Matrix mechanics

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Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg in 1925. ^[1]

Matrix mechanics was the first complete definition of quantum mechanics, its laws, and properties that described fully the behavior of subatomic particles by associating their properties with matrices. It has been shown to be exactly equivalent to the Schrödinger wave formulation of quantum mechanics and is the basis of the bra-ket notation used to summarize quantum mechanical wave functions.

1 Development of matrix mechanics
2 Mathematical details
3 Deriving Heisenberg's equation
4 Footnotes
5 See also
6 External links

[edit] Development of matrix mechanics

When it was introduced by Werner Heisenberg, Max Born and Pascual Jordan in 1925, matrix mechanics was not immediately accepted and was a source of great controversy.

Schrödinger's later introduction of wave mechanics was favored because there were no visual aids to fall back on in matrix mechanics and the mathematics were unfamiliar to most physicists.

Matrix mechanics consists of an array of quantities which when appropriately manipulated gave the observed frequencies and intensities of spectral lines. Heisenberg said himself that once and for all he had gotten rid of all electron orbits that did not exist. However, in Heisenberg's theory, the result of multiplication changes depending on its order. This means that the physical quantities in Heisenberg's theory are not ordinary variables but mathematical matrices. Heisenberg developed matrix mechanics by interpreting the electron as a particle with quantum behavior. It is based on sophisticated matrix computations which introduce discontinuities and quantum jumps.

In atomic physics, through spectroscopy, it was known that observational data related to atomic transitions arise from interactions of the atoms with light quanta. Heisenberg was the first to say that the atomic spectrum which showed spectral lines only in places where photons were being absorbed or emitted as electrons changed orbitals were the only relevant objects to be defined. Heisenberg recognized that the matrix formulation was built on the premise that all physical observables must be represented by matrices. The set of eigenvalues of the matrix representing an observable is the set of all possible values that could arise as outcomes of experiments conducted on a system to measure the observable. Since the outcome of an experiment to measure a real observable must be a real number, Hermitian matrices would represent such observables as their eigenvalues are real. If the result of a measurement is a certain eigenvalue, the corresponding eigenvector represents the state of the system immediately after the measurement.

Instead of using three dimensional orbitals, Heisenberg's matrix mechanics described the space in which the state of a quantum system inhabits as being one-dimensional as in the case of an anharmonic oscillator. To illustrate, consider the simple example of a point particle like an electron that is free to move on a line. An observable in this case could be the position of the particle, represented by the matrix X. Since the particle could be anywhere on the line, the possible outcome of a measurement of its position could be any one of an infinite set of eigenvalues of X, denoted by x. Thus X must be an infinite-dimensional matrix, and hence so is the corresponding linear vector space. Thus even one-dimensional motion could have an infinite-dimensional linear vector space associated with it. This made operators, functions, and spaces necessary to describe quantum mechanics.

The act of measurement in matrix mechanics is taken to 'collapse' the state of the system to that eigenvector (or eigenstate). Anyone familiar with Schroedinger's wave equation which came later in 1925 will be familiar with this concept in the form of wavefunction collapse. If one were to make simultaneous measurements of two or more observables, the system will collapse to a common eigenstate of these observables right after the measurement.

Further, from matrix theory we know that eigenvectors corresponding to distinct eigenvalues of a Hermitian matrix are orthogonal to each other which is analogous to the x, y, z axes of the Cartesian coordinate system except with an infinite number of distinct eigenvalues, and hence as many mutually orthogonal eigenvectors directed along different independent directions in the linear vector space.

Prior to measurement, the system could have been in a linear superposition of different eigenstates, with coefficients that might not be known. The Copenhagen interpretation is concerned only with outcomes of experiments.

Since a single measurement of any observable A yields one of the eigenvalues of A as the outcome, and collapses the state of the system to the corresponding eigenstate, subsequent measurements made immediately thereafter would continue to yield the same eigenvalue. So the correct thing to do would be to prepare a collection of a very large number of identical copies of the system and conduct a single trial on each copy. The arithmetic mean of all the results thus obtained is the average value we see, denoted by (A).

The Uncertainty Principle in matrix mechanics stems from the fact that, in general, two matrices A and B do not obey the arithmetical law of commutation. The commutator A B - B A = [A, B] does not equal 0. The famous commutation relation that is the basis for Quantum Mechanics and the later Uncertainty Principle is:

$\begin{matrix} \sum_{k}\end{matrix} p(n,k) q(k,n) - q(n,k) p(k,n) = h/2\pi i$

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

[edit] Mathematical details

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

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

In some sense, the Heisenberg picture is more natural and fundamental than the Schrödinger picture, especially for relativistic theories. Lorentz invariance is manifest in the Heisenberg picture.

Moreover, the similarity to classical physics is easily seen: by replacing the commutator above by the Poisson bracket, the Heisenberg equation becomes an equation in Hamiltonian mechanics.

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

[edit] Deriving Heisenberg's equation

Suppose we have an observable A (which is a Hermitian linear operator). The expectation value of A for a given state |ψ(t)> is given by:

$\lang A \rang _{t} = \lang \psi (t) | A | \psi(t) \rang$

or if we write following the Schrödinger equation

$| \psi (t) \rang = e^{-iHt / \hbar} | \psi (0) \rang$

(where H is the Hamiltonian and ħ is Planck's constant divided by 2*π) we get

$\lang A \rang _{t} = \lang \psi (0) | e^{iHt / \hbar} A e^{-iHt / \hbar} | \psi(0) \rang$

and so we define

$A(t) := e^{iHt / \hbar} A e^{-iHt / \hbar}$

Now,

${d \over dt} A(t) = {i \over \hbar} H e^{iHt / \hbar} A e^{-iHt / \hbar} + \left(\frac{\partial A}{\partial t}\right)_{classical} + {i \over \hbar}e^{iHt / \hbar} A \cdot (-H) e^{-iHt / \hbar}$

(differentiating according to the product rule),

$= {i \over \hbar } e^{iHt / \hbar} \left( H A - A H \right) e^{-iHt / \hbar} + \left(\frac{\partial A}{\partial t}\right)_{classical} = {i \over \hbar } \left( H A(t) - A(t) H \right) + \left(\frac{\partial A}{\partial t}\right)_{classical}$

(the last passage is valid since exp(-iHt/hbar) commutes with H)

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

(where [X,Y] is the commutator of two operators and defined as $[X, Y]: = X Y - Y X$ )

So we get

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

Though matrix mechanics does not include concepts such as the wave function of Erwin Schrödinger's wave equation, the two approaches were proven to be mathematically equivalent by the mathematician John von Neumann.

[edit] Footnotes

^ Especially since Werner Heisenberg was awarded the Nobel Prize in Physics in 1932 for the creation of quantum mechanics, the role of Max Born has been obfuscated. A 2005 biography of Born details his role as the creator of the matrix formulation of quantum mechanics. This was recognized in a paper by Heisenberg honoring Max Planck, in 1950. See: Nancy Thorndike Greenspan, "The End of the Certain World: The Life and Science of Max Born" (Basic Books, 2005), pp. 124 - 128, and 285 - 286.