Heisenberg picture

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In physics, the Heisenberg picture is a formulation of quantum mechanics in which the operators (observables and others) 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 two models only differ by a basis change with respect to time-dependency, which is the difference between active and passive transformation. The Heisenberg picture is the formulation of matrix mechanics in an arbitrary basis, in which the Hamiltonian is not necessarily diagonal.

1 Mathematical details
2 Deriving Heisenberg's equation
3 Commutator relations
4 See also
5 Further reading
6 External links

Mathematical details

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

$\frac{d}{dt}A(t)={i \over \hbar}[H,A(t)]%2B\left(\frac{\partial A}{\partial t}\right)(t),$

where H is the Hamiltonian and [·,·] is the commutator of A and H. 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.

This approach has a similarity to classical physics: 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

The expectation value of an observable A, which is a Hermitian linear operator, for a given state $|\psi(t)\rang$ is given by:

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

In general $| \psi(t) \rang = U(t)| \psi(0) \rang$ where $U(t)=U(t;0)$ is the time evolution operator. For an elementary derivation, we will take Hamiltonian to commute with itself at different times, and further, be independent of time, in which case it simplifies to

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

where H is the Hamiltonian and ħ is Planck's constant divided by $2 \pi$ . It follows that

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

With the definition,

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

it follows:

${d \over dt} A(t) = {i \over \hbar} H e^{iHt / \hbar} A e^{-iHt / \hbar} %2B e^{iHt / \hbar} \left(\frac{\partial A}{\partial t}\right) e^{-iHt / \hbar} %2B {i \over \hbar}e^{iHt / \hbar} A \cdot (-H) e^{-iHt / \hbar}$

(differentiating according to the product rule) noting that $\frac{\partial A}{\partial t}$ is the time derivative of A(t), the transformed operator, not the one we started with.

$= {i \over \hbar } e^{iHt / \hbar} \left( H A - A H \right) e^{-iHt / \hbar} %2B e^{iHt / \hbar} \left(\frac{\partial A}{\partial t}\right) e^{-iHt / \hbar} = {i \over \hbar } \left( H A(t) - A(t) H \right) %2B e^{iHt / \hbar} \left(\frac{\partial A}{\partial t}\right)e^{-iHt / \hbar}$

The last passage is valid since : $e^{-iHt/ \hbar}$ commutes with H. From this results the Heisenberg equation of motion:

${d \over dt} A(t) = {i \over \hbar } [ H , A(t) ] %2B \left(\frac{\partial A}{\partial t}\right)(t)$ ,

where [X, Y] is the commutator of two operators and defined as [X, Y] := XY − YX.

Now, using the operator identity

${e^B A e^{-B}} = A %2B [B,A] %2B \frac{1}{2!} [B,[B,A]] %2B \frac{1}{3!}[B,[B,[B,A]]]%2B\cdots$

one obtains for an observable A:

$A(t)=A%2B\frac{it}{\hbar}[H,A]-\frac{t^{2}}{2!\hbar^{2}}[H,[H,A]] - \frac{it^3}{3!\hbar^3}[H,[H,[H,A]]] %2B \dots.$

Due to the relationship between Poisson Bracket and Commutators this relation also holds for classical mechanics.

note: the relationship between Poisson Bracket and Commutators is

$[A,H]=i\hbar\{A,H\}$

in classical mechanics

$\{A,H\}={d\over dt}A$

so you can convince yourself that A(t) equation is the Taylor expansion on t=0

Commutator relations

Obviously, commutator relations are quite different than in the Schrödinger picture because of the time dependency of operators. For example, consider the operators $x(t_{1}), x(t_{2}), p(t_{1})$ and $p(t_{2})$ . The time evolution of those operators depends on the Hamiltonian of the system. For the one-dimensional harmonic oscillator