Heisenberg picture
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In quantum mechanics in the Heisenberg picture the state vector, |ψ> does not change with time, and an observable A satisfies
- <math>\frac{d}{dt}A=(i\hbar)^{-1}[A,H]+\left(\frac{\partial A}{\partial t}\right)_{classical}.<math>
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.
See also Schrödinger picture.
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:
- <math> \lang A \rang _{t} = \lang \psi (t) | A | \psi(t) \rang <math>
or if we write following the Schrödinger equation
- <math> | \psi (t) \rang = e^{-iHt / \hbar} | \psi (0) \rang <math>
(where H is the Hamiltonian and hbar is Planck's constant divided by 2*Pi) we get
- <math> \lang A \rang _{t} = \lang \psi (0) | e^{iHt / \hbar} A e^{-iHt / \hbar} | \psi(0) \rang <math>
and so we define
- <math> A(t) := e^{iHt / \hbar} A e^{-iHt / \hbar} <math>
Now,
- <math> {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} <math>
(differentiating according to the product rule),
- <math> = {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}<math>
(the last passage is valid since exp(-iHt/hbar) commutes with H)
- <math> = {i \over \hbar } [ H(t) , A ] + \left(\frac{\partial A}{\partial t}\right)_{classical} <math>
(where [X,Y] is the commutator of two operators and defined as <math> [X,Y] := XY - YX <math>)
So we get
- <math> {d \over dt} A(t) = {i \over \hbar } [ H(t) , A ] + \left(\frac{\partial A}{\partial t}\right)_{classical} <math>