Matrix mechanics

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 the classical position and momentum, X(t) and P(t), are sinusoidal.

Harmonic oscillator

In units where the mass and frequency of the oscillator are equal to one (see nondimensionalization), the energy of the oscillator is $${\displaystyle H=\frac{1}{2}\left(P^2+X^2\right).}$$

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

$X(t)=+\sqrt{2E}\cos (t)$ and $P(t)=-\sqrt{2E}\sin (t).$

The old quantum condition dictates 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 the Planck constant. The area of the circle of radius √2 E is 2π E . So $${\displaystyle E=\frac{nh}{2\pi }=n\hslash ,}$$ or, in natural units where ħ ≡ 1 , the energy becomes some whole number.

The Fourier components of X( t ) and P( t ) are simple, and even more so when they are re-expressed as a sum and difference of position X and momentum P :

$A(t)=X(t)+iP(t)=\sqrt{2E}{e}^{-it}$ and $A^{†}(t)=X(t)-iP(t)=\sqrt{2E}{e}^{+it}.$

Both A and A^† have only a single frequency, and X and P can be recovered from the similar sum and difference of A and A^† .

Since A( t ) has a classical Fourier series with only the lowest frequency, and the matrix element A_mn is the (m − n)th Fourier coefficient of the classical orbit, the matrix for A is nonzero only on the line just above the diagonal, where it is equal to √2E_n . The matrix for A^† is likewise only nonzero on the line below the diagonal, with the same elements. Thus, from A and A^† , reconstruction yields $${\displaystyle \sqrt{2}X(0)=\sqrt{\hslash }\left[\begin{array}{cccccc}0& \sqrt{1}& 0& 0& 0& \cdots \\ \sqrt{1}& 0& \sqrt{2}& 0& 0& \cdots \\ 0& \sqrt{2}& 0& \sqrt{3}& 0& \cdots \\ 0& 0& \sqrt{3}& 0& \sqrt{4}& \cdots \\ ⋮& ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right],}$$ and $${\displaystyle \sqrt{2}P(0)=\sqrt{\hslash }\left[\begin{array}{cccccc}0& -i\sqrt{1}& 0& 0& 0& \cdots \\ i\sqrt{1}& 0& -i\sqrt{2}& 0& 0& \cdots \\ 0& i\sqrt{2}& 0& -i\sqrt{3}& 0& \cdots \\ 0& 0& i\sqrt{3}& 0& -i\sqrt{4}& \cdots \\ ⋮& ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right],}$$ which, up to the choice of units, are the Heisenberg matrices for the harmonic oscillator. Both matrices are Hermitian, since they are constructed from the Fourier coefficients of real quantities.

Finding X( t ) and P( t ) is direct, since they are quantum Fourier coefficients so they evolve simply with time, as

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

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 X P + P X , while the imaginary part is proportional to the commutator, which is written as $${\displaystyle [X,P]\equiv XP-PX.}$$ In the special case of the harmonic oscillator, it is simple to verify explicitly that X P − P X is i ħ I , where I is the identity matrix.

It is likewise simple to verify that the matrix $${\displaystyle H=\frac{1}{2}\left(X^2+P^2\right)}$$ is a diagonal matrix, with eigenvalues E_i .

Conservation of energy

The harmonic oscillator is an important case. Finding the matrices is easier than determining the general conditions from these special forms. For this reason, Heisenberg investigated the anharmonic oscillator, with Hamiltonian $${\displaystyle H=\frac{1}{2}{P}^2+\frac{1}{2}{X}^2+\epsilon X^3.}$$

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, $${\displaystyle \frac{dX}{dt}=P,\frac{dP}{dt}=-X-3\epsilon X^2.}$$

He noticed that if this could be done, then H, considered as a matrix function of X and P, will have zero time derivative. $${\displaystyle \frac{dH}{dt}=P\ast \frac{dP}{dt}+\left(X+3\epsilon X^2\right)\ast \frac{dX}{dt}=0,}$$ where A∗B is the anticommutator, $${\displaystyle A\ast B=\frac{1}{2}(AB+BA).}$$

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. The Planck constant does not appear in the classical equations, so that the matrices could be constructed for many different values of ħ 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, $${\displaystyle {\int }_0^T\sum _k{P}_{mk}(t)\frac{d{X}_{kn}}{dt}dt\stackrel{?}{\approx }{J}_{mn}.}$$

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 be used? Semiclassically, it should be either m or n, but the difference is order ħ, and an answer to order ħ is sought. The quantum condition tells us that J_mn is 2πn on the diagonal, so 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, guided by 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. $${\displaystyle \begin{array}{cc}\frac{d}{dJ}{\int }_0^{T}PdX& =1\\ & ={\int }_0^{T}dt\left(\frac{dP}{dJ}\frac{dX}{dt}+P\frac{d}{dJ}\frac{dX}{dt}\right)\\ & ={\int }_0^{T}dt\left(\frac{dP}{dJ}\frac{dX}{dt}-\frac{dP}{dt}\frac{dX}{dJ}\right)\end{array}}$$ where the derivatives ⁠dP/dJ⁠ and ⁠dX/dJ⁠ should be interpreted as differences with respect to J at corresponding times on nearby orbits, exactly what would be obtained if the Fourier coefficients of the orbital motion were differentiated. (These derivatives are symplectically orthogonal in phase space to the time derivatives ⁠dP/dt⁠ and ⁠dX/dt⁠).

The final expression is clarified by introducing the variable canonically conjugate to J, which is called the angle variable θ: The derivative with respect to time is a derivative with respect to θ, up to a factor of 2πT, $${\displaystyle \frac{2\pi }{T}{\int }_0^{T}dt\left(\frac{dP}{dJ}\frac{dX}{d\theta }-\frac{dP}{d\theta }\frac{dX}{dJ}\right)=1.}$$ 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, unlike Dirac, were not familiar with the theory of Poisson brackets, so, for them, the differentiation effectively evaluated {X, P} in J,θ coordinates.

The Poisson Bracket, unlike the action integral, does have a simple translation to matrix mechanics – it normally corresponds to the imaginary part of the product of two variables, the commutator.

To see this, examine the (antisymmetrized) 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 it is possible to shift any matrix element diagonally through the correspondence, $${\displaystyle A_{(m+r)(n+r)}-A_{mn}\approx r{\left(\frac{dA}{dJ}\right)}_{mn}}$$ where the right hand side is really only the (m − n)th Fourier component of ⁠dA/dJ⁠ at the 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, $${\displaystyle ik{A}_{m(m+k)}\approx {\left(\frac{T}{2\pi }\frac{dA}{dt}\right)}_{m(m+k)}={\left(\frac{dA}{d\theta }\right)}_{m(m+k)}.}$$ since the coefficient A_m(m+k) is semiclassically the kth Fourier coefficient of the mth 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 (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 rth Fourier components of classical quantities, $${\displaystyle \begin{array}{cc}(AB-BA)[0,k]& =\sum _{r=-\mathrm{\infty }}^{\mathrm{\infty }}(A[0,r]B[r,k]-A[r,k]B[0,r])\\ & =\sum _r\left(A[-r+k,k]+(r-k)\frac{dA}{dJ}[r]\right)\left(B[0,k-r]+r\frac{dB}{dJ}[r-k]\right)-\sum _{r}A[r,k]B[0,r].\end{array}}$$

Flipping the summation variable in the first sum from r to r′ = k − r, the matrix element becomes, $${\displaystyle \sum _{r^{\prime }}\left(A[r^{\prime },k]-r^{\prime }\frac{dA}{dJ}[k-r^{\prime }]\right)\left(B[0,r^{\prime }]+(k-r^{\prime })\frac{dB}{dJ}[r^{\prime }]\right)-\sum _{r}A[r,k]B[0,r]}$$ and it is clear that the principal (classical) part cancels.

The leading quantum part, neglecting the higher order product of derivatives in the residual expression, is then equal to $${\displaystyle \sum _{r^{\prime }}\left(\frac{dB}{dJ}[r^{\prime }](k-r^{\prime })A[r^{\prime },k]-\frac{dA}{dJ}[k-r^{\prime }]r^{\prime }B[0,r^{\prime }]\right)}$$ so that, finally, $${\displaystyle (AB-BA)[0,k]=\sum _{r^{\prime }}\left(\frac{dB}{dJ}[r^{\prime }]i\frac{dA}{d\theta }[k-r^{\prime }]-\frac{dA}{dJ}[k-r^{\prime }]i\frac{dB}{d\theta }[r^{\prime }]\right)}$$ which can be identified with i times the kth classical Fourier component of the Poisson bracket.

Heisenberg's original differentiation trick 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 $${\displaystyle i\hslash \{X,P{\}}_{\mathrm{P}\mathrm{B}}⟼[X,P]\equiv XP-PX=i\hslash ,}$$ 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. (A full quantum treatment, however, for more elaborate arguments of the brackets, was appreciated in the 1940s to amount to extending Poisson brackets to Moyal brackets.)

State vectors and the Heisenberg equation

To make the transition to standard quantum mechanics, the most important further addition was the quantum state vector, now written |ψ⟩, 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 ψ_m, was furnished by Born. This 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 $${\displaystyle \sum _{mn}{\psi }_m^{\ast }{A}_{mn}{\psi }_n.}$$ Alternatively, and equivalently, the state vector gives the probability amplitude ψ_n for the quantum system to be in the energy state n.

Once the state vector was introduced, matrix mechanics could be rotated to any basis, where the H matrix need no longer be diagonal. The Heisenberg equation of motion in its original form states that A_mn evolves in time like a Fourier component, $${\displaystyle A_{mn}(t)=e^{i(E_m-E_n)t}{A}_{mn}(0),}$$ which can be recast in differential form $${\displaystyle \frac{d{A}_{mn}}{dt}=i(E_m-E_n)A_{mn},}$$ 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, $${\displaystyle \frac{dA}{dt}=i(HA-AH).}$$ This is now a matrix equation, so it holds in any basis. This is the modern form of the Heisenberg equation of motion.

Its formal solution is: $${\displaystyle A(t)=e^{iHt}A(0)e^{-iHt}.}$$

All these forms of the equation of motion above say the same thing, that A(t) is equivalent to A(0), through a basis rotation by the unitary matrix e^iHt, a systematic picture elucidated by Dirac in his bra–ket notation.

Conversely, by rotating the basis for the state vector at each time by e^iHt, the time dependence in the matrices can be undone. The matrices are now time independent, but the state vector rotates, $${\displaystyle |\psi (t)⟩=e^{-iHt}|\psi (0)⟩,\frac{d|\psi ⟩}{dt}=-iH|\psi ⟩.}$$ This is the Schrödinger equation for the state vector, and this time-dependent change of basis amounts to transformation to the Schrödinger picture, with ⟨x|ψ⟩ = ψ(x).

In quantum mechanics in the Heisenberg picture the state vector, |ψ⟩ does not change with time, while an observable A satisfies the Heisenberg equation of motion,

${\displaystyle \frac{dA}{dt}=\frac{i}{\hslash }[H,A]+\frac{\mathrm{\partial }A}{\mathrm{\partial }t}.}$

The extra term is for operators such as $${\displaystyle A=\left(X+t^{2}P\right)}$$ which have an explicit time dependence, in addition to the time dependence from the unitary evolution discussed.

The Heisenberg picture does not distinguish time from space, so it is better suited to relativistic theories than the Schrödinger equation. Moreover, the similarity to classical physics is more manifest: the Hamiltonian equations of motion for classical mechanics are recovered by replacing the commutator above by the Poisson bracket (see also below). By the Stone–von Neumann theorem, the Heisenberg picture and the Schrödinger picture must be unitarily equivalent, as detailed below.

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.

The operator identity $${\displaystyle [a,bc]=abc-bca=abc-bac+bac-bca=[a,b]c+b[a,c]}$$ allows the evaluation of the commutator of P with any power of X, and it implies that $${\displaystyle \left[P,X^n\right]=-in{X}^{n-1}}$$ which, together with linearity, implies that a P-commutator effectively differentiates any analytic matrix function of X.

Assuming limits are defined sensibly, this extends to arbitrary functions−but the extension need not be made explicit until a certain degree of mathematical rigor is required,

${\displaystyle [P,f(X)]=-i{f}^{\prime }(X).}$

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, $${\displaystyle |\psi ⟩={\int }_x\psi (x)|x⟩,}$$ so that ψ(x) = ⟨x|ψ⟩, and the operator X multiplies each eigenvector by x, $${\displaystyle X|\psi ⟩={\int }_{x}x\psi (x)|x⟩.}$$

Define a linear operator D which differentiates ψ, $${\displaystyle D{\int }_x\psi (x)|x⟩={\int }_x{\psi }^{\prime }(x)|x⟩,}$$ and note that $${\displaystyle (DX-XD)|\psi ⟩={\int }_x\left[{\left(x\psi (x)\right)}^{\prime }-x{\psi }^{\prime }(x)\right]|x⟩={\int }_x\psi (x)|x⟩=|\psi ⟩,}$$ so that the operator −iD obeys the same commutation relation as P. Thus, the difference between P and −iD must commute with X, $${\displaystyle [P+iD,X]=0,}$$ 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, $${\displaystyle (P+iD)|x⟩=f(x)|x⟩,}$$ rotating each state |x⟩ by a phase f(x), that is, redefining the phase of the wavefunction: $${\displaystyle \psi (x)\to e^{-if(x)}\psi (x).}$$ The operator iD is redefined by an amount: $${\displaystyle iD\to iD+f(X),}$$ which means that, in the rotated basis, P is equal to −iD.

Hence, there is always a basis for the eigenvalues of X where the action of P on any wavefunction is known: $${\displaystyle P{\int }_x\psi (x)|x⟩={\int }_x-i{\psi }^{\prime }(x)|x⟩,}$$ and the Hamiltonian in this basis is a linear differential operator on the state-vector components, $${\displaystyle \left[\frac{P^2}{2m}+V(X)\right]{\int }_x{\psi }_x|x⟩={\int }_x\left[-\frac{1}{2m}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+V(x)\right]{\psi }_x|x⟩}$$

Thus, the equation of motion for the state vector is but a celebrated differential equation,

${\displaystyle i\frac{\mathrm{\partial }}{\mathrm{\partial }t}{\psi }_t(x)=\left[-\frac{1}{2m}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+V(x)\right]{\psi }_t(x).}$

Since D is a differential operator, in order for it to be sensibly defined, there must be eigenvalues of X which neighbors 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 a 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.

Matrix mechanics easily extends to many degrees of freedom in a natural way. Each degree of freedom has a separate X operator and a separate effective differential operator P, and the wavefunction is a function of all the possible eigenvalues of the independent commuting X variables. $${\displaystyle \begin{array}{cc}\left[X_i,X_j\right]& =0\\ \left[P_i,P_j\right]& =0\\ \left[X_i,P_j\right]& =i{\delta }_{ij}.\end{array}}$$

In particular, this means that a system of N interacting particles in 3 dimensions is described by one vector whose components in a basis where all the X are diagonal is a mathematical function of 3N-dimensional space describing all their possible positions, effectively a much bigger collection of values than the mere collection of N three-dimensional wavefunctions in one physical space. Schrödinger came to the same conclusion independently, and eventually proved the equivalence of 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 quantum particles has them 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 just two positions, the wavefunction for N particles requires 2^N complex numbers, one for each total configuration of positions. This is exponentially many numbers in N, so simulating quantum mechanics on a computer requires exponential resources. Conversely, 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. This is the aspiration behind quantum computing.

Ehrenfest theorem

For the time-independent operators X and P, ⁠∂A/∂t⁠ = 0 so the Heisenberg equation above reduces to: $${\displaystyle i\hslash \frac{dA}{dt}=[A,H]=AH-HA,}$$ where the square brackets [ , ] denote the commutator. For a Hamiltonian which is ⁠p²/2m⁠ + V(x), the X and P operators satisfy: $${\displaystyle \frac{dX}{dt}=\frac{P}{m},\frac{dP}{dt}=-\mathrm{\nabla }V,}$$ where the first is classically the velocity, and second is classically the force, or potential gradient. These reproduce Hamilton's form of Newton's laws of motion. In the Heisenberg picture, the X and P operators satisfy the classical equations of motion. You can take the expectation value of both sides of the equation to see that, in any state |ψ⟩: $${\displaystyle \begin{array}{cc}\frac{d}{dt}⟨X⟩& =\frac{d}{dt}⟨\psi |X|\psi ⟩=\frac{1}{m}⟨\psi |P|\psi ⟩=\frac{1}{m}⟨P⟩\\ \frac{d}{dt}⟨P⟩& =\frac{d}{dt}⟨\psi |P|\psi ⟩=⟨\psi |(-\mathrm{\nabla }V)|\psi ⟩=-⟨\mathrm{\nabla }V⟩.\end{array}}$$

So Newton's laws are exactly obeyed by the expected values of the operators in any given state. This is Ehrenfest's theorem, which is an obvious corollary of the Heisenberg equations of motion, but is less trivial in the Schrödinger picture, where Ehrenfest discovered it.

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 the canonical transformation: $${\displaystyle \begin{array}{cc}x& \to x+dx=x+\frac{\mathrm{\partial }H}{\mathrm{\partial }p}dt\\ p& \to p+dp=p-\frac{\mathrm{\partial }H}{\mathrm{\partial }x}dt.\end{array}}$$

Since the Hamiltonian can be an arbitrary function of x and p, there are such infinitesimal canonical transformations corresponding to every classical quantity G, where G serves as the Hamiltonian to generate a flow of points in phase space for an increment of time s, $${\displaystyle \begin{array}{cc}dx& =\frac{\mathrm{\partial }G}{\mathrm{\partial }p}ds=\left\{G,X\right\}ds\\ dp& =-\frac{\mathrm{\partial }G}{\mathrm{\partial }x}ds=\left\{G,P\right\}ds.\end{array}}$$

For a general function A(x,p) on phase space, its infinitesimal change at every step ds under this map is $${\displaystyle dA=\frac{\mathrm{\partial }A}{\mathrm{\partial }x}dx+\frac{\mathrm{\partial }A}{\mathrm{\partial }p}dp=\{A,G\}ds.}$$ The quantity G is called the infinitesimal generator of the canonical transformation.

In quantum mechanics, the quantum analog G is now a Hermitian matrix, and the equations of motion are given by commutators, $${\displaystyle dA=i[G,A]ds.}$$

The infinitesimal canonical motions can be formally integrated, just as the Heisenberg equation of motion were integrated, $${\displaystyle A^{\prime }=U^{†}AU}$$ where U = e^iGs and s is an arbitrary parameter.

The definition of a quantum canonical transformation is thus 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, $${\displaystyle U^{†}=U^{-1}.}$$ These transformations leave the sum of the absolute square of the wavefunction components invariant, while they 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.

For example, the motion generated by P can be found by solving the Heisenberg equation of motion using P as a Hamiltonian, $${\displaystyle \begin{array}{cc}dX& =i[X,P]ds=ds\\ dP& =i[P,P]ds=0.\end{array}}$$ These are translations of the matrix X by a multiple of the identity matrix, $${\displaystyle X\to X+sI.}$$ This is the interpretation of the derivative operator D: e^iPs = e^D, the exponential of a derivative operator is a translation (so Lagrange's shift operator).

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 X² + P² generates rotations in phase space.

When a transformation, like a rotation in physical space, commutes with the Hamiltonian, the transformation is called a symmetry (behind a degeneracy) of the Hamiltonian – 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 symmetry generator L vanishes, $${\displaystyle \frac{dH}{ds}=i[L,H]=0.}$$

It then follows that the change in the generator under time translation also vanishes, $${\displaystyle \frac{dL}{dt}=i[H,L]=0}$$ so that the matrix L is constant in time: it is conserved.

The one-to-one association of infinitesimal symmetry generators and conservation laws was discovered by Emmy Noether for classical mechanics, where the commutators are Poisson brackets, but the quantum-mechanical reasoning is identical. In quantum mechanics, any unitary symmetry transformation yields a conservation law, since if the matrix U has the property that $${\displaystyle U^{-1}HU=H}$$ so it follows that $${\displaystyle UH=HU}$$ and that the time derivative of U is zero – it is conserved.

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 up to an integer multiple of 2π. 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 transformations 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 yield 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 q X(t), and the time variation of this moment translates directly into the space-time variation of the vector potential, which yields nested outgoing spherical waves.

For atoms, the wavelength of the emitted light is about 10,000 times the atomic radius, and the dipole moment is the only contribution to the radiative field, while 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, $${\displaystyle P(\omega )=\frac{2}{3}{\omega }^4|d_i{|}^2.}$$

Now, in Heisenberg's representation, the Fourier coefficients of the dipole moment are the matrix elements of X. This 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, $${\displaystyle P_{ij}=\frac{2}{3}{\left(E_i-E_j\right)}^4{\left|X_{ij}\right|}^2.}$$

This then 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_ij is zero, the corresponding transition should be absent. These were called the selection rules, which were a puzzle until the advent of 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 $${\displaystyle L_i={\epsilon }_{ijk}{X}^j{P}^k}$$ where 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 all three coordinate matrices X, Y, Z (or with any vector) are easy to find, $${\displaystyle \left[L_i,X_j\right]=i{\epsilon }_{ijk}{X}_k,}$$ which confirms that the operator L generates rotations between the three components of the vector of coordinate matrices X.

From this, the commutator of L_z and the coordinate matrices X, Y, Z can be read off, $${\displaystyle \begin{array}{cc}\left[L_z,X\right]& =iY,\\ \left[L_z,Y\right]& =-iX.\end{array}}$$

This means that the quantities X + iY and X − iY have a simple commutation rule, $${\displaystyle \begin{array}{cc}\left[L_z,X+iY\right]& =(X+iY),\\ \left[L_z,X-iY\right]& =-(X-iY).\end{array}}$$

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, $${\displaystyle L_z((X+iY)|m⟩)=(X+iY)L_z|m⟩+(X+iY)|m⟩=(m+1)(X+iY)|m⟩}$$ 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, while Z does not change the value of m.

So, in a basis of |l,m⟩ states where L² 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′ < l − 1: Dipole transitions are forbidden with a change in angular momentum of more than one unit.

Sum rules

The Heisenberg equation of motion determines the matrix elements of P in the Heisenberg basis from the matrix elements of X. $${\displaystyle P_{ij}=m\frac{d}{dt}{X}_{ij}=im\left(E_i-E_j\right)X_{ij},}$$ which turns the diagonal part of the commutation relation into a sum rule for the magnitude of the matrix elements: $${\displaystyle \sum _j{P}_{ij}{x}_{ji}-X_{ij}{p}_{ji}=i\sum _{j}2m\left(E_i-E_j\right){\left|X_{ij}\right|}^2=i.}$$

This yields 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: $${\displaystyle \sum _{j}2m\left(E_i-E_j\right){\left|X_{ij}\right|}^2=1.}$$