Definitions

matrix mechanics

matrix mechanics: see quantum theory.
Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.

Matrix mechanics was the first complete and correct definition of quantum mechanics. It extended the Bohr Model by describing how the quantum jumps occur. It did so by interpreting the physical properties of particles as matrices which evolve in time. It is equivalent to the Schrödinger wave formulation of quantum mechanics, and is the basis of Dirac's bra-ket notation for the wave function.

Development of matrix mechanics

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

Epiphany at Heligoland

In 1925 Werner Heisenberg was working, in Göttingen, on the problem of calculating the spectral lines of hydrogen. By May 1925 Heisenberg began trying to describe atomic systems by observables only. On June 7, to escape the effects of a bad attack of hay fever, Heisenberg left for the pollen free North Sea island of Heligoland. While there Heisenberg, in between mountain climbing and learning by heart poems from Goethe's West-östlicher Diwan, continued to ponder the spectral issue and eventually realised that adopting non-commuting observables might solve the problem, and he later wrote

The Three Papers

After Heisenberg returned to Göttingen, he showed Wolfgang Pauli his calculations, commenting at one point: On July 9 Heisenberg gave the same paper of his calculations to Max Born, saying: prior to publication. Heisenberg then departed for a while, leaving Born to analyse the paper.

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

Heisenberg, after a collaboration with Kramers, began to understand that the transition probabilities were not quite classical quantities, because the only frequencies that appear in the Fourier series should be the ones that are 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 the Fourier series. Classically, the Fourier coefficients give the intensity of the emitted radiation, so in quantum mechanics the magnitude of the matrix elements were the intensities of spectral lines.

The quantities in Heisenberg's formulation were the classical position and momentum, but now they were no longer sharply defined. Each quantity was represented by a collection of Fourier coefficients with two indices, corresponding to the initial and final states. When Born read the paper, he recognized the formulation as one which could be transcribed and extended to the systematic language of matrices, which he had learned from his study under Jakob Rosanes at Breslau University. Born, with the help of his assistant and former student Pascual Jordan, began immediately to make the transcription and extension, and they submitted their results for publication; the paper was received for publication just 60 days after Heisenberg’s paper. A follow-on paper was submitted for publication before the end of the year by all three authors. (A brief review of Born’s role in the development of the matrix mechanics formulation of quantum mechanics along with a discussion of the key formula involving the non-commutivity of the probability amplitudes can be found in an article by Jeremy Bernstein. A detailed historical and technical account can be found in Mehra and Rechenberg’s book The Historical Development of Quantum Theory. Volume 3. The Formulation of Matrix Mechanics and Its Modifications 1925–1926.)

Up until this time, matrices were seldom used by physicists, they were considered to belong to the realm of pure mathematics. Gustav Mie had used them in a paper on electrodynamics in 1912 and Born had used them in his work on the lattices theory of crystals in 1921. While matrices were used in these cases, the algebra of matrices with their multiplication did not enter the picture as they did in the matrix formulation of quantum mechanics. Born, however, had learned matrix algebra from Rosanes, as already noted, but Born had also learned Hilbert’s theory of integral equations and quadratic forms for an infinite number of variables as was apparent from a citation by Born of Hilbert’s work Grundzüge einer allgemeinen Theorie der Linearen Integralgleichungen published in 1912. Jordan, too was well equipped for the task. For a number of years, he had been an assistant to Richard Courant at Göttingen in the preparation of Courant and David Hilbert’s book Methoden der mathematischen Physik I, which was published in 1924. This book, fortuitously, contained a great many of the mathematical tools necessary for the continued development of quantum mechanics. In 1926, John von Neumann became assistant to David Hilbert, and he would coin the term Hilbert space to describe the algebra and analysis which were used in the development of quantum mechanics.

Heisenberg's reasoning

Before matrix mechanics, the old quantum theory described the motion of a particle by a classical orbit $X\left(t\right),P\left(t\right)$ with the restriction that the time integral over one period T of the momentum times the velocity must be a positive integer multiple of Planck's constant

$int_0^T P dX = n h$

While this restriction correctly selects orbits with more or less the right energy values $E_n$, 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 neglected, it will emit radiation in a pattern which repeats itself every orbital period. The frequencies which make up the outgoing wave are then integer multiples of the orbital frequency, and this is a reflection of the fact that X(t) is periodic, so that its Fourier representation has frequencies $2pi n/T$ only.


X(t) = sum_{n=-infty}^infty e^{2pi i nt over T} X_n

The coefficients $X_n$ are complex numbers. The ones with negative frequencies must be the complex conjugates of the ones with positive frequencies, so that X(t) will always be real,

$X_n = X_\left\{-n\right\}^*$.

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 number m, the energy of the photon is $E_n - E_m$, which means that its frequency is $\left(E_n - E_m\right)/h$.

For large n and m, but with n-m relatively small, these are the classical frequencies by Bohr's correspondence principle

$E_n-E_m approx h\left(n-m\right)/T$

In the formula above, T is the classical period of either orbit n or orbit m, since the difference between them is higher order in h. But for n and m small, or if $n-m$ is large, the frequencies are not integer multiples of any single frequency.

Since the frequencies which the particle emits are the same as the frequencies in the fourier description of its motion, this suggests that something in the time-dependent description of the particle is oscillating with frequency $\left(E_n - E_m\right)/h$ . Heisenberg called this quantity $X_\left\{nm\right\}$, and demanded that it should reduce to the classical Fourier coefficients in the classical limit. For large values of n, m but with n-m relatively small, $X_\left\{nm\right\}$ is the (n-m)th fourier coefficient of the classical motion at orbit n. Since $X_\left\{nm\right\}$ has opposite frequency to $X_\left\{mn\right\}$, the condition that X is real becomes:

$X_\left\{nm\right\}=X_\left\{mn\right\}^*$.

By definition, $X_\left\{nm\right\}$ only has the frequency $\left(E_n - E_m\right)/h$, so its time evolution is simple:

$X_\left\{nm\right\}\left(t\right) = e^\left\{2pi i\left(E_n - E_m\right)t/h\right\} X_\left\{nm\right\}\left(0\right)$.

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

Given two arrays $X_\left\{nm\right\}$ and $P_\left\{nm\right\}$ describing two physical quantities, Heisenberg could form a new array of the same type by combining the terms $X_\left\{nk\right\}P_\left\{km\right\}$, 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 separately, the correspondence with Fourier series allowed Heisenberg to deduce the rule by which the arrays should be multiplied:


(XP)_{mn} = sum_{k=0}^infty X_{mk} P_{kn}

Born pointed out that this is the law of matrix multiplication, so that the position, the momentum, the energy, all the observable quantities in the theory, are interpreted as matrices. Because of the multiplication rule, the product 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 frequencies in the quantum motion are not multiples of a common frequency, the matrix elements cannot be interpreted as the Fourier coefficients of a sharp classical trajectory. Nevertheless, as matrices, $X\left(t\right)$ and $P\left(t\right)$ satisfy the classical equations of motion.

Further discussion

When it was introduced by Werner Heisenberg, Max Born and Pascual Jordan in 1925, matrix mechanics was not immediately accepted and was a source of great controversy. Schrödinger's later introduction of wave mechanics was favored.

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

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

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 measurable by spectroscopy should appear in the theory. These quantities include the energy levels and their intensities but they do not include the exact location of a particle in its Bohr orbit. It is very hard to imagine an experiment which could determine whether an electron in the ground state of a hydrogen atom is to the right or to the left of 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 whose elements are indexed by two different energy levels. The set of eigenvalues of the matrix were eventually understood to be the set of all possible values that the observable can have. Since Heisenberg's matrices are Hermitian, the eigenvalues are real.

If an observable is measured and the result is a certain eigenvalue, the corresponding eigenvector is the state of the system immediately after the measurement. The act of measurement in matrix mechanics 'collapses' the state of the system. If you measure two observables simultaneously, the state of the system should collapse to a common eigenvector of the two observables. Since most matrices don't have any eigenvectors in common, most observables 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 are both diagonal, it is clear that their product does not depend on their order because multiplication of diagonal matrices is just multiplication of numbers. The Uncertainty Principle then is a consequence of the fact that two matrices A and B do not always commute, A B - B A does not necessarily equal 0. The famous commutation relation of matrix mechanics:

$sum_k \left(q_\left\{nk\right\} p_\left\{km\right\} - p_\left\{nk\right\} q_\left\{km\right\}\right) = \left\{ihover 2pi\right\} delta_\left\{nm\right\}$

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

In 1925, Werner Heisenberg was not yet 24 years old.

Nobel Prize

In 1928, Albert Einstein nominated Heisenberg, Born, and Jordan for the Nobel Prize in Physics, but it was not to be. The announcement of the Nobel Prize in Physics for 1932 was delayed until November 1933. It was 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” and Erwin Schrödinger and Paul Adrien Maurice Dirac shared the 1933 Prize "for the discovery of new productive forms of atomic theory". One can rightly ask why Born was not awarded the Prize in 1932 along with Heisenberg, and Bernstein gives some speculations on this matter. One of them is related to Jordan joining the Nazi Party on May 1, 1933 and becoming a Storm Trooper. Hence, Jordan’s Party affiliations and Jordan’s links to Born may have affected Born’s chance at the Prize at that time. Bernstein also notes that when Born won the Prize in 1954, Jordan was still alive, and the Prize was awarded for the statistical interpretation of quantum mechanics, attributable alone to Born.

Heisenberg’s reaction to Born for Heisenberg receiving the Prize for 1932 and to Born for Born receiving the Prize in 1954 are also instructive in evaluating whether Born should have shared the Prize with Heisenberg. On November 25, 1933 Born received a letter from Heisenberg in which he said he had been delayed in writing due to a “bad conscience” that he alone had received the Prize “for work done in Göttingen in collaboration – you, Jordan and I.” Heisenberg went on to say that Born and Jordan’s contribution to quantum mechanics cannot be changed by “a wrong decision from the outside.” In 1954, Heisenberg wrote an article honoring Max Planck for his insight in 1900. In the article, Heisenberg credited Born and Jordan for the final mathematical formulation 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.”

Mathematical development

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

Harmonic oscillator

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

$H = \left\{1 over 2\right\} \left(P^2 + X^2\right)$

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

$X\left(t\right)= sqrt\left\{2E\right\}cos\left(t\right) ;;;; P\left(t\right) = sqrt\left\{2E\right\}sin\left(t\right)$

The old quantum condition says that the integral of P dX over an orbit, which is the area of the circle in phase space, must be an integer multiple of Planck's constant. The area of the circle of radius $scriptstyle sqrt\left\{2E\right\}$ is $2pi E$ So

$E = \left\{n h over 2pi\right\}$

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

The Fourier components of X(t) and P(t) are very simple, even more so if they are combined into the quantities:


A(t) = X(t) + i P(t) = sqrt{2E},e^{it}

A^dagger(t) = X(t) - i P(t) = sqrt{2E},e^{-it}

both $scriptstyle A$ and $scriptstyle A^dagger$ have only a single frequency, and $X$ and $P$ can be recovered from their sum and difference.

Since $A\left(t\right)$ has a classical fourier series with only the lowest frequency, and the matrix element $A_\left\{mn\right\}$ is the (m-n)th fourier coefficient of the classical orbit, the matrix for $scriptstyle A$ is nonzero only on the line just above the diagonal, where it is equal to $scriptstyle sqrt\left\{2E_n\right\}$. The matrix for $scriptstyle A^dagger$ is likewise only nonzero on the line below the diagonal, with the same elements. Reconstructing X and P from $scriptstyle A$ and $scriptstyle A^dagger$:


sqrt{2} X(0)= begin{bmatrix} 0 & sqrt{1} & 0 & 0 & ldots sqrt{1} & 0 & sqrt{2} & 0 & 0 & ldots 0 & sqrt{2} & 0 & sqrt{3} & 0 & ldots 0 & 0 & sqrt{3} & 0 & sqrt{4} & ldots vdots & vdots & & ddots & ddots & ddots end{bmatrix}


sqrt{2} P(0) = begin{bmatrix} 0 & isqrt{1} & 0 & 0 & ldots -isqrt{1} & 0 & isqrt{2} & 0 & 0 & ldots 0 & -isqrt{2} & 0 & isqrt{3} & 0 & ldots 0 & 0 & -isqrt{3} & 0 & isqrt{4} & ldots vdots & vdots & & ddots & ddots & ddots end{bmatrix}

which, up to the choice of units, are the Heisenberg matrices for the harmonic oscillator. Notice that both matrices are hermitian, since they are constructed from the Fourier coefficients of real quantities. To find $X\left(t\right)$ and $P\left(t\right)$ is simple, since they are quantum Fourier coefficients so they evolve simply with time.


X_{mn}(t) = X_{mn}(0) e^{i(E_m - E_n)t} ;;;; P_{mn}(t) = P_{mn}(0) e^{i(E_m -E_n)t}

The matrix product of $X$ and $P$ is not hermitian, but has a real and imaginary part. The real part is one half the symmetric expression $\left(XP + PX\right)$, while the imaginary part is proportional to the commutator $\left[X,P\right]=\left(XP - PX\right)$. It is easy to verify explicitly that $\left(X P -P X\right)$ in the case of the harmonic oscillator, is i, multiplied by the identity. It is also easy to verify that the matrix


H ={1over 2}(X^2 + P^2) is a diagonal matrix, with eigenvalues E_i.

Conservation of energy

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


H = {1over 2} P^2 + {1over 2} X^2 + epsilon X^3

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


{dX over dt} = P ;;;;;;;; {dP over dt} = - X - 3 epsilon X^2

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


{dHover dt} = P*{dPover dt} + (X + 3 epsilon X^2)*{dXover dt} = 0 Where $A*B$ is the symmetric product.


A*B = {1over 2}(AB+BA) .

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

The process of emission and absorption of photons seemed to demand that the conservation of energy will hold at best on average. If a wave containing exactly one photon passes over some atoms, and one of them absorbs it, that atom needs to tell the others that they can't absorb the photon anymore. But if the atoms are far apart, any signal cannot reach the other atoms in time, and they might end up absorbing the same photon anyway and dissipating the energy to the environment. When the signal reached them, the other atoms would have to somehow recall that energy. This paradox led Bohr, Kramers and Slater to abandon exact conservation of energy. Heisenberg's formalism, when extended to include the electromagnetic field, was obviously going to sidestep 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 determine the matrix elements. Planck's constant does not appear in the classical equations, so that the matrices could be constructed for many different values of $scriptstyle hbar$ and still satisfy the equations of motion, but with different energy levels.

So in order to implement his program, Heisenberg needed to use the old quantum condition to fix the energy levels, then fill in the matrices with Fourier coefficients of the classical equations, then alter the matrix coefficients 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 not exist in the new formalism.

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


int_0^T sum_k P_{mk}(t) {dX_{kn} over dt} dt approx ? J_{mn} ,

There are several problems with this integral, all stemming from the incompatibility of the matrix formalism with the old picture of orbits. Which period T should you use? Semiclassically, it should be either m or n, but the difference is order h and we want an answer to order h. The quantum condition tells us that $J_\left\{mn\right\}$ is $2pi n$ 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 complete sense in the classical limit, where n is not an integer but the continuous action variable J, but Heisenberg performed analogous manipulations with matrices, where the intermediate expressions are sometimes discrete differences and sometimes derivatives. In the following discussion, for the sake of clarity, the differentiation will be performed on the classical variables, and the transition to matrix mechanics will be done afterwards using the correspondence principle.

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


{d over dJ } int_0^T P dX = 1 ,

$= int_0^T dt left\left(\left\{dpover dJ\right\} \left\{dXover dt\right\} + p\left\{dover dJ\right\}\left\{dXover dt\right\} right\right) = int_0^T dt left\left(\left\{dpover dJ\right\} \left\{dXover dt\right\} - \left\{dpover dt\right\}\left\{dXover dJ\right\} right\right) ,$

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


{2piover T} int_0^T dt ;; {dp over dJ} {dXover dtheta} - {dP over dtheta} {dXover dJ} ,

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

The Poisson Bracket, unlike the action integral, has a simple translation to matrix mechanics--- it is the imaginary 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 the index, keeping in mind that the answer is zero classically.

In the correspondence limit, when indices m n are large and nearby, while k,r are small, the rate of change of the matrix elements in the diagonal direction is the matrix element of the J derivative of the corresponding classical quantity. So we can shift any matrix element diagonally using the formula:


A_{(m+r) (n+r)} - A_{mn} approx r; ({dAover dJ})_{m n} ,

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 the distance from the diagonal,


ik A_{m (m+k)} approx ({Tover 2pi} {dAover dt})_{m (m+k)} =({dAover dtheta})_{m (m+k)} ,

Since the coefficient $A_\left\{m \left(m+k\right)\right\}$ 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 the shifting. Since all the matrix elements are at indices which have a small distance from the large index position $\left(m,m\right)$, it helps to introduce two temporary notations: $A\left[r,k\right] = A_\left\{\left(m+r\right)\left(m+k\right)\right\}$, for the matrices, and $\left(dA/dJ\right)\left[r\right]$ for the r'th Fourier components of classical quantities.


(AB - BA)(0,k) = sum_{r=-infty}^{infty} (A[0,r] B[r,k] - A[r,k] B[0,r] )


= sum_r (; A[-r+k,k] + (r-k){dA over dJ}[r]; ) (; B[0,k-r] + r {dBover dJ}[r-k] ; ) - sum_r A(r,k)B(0,r) ,

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

$sum_\left\{r\text{'}\right\} \left(;A\left[r\text{'},k\right] - r\text{'} \left\{dA over dJ\right\}\left[k-r\text{'}\right];\right)\left(; B\left[0,r\text{'}\right] +\left(k-r\text{'}\right)\left\{dBover dJ\right\}\left[r\text{'}\right];\right)- sum_r A\left[r,k\right] B\left[0,r\right]$
,

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

$sum_\left\{r\text{'}\right\} \left(; \left\{dBover dJ\right\}\left[r\text{'}\right]\left(k-r\text{'}\right)A\left[r\text{'},k\right] - \left\{dAover dJ\right\}\left[k-r\text{'}\right] r\text{'} B\left[0,r\text{'}\right]\right)$

$sum_\left\{r\text{'}\right\} \left(; \left\{dBover dJ\right\}\left[r\text{'}\right]i\left\{dAover dtheta\right\}\left[k-r\text{'}\right] - \left\{dAover dJ\right\}\left[k-r\text{'}\right]i\left\{dBover dtheta\right\}\left[r\text{'}\right]\right)$
,

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

Once they were able to establish that:


[X , P ] = XP - PX = i {X,P}_mathrm{PB} = i ,

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 was assumed to be universally true, even though the derivation from the old quantum theory required semiclassical reasoning.

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 $| psi rangle$, which is the vector that the matrices act on. Without the state vector, it is not clear which particular motion the Heisenberg matrices are describing, since they include all the motions somewhere.

The interpretation of the state vector, whose components are written $psi_m$, was given by Born. The interpretation is statistical: the result of a measurement of the physical quantity corresponding to the matrix A is random, with an average value equal to


sum_{mn} psi_m^* A_{mn} psi_n

Alternatively and equivalently, the state vector gives the probability amplitude $psi_i$ for the quantum system to be in the energy state i. Once the state vector was introduced, matrix mechanics could be rotated to any basis, where the H matrix was no longer diagonal. The Heisenberg equation of motion in its original form states that $A_\left\{mn\right\}$ evolves in time like a Fourier component


A_{mn}(t) = e^{i(E_m - E_n)t} A_{mn} (0)

which can be recast in differential form


{dA_{mn}over dt} = i(E_m - E_n ) A_{mn}

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


{dAover dt} = i(H A - A H )

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

$,$
A(t) = e^{iHt} A(0) e^{-iHt}

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 $e^\left\{iHt\right\}$. By rotating the basis for the state vector at each time by $e^\left\{iHt\right\}$, you can undo the time dependence in the matrices. The matrices are now time independent, but the state vector rotates:


| psi(t) rangle = e^{-iHt} | psi(0) rangle, ;;;; {d |psi rangle over dt} = - i H | psi rangle

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

In quantum mechanics in the Heisenberg picture the state vector, $| psi rangle$ does not change with time, and an observable A satisfies

$frac\left\{dA\right\}\left\{dt\right\} = \left\{i over hbar \right\} \left[H , A\left(t\right) \right] + frac\left\{partial A\right\}\left\{partial t\right\}$

The extra term is for operators like $A=\left(X+ t^2 P\right)$ which have an explicit time dependence in addition to the time dependence from unitary evolution. The Heisenberg picture does not distinguish time from space, so it is nicer for relativistic theories.

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

By the Stone-von Neumann theorem, the Heisenberg picture and the Schrödinger picture are unitarily equivalent.

Further results

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

Wave mechanics

Jordan noted that the commutation relations ensure that p acts as a differential operator, and came very close to formulating the Schrödinger equation. The identity


[a,bc] = abc - bca = abc - bac + bac - bca = [a,b]c + b[a,c] ,

allows the evaluation of the commutator of p with any power of x, and it implies that


[p,x^n] = - i nx^{n-1} ,

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


[p,f(x)] = -i f'(x) ,

Since x is a Hermitian matrix, it should be diagonalizable, and it will be clear from the eventual form of p that every real number can be an eigenvalue. This makes some of the mathematics subtle, since there is a separate eigenvector for every point in space. In the basis where x is diagonal, an arbitrary state can be written as a superposition of states with eigenvalues x:


|psirangle = int_x psi(x)|xrangle ,

and the operator x multiplies each eigenvector by x.


x |psi> = int_x x psi(x) |xrangle ,

Define a linear operator D which differentiates $psi$:


D int_x psi(x) | xrangle = int_x psi'(x) |xrangle , and note that:

(D x - x D) |psirangle = int_x ((x psi(x))' - x psi'(x) ) |xrangle = int_x psi(x) |xrangle = |psirangle ,

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


[p+iD,x]=0 , so it may be simultaneously diagonalized with x: its value acting on any eigenstate of x is some function f of the eigenvalue x. This function must be real, because both p and -iD are Hermitian:

(p+iD ) |xrangle = f(x) |xrangle ,

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


psi(x) rightarrow e^{-if(x)} psi(x) ,

the operator iD is redefined by an amount:


iD rightarrow iD + f(x) ,

which means that in the rotated basis, p is equal to -iD. So there is always a basis for the eigenvalues of x where the action of p on any wavefunction is known:


p int_x psi(x) |xrangle = int_x - i psi'(x) |xrangle ,

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


({p^2over 2m} + V(x) ) int_x psi_x |xrangle = int_x (-{1over 2m}{partial^2 over partial x^2} + V(x)) psi_x |xrangle

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


i{partial over partial t} psi_t(x) = (-{1over 2m} {partial^2 over partial x^2} + V(x)) psi_t(x) ,

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

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


[X_i ,X_j ] = 0 ,

[P_i, P_j ] = 0 ,

[X_i ,P_j ] = idelta_{ij} ,

In particular, this means that a system of N interacting particles in 3 dimensions are described by one vector whose components in a basis where all the X are diagonal is a mathematical function of 3N dimensional space which describes all their possible positions, which is a much bigger collection of values than N three dimensional wavefunctions in physical space. Schrödinger came to the same conclusion independently, and eventually 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 entangled. This entanglement leads to strange correlations between distant particles which violate the classical Bell's inequality.

Even if the particles can only be in two positions, the wavefunction for N particles requires $2^N$ complex numbers, one for each configuration of positions. This is exponentially many numbers in N, so simulating quantum 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 $2^N$ bits to solve, which is the motivation for quantum computing.

Transformation theory

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

The Hamiltonian flow is then the canonical canonical transformation:


xrightarrow x+dx = x + {partial H over partial p} dt

p rightarrow p+dp = p -{partial H over partial x} dt

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


dx = {partial G over partial p} ds = { G,X } ds ,

dp = -{partial G over partial x} ds = { G,P } ds ,

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


dA = {partial A over partial x} dx + {partial Aover partial p} dp = { A,G} ds ,

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:


dA = i [G,A] ds ,

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


A' = U^{dagger} A U ,

where $U=e^\left\{iGs\right\}$ and s is an arbitrary parameter. The definition of a canonical transformation is an arbitrary unitary change of basis on the space of all state vectors. U is an arbitrary unitary matrix, a complex rotation in phase space.


U^{dagger}= U^{-1} , these transformations leave the sum of the absolute square of the wavefunction components invariant, and take states which are multiples of each other (including states which are imaginary multiples of each other ) to states which 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 motion generated by P can be found by solving the Heisenberg equation of motion using P as the Hamiltonian:


dX = i[X,P] ds = ds ,

dP = i[P,P] ds = 0 ,

They are translations of the matrix X which add a multiple of the identity: $Xrightarrow X+s$. This is also the interpretation of the derivative operator D $e^\left\{iPs\right\} = e^D$, the exponential of a derivative operator is a translation. The X operator likewise generates translations in P. The Hamiltonian generates translations in time, the angular momentum generates rotations in physical space, and the operator $X^2+P^2$ generates rotations in phase space.

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


{dHover ds} = i[L,H] = 0 ,

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


{dLover dt} = i[H,L] = 0 ,

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

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


U^{-1} H U = H , it follows that $UH=HU$ and 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 a complex number of unit magnitude, not a real number. Another way of saying this is that a unitary matrix is the exponential of i times a Hermitian matrix, so that the additive conserved real quantity, the phase, is only well defined only up to an integer multiple of 2pi. Only when the unitary symmetry matrix is part of a family that comes arbitrarily close to the identity are the conserved real quantities single-valued, and then the demand that they 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 are discrete, and the operation of space-inversion, or parity, and charge conjugation are examples.

The interpretation of the matrices as generators of canonical trasformations is due to Paul Dirac. The correspondence between symmetries and matrices was shown by Eugene Wigner to be complete, if antiunitary matrices 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 fourier coefficients 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 equal and 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 outgoing spherical waves. For atoms the wavelength of the emitted light is about 10,000 times the atomic radius, the dipole moment is the only contribution to the radiative field and all other details of the atomic charge distribution can be ignored.

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


P(omega) = {2over 3} {omega^4} |d_i|^2 ,

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


P_{ij} = {2 over 3} (E_i -E_j)^4 |X_{ij}|^2 ,

This allowed the magnitude of the matrix elements to be interpreted statistically--- they give the intensity of the 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, wherever $X_\left\{ij\right\}$ is zero, the corresponding transition should be absent. These were called the selection rules, and they were a puzzle before matrix mechanics.

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


L_i = epsilon_{ijk} x^j p^k , and the products in this expression are independent of order and real, because different components of x and p commute.

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


[L_i, x_j] = iepsilon_{ijk} x_k , 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,


[L_z , x] = iy ,

[L_z,y] = -ix ,

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


[L_z,x+iy] = (x+iy) ,

[L_z,x-iy] = -(x-iy) ,

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


L_z ((x+iy)|mrangle )= (x+iy)L_z|mrangle + (x+iy) |mrangle = (m+1) (x+iy)|mrangle ,

meaning that the matrix (x+iy) takes an eigenvector of $L_z$ with eigenvalue m to an eigenvector with eigenvalue 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 where $L^2$ and $L_z$ have definite values, the matrix elements of any of the three components 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 angular momentum 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 that the final state is |l',l'>, the value of l' can be at most one bigger than the biggest value of l that occurs in the initial state. So l' is at most l+1. The matrix elements vanish for l'>l+1, and the reverse matrix element is determined by Hermiticity, so these vanish also when l' Sum rules

The Heisenberg equation of motion determine the matrix elements of p in the Heisenberg basis from the matrix elements of x.


p_{ij} = m{dover dt} x_{ij} = im (E_i - E_j) x_{ij} ,

which turns the diagonal part of the commutation relation into a sum rule for the magnitude of the matrix elements:


sum_j p_{ij}x_{ji} - x_{ij}p_{ji} = i sum_j 2m(E_i - E_j) |x_{ij}|^2 = i. ,

This gives a relation for the sum of the spectroscopic intensities to and from any given state, although to be absolutely correct, contributions from the radiative capture probability for unbound scattering states must be included in the sum:


sum_j 2m(E_i - E_j) |x_{ij}|^2 = 1. ,

The three formulating papers

• W. Heisenberg, Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Zeitschrift für Physik, 33, 879-893, 1925 (received July 29, 1925). [English translation in: B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1 (English title: Quantum-Theoretical Re-interpretation of Kinematic and Mechanical Relations).]
• M. Born and P. Jordan, Zur Quantenmechanik, Zeitschrift für Physik, 34, 858-888, 1925 (received September 27, 1925). [English translation in: B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1 (English title: On Quantum Mechanics).]
• M. Born, W. Heisenberg, and P. Jordan, Zur Quantenmechanik II, Zeitschrift für Physik, 35, 557-615, 1925 (received November 16, 1925). [English translation in: B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1 (English title: On Quantum Mechanics II).]

Bibliography

• Jeremy Bernstein Max Born and the Quantum Theory, Am. J. Phys. 73 (11) 999-1008 (2005). Department of 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" (Basic Books, 2005) ISBN 0-7382-0693-8. Also published in Germany: Max Born - Baumeister der Quantenweld. Eine Biographie (Spektrum Akademischer Verlag, 2005), ISBN 3-8274-1640-X.
• Max Jammer The Conceptual Development of Quantum Mechanics (McGraw-Hill, 1966)
• Jagdesh Mehra and Helmut Rechenberg The Historical Development of Quantum Theory. Volume 3. The Formulation of Matrix Mechanics and Its Modifications 1925–1926. (Springer, 2001) ISBN 0-387-95177-6
• B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1