernstein, 2005, p. 1004.22. Bernstein, 2005, p. 1006.

23. Greenspan, 2005, p. 191.

24. Greenspan, 2005, pp. 285-286.

25. P.A.M. Dirac The Principles of Quantum Mechanics, Oxford University Press

External links

An Overview of Matrix MechanicsMatrix Methods in Quantum Mechanics

Heisenberg Quantum Mechanics (The theory's origins and its historical developing 1925-27)Werner Heisenberg 1970 CBC radio InterviewWerner Karl Heisenberg Co-founder of Quantum MechanicsIan J. R. Aitchison, David A. MacManus, Thomas M. Snyder. Understanding Heisenberg's `magical'paper of July 1925: a new look at the calculational details.

Copyright Information

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