Heisenberg picture

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In physics, the Heisenberg picture is that formulation of quantum mechanics where the operators are time-dependent and the statevectors time-independent.

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The "Heisenberg Picture" of quantum mechanics is also known as Matrix mechanics.

When it was introduced by Werner Heisenberg in 1925 it was not immediately accepted and was a source of great controversy. A great assessment of the situation at that time, which is hard to improve on, can be found on the website of the American Institute of Physics.

"Most physicists were slow to accept "matrix mechanics" because of its abstract nature and its unfamiliar mathematics. They gladly welcomed Schrödinger's alternative wave mechanics when it appeared in early 1926, since it entailed more familiar concepts and equations, and it seemed to do away with quantum jumps and discontinuities. French physicist Louis de Broglie had suggested that not only light but also matter might behave like a wave. Drawing on this idea, to which Einstein had lent his support, Schrödinger attributed the quantum energies of the electron orbits in the old quantum theory of the atom to the vibration frequencies of electron "matter waves" around the atom's nucleus. Just as a piano string has a fixed tone, so an electron-wave would have a fixed quantum of energy. This led to much easier calculations and more familiar visualizations of atomic events than did Heisenberg's matrix mechanics, where the energy was found in an abstruse calculation." Heisenberg Quantum Mechanics

Given the above statement do not feel bad if you do not understand the following, as not many physicists did at the time. As a matter of fact if one googles the term "quantum mechanics" they will find many, many, many more hits that refer to the Schrödinger equation than to the Heisenberg equation. Matrix mechanics is disfavored not because it is wrong or has been superseded but because it involves a form of mathematics that most physicists are less familiar with.

Mathematical details

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

$\frac{d}{dt}A=(i\hbar)^{-1}[A,H]+\left(\frac{\partial A}{\partial t}\right)_{classical}.$

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.

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 hbar is Planck's constant divided by 2*pi) 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),

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