Matrix mechanics
Encyclopedia
Matrix mechanics is a formulation of quantum mechanics
created by Werner Heisenberg
, Max Born
, and Pascual Jordan
in 1925.
Matrix mechanics was the first conceptually autonomous and logically consistent formulation 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
that 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.
, Max Born
, and Pascual Jordan
formulated the matrix mechanics representation of quantum mechanics.
on the problem of calculating the spectral line
s of hydrogen
. By May 1925 he began trying to describe atomic systems by observable
s 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 Helgoland. While there, in between climbing and learning by heart poems from Goethe
's West-östlicher Diwan
, he continued to ponder the spectral issue and eventually realised that adopting non-commuting
observables might solve the problem, and he later wrote
, 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,
"...he had written a crazy paper and did not dare to send it in for publication, and that Born should read it and advise him on it..."
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 of the position operator
were the intensity of radiation in the bright-line spectrum.
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.
described the motion of a particle by a classical orbit, with well defined position and momentum
, 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
While this restriction correctly selects orbits with more or less the
right energy values
, 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
only.
The coefficients
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,
.
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
,
which means that its frequency is
.
For large n and m, but with
relatively small, these are the classical frequencies by Bohr
's
correspondence principle
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
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
. Heisenberg called this quantity 
,
and demanded that it should reduce to the classical Fourier coefficients
in the classical limit. For large values of n, m but with
relatively small,
is the 
th fourier coefficient of the classical motion at orbit n. Since 
has opposite frequency to
, the condition that X is real becomes:
.
By definition,
only has the frequency
, so its time evolution is simple:
.
This is the original form of Heisenberg's equation of motion.
Given two arrays
and 
describing two
physical quantities, Heisenberg could form a new array of the same type by
combining the terms
, 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:
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:
is
different from
.
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,
and 
satisfy the classical equations of motion.
, 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,i.e,that A B - B A does not necessarily equal 0. The famous commutation relation of matrix mechanics:
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.
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 reactions to Born for Heisenberg receiving the Prize for 1932 and 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.”
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.
one, the energy of the oscillator is
The level set
s 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 phase space, must be an integer multiple of Planck's constant. The area of the circle of radius
is
So
or, in units of length where
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:
both
and 
have only a single frequency, and 
and 
can be recovered from their sum and difference.
Since
has a classical Fourier series with only the lowest frequency, and the matrix element 
is the (m-n)th Fourier coefficient of the classical orbit, the matrix for 
is nonzero only on the line just
above the diagonal, where it is equal to
. The matrix for 
is likewise only nonzero on the line below the diagonal, with
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 both matrices are hermitian, since they are constructed from the Fourier coefficients of real quantities. To find
and
is simple, since they are quantum Fourier coefficients so they evolve simply with time.
The matrix product of
and 
is not hermitian, but has a real and imaginary part. The real part is one half the symmetric expression 
, while the imaginary part is proportional to the commutator 
. It is easy to verify explicitly that 
in the case of the harmonic oscillator, is ih/2π, multiplied by the identity
.
It is also easy to verify that the matrix
is a diagonal matrix
, with eigenvalues
.
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
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:
He noticed that if this could be done then H considered as a matrix function of X and P, will have zero time derivative.
Where
is the symmetric product
.
.
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
.
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:
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
is 
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.
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
.
The derivative with respect to time is a derivative with respect to
,
up to a factor of
.
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
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:
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,
Since the coefficient
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
, it helps to introduce two temporary notations:
, for the matrices, and
for the r'th Fourier components of classical quantities.
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 of derivatives, is
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:
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.
, now written
,
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
, 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
Alternatively and equivalently, the state vector gives the probability amplitude
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 
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 diagonal values
:
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:
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
. By rotating the basis for the state vector at each time by 
, you can undo the 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 the transformation to the Schroedinger picture.
In quantum mechanics
in the Heisenberg picture
the state vector,
does not change with time, and an observable A
satisfies
The extra term is for operators like
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.
See also Schrödinger picture
.
allows the evaluation of the commutator of p with any power of x, and it implies that
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.
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:
and the operator x multiplies each eigenvector by x.
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 commute with x.
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:
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 where 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 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.
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 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 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
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 
bits to solve, which is the motivation for quantum computing.
The Hamiltonian flow is then the canonical
canonical transformation
:
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.
For a general function A(x,p) on phase space, the infinitesimal change at every step ds under the map is:
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 were integrated:
where
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.
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:
They are translations of the matrix X which add a multiple of the identity:
. This is also the interpretation of the derivative operator 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 
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:
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 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
it follows that
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 up to an integer multiple of 2\pi. 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.
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:
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:
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
is zero, the corresponding transition should be absent. These were called the selection rule
s, 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:
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:
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 law implies that these operators only have certain off diagonal matrix elements in states of definite m.
meaning that the matrix (x+iy) takes an eigenvector of
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
and 
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'
which turns the diagonal part of the commutation relation into a sum rule for the magnitude of the matrix elements:
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:
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
created by Werner Heisenberg
Werner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
, Max Born
Max Born
Max Born was a German-born physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s...
, and Pascual Jordan
Pascual Jordan
-Further reading:...
in 1925.
Matrix mechanics was the first conceptually autonomous and logically consistent formulation of quantum mechanics. It extended the Bohr Model
Bohr model
In atomic physics, the Bohr model, introduced by Niels Bohr in 1913, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with electrostatic forces providing attraction,...
by describing how the quantum jumps occur. It did so by interpreting the physical properties of particles as matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
that evolve in time. It is equivalent to the Schrödinger wave formulation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
of quantum mechanics, and is the basis of Dirac
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
's bra-ket notation
Bra-ket notation
Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...
for the wave function.
Development of matrix mechanics
In 1925, Werner HeisenbergWerner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
, Max Born
Max Born
Max Born was a German-born physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s...
, and Pascual Jordan
Pascual Jordan
-Further reading:...
formulated the matrix mechanics representation of quantum mechanics.
Epiphany at Helgoland
In 1925 Werner Heisenberg was working in GöttingenGöttingen
Göttingen is a university town in Lower Saxony, Germany. It is the capital of the district of Göttingen. The Leine river runs through the town. In 2006 the population was 129,686.-General information:...
on the problem of calculating the spectral line
Spectral line
A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from a deficiency or excess of photons in a narrow frequency range, compared with the nearby frequencies.- Types of line spectra :...
s of hydrogen
Hydrogen
Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...
. By May 1925 he began trying to describe atomic systems by observable
Observable
In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...
s only. On June 7, to escape the effects of a bad attack of hay fever
Hay Fever
Hay Fever is a comic play written by Noël Coward in 1924 and first produced in 1925 with Marie Tempest as the first Judith Bliss. Laura Hope Crews played the role in New York...
, Heisenberg left for the pollen free North Sea
North Sea
In the southwest, beyond the Straits of Dover, the North Sea becomes the English Channel connecting to the Atlantic Ocean. In the east, it connects to the Baltic Sea via the Skagerrak and Kattegat, narrow straits that separate Denmark from Norway and Sweden respectively...
island of Helgoland. While there, in between climbing and learning by heart poems from Goethe
Johann Wolfgang von Goethe
Johann Wolfgang von Goethe was a German writer, pictorial artist, biologist, theoretical physicist, and polymath. He is considered the supreme genius of modern German literature. His works span the fields of poetry, drama, prose, philosophy, and science. His Faust has been called the greatest long...
's West-östlicher Diwan
West-östlicher Diwan
West-östlicher Diwan or West-östlicher Divan or West-Eastern Divan is a diwan, or collection of lyrical poems by the German poet Johann Wolfgang von Goethe. The work was inspired by the Persian poet Hafez....
, he continued to ponder the spectral issue and eventually realised that adopting non-commuting
Commutativity
In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
observables might solve the problem, and he later wrote
The Three Papers
After Heisenberg returned to GöttingenGöttingen
Göttingen is a university town in Lower Saxony, Germany. It is the capital of the district of Göttingen. The Leine river runs through the town. In 2006 the population was 129,686.-General information:...
, he showed Wolfgang Pauli
Wolfgang Pauli
Wolfgang Ernst Pauli was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after being nominated by Albert Einstein, he received the Nobel Prize in Physics for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle or...
his calculations, commenting at one point:
On July 9 Heisenberg gave the same paper of his calculations to Max Born
Max Born
Max Born was a German-born physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s...
, saying,
"...he had written a crazy paper and did not dare to send it in for publication, and that Born should read it and advise him on it..."
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
Bohr model
In atomic physics, the Bohr model, introduced by Niels Bohr in 1913, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with electrostatic forces providing attraction,...
by interpreting the Fourier coefficients
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
of the orbits as intensities. But his answer, like all other calculations in the old quantum theory
Old quantum theory
The old quantum theory was a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was a collection of heuristic prescriptions which are now understood to be the first quantum corrections to classical mechanics...
, was only correct for large orbits
Correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....
.
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
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
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 of the position operator
Position operator
In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle. Consider, for example, the case of a spinless particle moving on a line. The state space for such a particle is L2, the Hilbert space of complex-valued and square-integrable ...
were the intensity of radiation in the bright-line spectrum.
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
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
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
Pascual Jordan
-Further reading:...
, 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
Gustav Mie
Gustav Adolf Feodor Wilhelm Ludwig Mie was a German physicist.-Biography:Mie was born in Rostock. From 1886 he studied mathematics and physics at the University of Rostock. In addition to his major subjects, he also attended lectures in chemistry, zoology, geology, mineralogy, astronomy as well as...
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
Richard Courant
Richard Courant was a German American mathematician.- Life :Courant was born in Lublinitz in the German Empire's Prussian Province of Silesia. During his youth, his parents had to move quite often, to Glatz, Breslau, and in 1905 to Berlin. He stayed in Breslau and entered the university there...
at Göttingen in the preparation of Courant and David Hilbert’s
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
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
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
became assistant to David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
, and he would coin the term Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
to describe the algebra and analysis which were used in the development of quantum mechanics.
Heisenberg's reasoning
Before matrix mechanics, the old quantum theoryOld quantum theory
The old quantum theory was a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was a collection of heuristic prescriptions which are now understood to be the first quantum corrections to classical mechanics...
described the motion of a particle by a classical orbit, with well defined position and momentum
, 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 constantWhile this restriction correctly selects orbits with more or less the
right energy values
, 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
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
has frequencies
only.The coefficients
are complex numbers. The ones withnegative frequencies must be the complex conjugates 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 number m, the energy of the photon is
,which means that its frequency is
.For large n and m, but with
relatively small, these are the classical frequencies by BohrNiels Bohr
Niels Henrik David Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in...
's
correspondence principle
Correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....
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
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
. Heisenberg called this quantity ,and demanded that it should reduce to the classical Fourier coefficients
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
in the classical limit. For large values of n, m but with
relatively small, is the th fourier coefficient of the classical motion at orbit n. Since has opposite frequency to, the condition that X is real becomes:.By definition,
only has the frequency, so its time evolution is simple:.This is the original form of Heisenberg's equation of motion.
Given two arrays
and describing twophysical quantities, Heisenberg could form a new array of the same type by
combining the terms
, which also oscillate with the right frequency. Since the Fourier coefficientsFourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
of the product of two quantities is the convolution
Convolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
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:
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:
isdifferent from
.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
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
of a sharp classical trajectory. Nevertheless, as matrices,
and satisfy the classical equations of motion.Further discussion
When it was introduced by Werner HeisenbergWerner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
, Max Born
Max Born
Max Born was a German-born physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s...
and Pascual Jordan
Pascual Jordan
-Further reading:...
in 1925, matrix mechanics was not immediately accepted and was a source of great controversy.
Schrödinger's later introduction of wave mechanics
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
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
Spectroscopy
Spectroscopy is the study of the interaction between matter and radiated energy. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, e.g., by a prism. Later the concept was expanded greatly to comprise any interaction with radiative...
gave observational data on atomic transitions arising from the interactions of atoms with light quanta
Quantum
In physics, a quantum is the minimum amount of any physical entity involved in an interaction. Behind this, one finds the fundamental notion that a physical property may be "quantized," referred to as "the hypothesis of quantization". This means that the magnitude can take on only certain discrete...
. 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
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
.
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,i.e,that A B - B A does not necessarily equal 0. The famous commutation relation of matrix mechanics:
shows that there are no states which simultaneously have a definite position and momentum. But the principle of uncertainty (also called complementarity
Complementarity (physics)
In physics, complementarity is a basic principle of quantum theory proposed by Niels Bohr, closely identified with the Copenhagen interpretation, and refers to effects such as the wave–particle duality...
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.
Nobel Prize
In 1928, Albert EinsteinAlbert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
nominated Heisenberg, Born, and Jordan for the Nobel Prize in Physics
Nobel Prize in Physics
The Nobel Prize in Physics is awarded once a year by the Royal Swedish Academy of Sciences. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895 and awarded since 1901; the others are the Nobel Prize in Chemistry, Nobel Prize in Literature, Nobel Peace Prize, and...
, 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
Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...
and Paul Adrien Maurice Dirac
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
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
Sturmabteilung
The Sturmabteilung functioned as a paramilitary organization of the National Socialist German Workers' Party . It played a key role in Adolf Hitler's rise to power in the 1920s and 1930s...
. 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 reactions to Born for Heisenberg receiving the Prize for 1932 and 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
Max Planck
Max Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...
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 theirmatrix elements in special cases by guesswork, guided by the
correspondence principle
Correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....
. Since the matrix elements are the quantum mechanical
analogs of Fourier coefficients
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
of the classical orbits, the simplest case is the
harmonic oscillator
Harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....
, where X(t) and P(t) are sinusoidal.
Harmonic oscillator
In units where the mass and frequency of the oscillator are equal toone, the energy of the oscillator is
The level set
Level set
In mathematics, a level set of a real-valued function f of n variables is a set of the formthat is, a set where the function takes on a given constant value c....
s of H are the orbits, and they are nested circles. The classical orbit with energy E is:
The old quantum condition
Old quantum theory
The old quantum theory was a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was a collection of heuristic prescriptions which are now understood to be the first quantum corrections to classical mechanics...
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
is
Soor, in units of length where
is one, the energy is aninteger.
The Fourier components
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
of X(t) and P(t) are very simple, even more so if they are combined into the quantities:
both
and have only a single frequency, and and can be recovered from their sum and difference.Since
has a classical Fourier series with only the lowest frequency, and the matrix element is the (m-n)th Fourier coefficient of the classical orbit, the matrix for is nonzero only on the line justabove the diagonal, where it is equal to
. The matrix for is likewise only nonzero on the line below the diagonal, withthe 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 both matrices are hermitian, since they are constructed from the Fourier coefficients of real quantities. To find
and is simple, since they are quantum Fourier coefficients so they evolve simply with time.The matrix product of
and is not hermitian, but has a real and imaginary part. The real part is one half the symmetric expression , while the imaginary part is proportional to the commutator . It is easy to verify explicitly that in the case of the harmonic oscillator, is ih/2π, multiplied by the identityIdentity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
.
It is also easy to verify that the matrix
is a diagonal matrix
Diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...
, with eigenvalues
.Conservation of energy
The harmonic oscillator is too special. It is too easy to find thematrices exactly, and it is too hard to discover general conditions from these special forms. For this reason, Heisenberg investigated the anharmonic oscillator, with Hamiltonian
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
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
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
at every classical frequency. To determine the matrix elements, Heisenberg required that the classical equations of motion be obeyed as matrix
equations:
He noticed that if this could be done then H considered as a matrix function of X and P, will have zero time derivative.
Where
is the symmetric productJordan algebra
In abstract algebra, a Jordan algebra is an algebra over a field whose multiplication satisfies the following axioms:# xy = yx # = x ....
.
.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
Product recall
A product recall is a request to return to the maker a batch or an entire production run of a product, usually due to the discovery of safety issues. The recall is an effort to limit liability for corporate negligence and to improve or avoid damage to publicity...
that energy. This paradox led Bohr, Kramers and Slater
BKS theory
The Bohr-Kramers-Slater theory was perhaps the final attempt at understanding the interaction of matter and electromagnetic radiation on the basis of the so-called Old quantum theory, in which quantum phenomena are treated by imposing quantum restrictions on classically describable behaviour...
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
Wavefunction collapse
In quantum mechanics, wave function collapse is the phenomenon in which a wave function—initially in a superposition of several different possible eigenstates—appears to reduce to a single one of those states after interaction with an observer...
.
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 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
Old quantum theory
The old quantum theory was a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was a collection of heuristic prescriptions which are now understood to be the first quantum corrections to classical mechanics...
to fix the energy levels, then fill in the matrices with Fourier coefficients
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
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
Old quantum theory
The old quantum theory was a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was a collection of heuristic prescriptions which are now understood to be the first quantum corrections to classical mechanics...
into a simple 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 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
is 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.
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
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
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
.The derivative with respect to time is a derivative with respect to
,up to a factor of
.So the quantum condition integral is the average value over one cycle of the
Poisson bracket
Poisson bracket
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time-evolution of a Hamiltonian dynamical system...
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
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:
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,
Since the coefficient
is semiclassically the k'th Fouriercoefficient 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
, it helps to introduce two temporary notations:, for the matrices, and for the r'th Fourier components of classical quantities.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 of derivatives, is
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:
this condition replaced and extended the old quantization
Old quantum theory
The old quantum theory was a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was a collection of heuristic prescriptions which are now understood to be the first quantum corrections to classical mechanics...
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
Old quantum theory
The old quantum theory was a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was a collection of heuristic prescriptions which are now understood to be the first quantum corrections to classical mechanics...
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 vectorWavefunction
Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...
, now written
,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
, 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 toAlternatively and equivalently, the state vector gives the probability amplitude
Probability amplitude
In quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density.For example, if the probability amplitude of a quantum state is \alpha, the probability of measuring that state is |\alpha|^2...
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 evolves in time like a Fourier componentwhich 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 diagonal values
: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:
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
Unitary
Unitary may refer to:* Unitary construction, in automotive design, another common term for a unibody or monocoque construction**Unitary as chemical weapons opposite of Binary...
matrix
. By rotating the basis for the state vector at each time by , you can undo the 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 the transformation to the Schroedinger picture.
In quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
in the Heisenberg picture
Heisenberg picture
In physics, the Heisenberg picture is a formulation of quantum mechanics in which the operators incorporate a dependency on time, but the state vectors are time-independent. It stands in contrast to the Schrödinger picture in which the operators are constant and the states evolve in time...
the state vector,
does not change with time, and an observable Asatisfies
The extra term is for operators like
which have an explicit time dependence in addition to the time dependence from unitary evolution. The HeisenbergWerner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
picture does not distinguish time from space, so it is nicer for relativistic
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
theories.
Moreover, the similarity to classical physics
Classical physics
What "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...
is more obvious: the Hamiltonian equations of motion for classical mechanics are recovered by replacing the commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
above by the Poisson bracket
Poisson bracket
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time-evolution of a Hamiltonian dynamical system...
.
By the Stone-von Neumann theorem, the Heisenberg picture and the Schrödinger picture
Schrödinger picture
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators are constant. This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time...
are unitarily equivalent.
See also Schrödinger picture
Schrödinger picture
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators are constant. This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time...
.
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 identityallows the evaluation of the commutator of p with any power of x, and it implies that
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.
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:
and the operator x multiplies each eigenvector by x.
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 commute with x.
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:
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 where 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 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
Schrödinger picture
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators are constant. This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time...
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.
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 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 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
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
. 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
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 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
Canonical is an adjective derived from canon. Canon comes from the greek word κανών kanon, "rule" or "measuring stick" , and is used in various meanings....
canonical transformation
Canonical transformation
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates → that preserves the form of Hamilton's equations , although it...
:
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.
For a general function A(x,p) on phase space, the infinitesimal change at every step ds under the map is:
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 were integrated:
where
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.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:
They are translations of the matrix X which add a multiple of the identity:
. This is also the interpretation of the derivative operator 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 generates rotations in phase space.When a transformation, like a rotation in physical space, commutes with the Hamiltonian, the transformation is called a symmetry
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
. 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:
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 and conservation laws was first discovered by Emmy Noether
Emmy Noether
Amalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...
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
it follows that
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 up to an integer multiple of 2\pi. 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
Parity (physics)
In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...
, and charge conjugation are examples.
The interpretation of the matrices as generators of canonical transformations is due to Paul Dirac
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
. 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:
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:
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
is zero, the corresponding transition should be absent. These were called the selection ruleSelection rule
In physics and chemistry a selection rule, or transition rule, formally constrains the possible transitions of a system from one state to another. Selection rules have been derived for electronic, vibrational, and rotational transitions...
s, 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
Pseudovector
In physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image...
are:
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:
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 law implies that these operators only have certain off diagonal matrix elements in states of definite m.
meaning that the matrix (x+iy) takes an eigenvector of
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
and 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.which turns the diagonal part of the commutation relation into a sum rule for the magnitude of the matrix elements:
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:
The three formulating papers
- W. Heisenberg, Über quantentheoretische Umdeutung kinematischer und mechanischer BeziehungenÜber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen"Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen" was a breakthrough paper in quantum mechanics written by Werner Heisenberg...
, 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, 1926 (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).]
See also
- Interaction pictureInteraction pictureIn quantum mechanics, the Interaction picture is an intermediate between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of...
- Quantum mechanicsQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
- Schrödinger equationSchrödinger equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
- Bra-ket notationBra-ket notationBra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...
- Introduction to quantum mechanicsIntroduction to quantum mechanicsQuantum mechanics is the body of scientific principles that explains the behavior of matter and its interactions with energy on the scale of atoms and atomic particles....
External links
- An Overview of Matrix Mechanics (more of a review; certainly not for beginners. "Obviously" is in the first sentence. Obvious? Maybe to somebody who has already studied it. In which case, not obvious at all. A real teacher never uses that word.)
- Matrix Methods in Quantum Mechanics
- Heisenberg Quantum Mechanics (The theory's origins and its historical developing 1925-27)
- Werner Heisenberg 1970 CBC radio Interview
- Werner Karl Heisenberg Co-founder of Quantum Mechanics
- Ian J. R. Aitchison, David A. MacManus, Thomas M. Snyder. Understanding Heisenberg's `magical' paper of July 1925: a new look at the calculational details.