# Heisenberg Picture

Operators have time dependence. $$|\alpha,t_0;t\rangle_H = |\alpha,t_0\rangle$$. $$A_H(0) = A_S(0)$$. $$A_H(t) = \hat{U}^\dagger(t,t_0)A_S(0)\hat{U}(t,t_0)$$.

$$\langle b|A'|a\rangle = \langle b|U^\dagger AU|a\rangle$$.

## Equation of Motion

$$i\hbar\frac{\partial}{\partial t}(\hat{U})|\Psi\rangle = \hat{H}\hat{U}|\Psi\rangle \Rightarrow -i\hbar\frac{\partial}{\partial t}(\hat{U}^\dagger) = \hat{U}^\dagger\hat{H}$$

$$\frac{d}{dt}A_H = \frac{\partial}{\partial t}(\hat{U}^\dagger)A_S\hat{U} + \hat{U}^\dagger\frac{\partial}{\partial t}(A_S)\hat{U} + \hat{U}^\dagger A_S\frac{\partial}{\partial t}(\hat{U}) = \frac{\partial}{\partial t}(\hat{U}^\dagger)A_S\hat{U} + \hat{U}^\dagger A_S\frac{\partial}{\partial t}(\hat{U}) = \frac{1}{i\hbar}(-\hat{H}\hat{U}^\dagger A_S\hat{U} + \hat{U}^\dagger A_S\hat{U}\hat{H}) = -\frac{i}{\hbar}[A_H,H]$$ (Note that the time evolution operator uses the Hamiltonian as its generator, i.e. it is a function of the Hamiltonian and because any function of an operator commutes with the operator).

$$\frac{d}{dt}A_H = -\frac{i}{\hbar}[A_H,H]$$.

Created: 2023-06-25 Sun 02:33

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