Heisenberg dynamics

In Schrödinger’s formulation of time evolution we have a time evolution operator $U(t,t_0)$ that acts on a state $\ket{\psi,t_0}$ to produce $\ket{\psi,t}$ . In Heisenberg’s formulation, we absorb the time evolution into the observable operator instead.

\begin{align*} \braket{\psi,t|A_S|\psi,t} = \braket{\psi | U\adj(t) A_S U(t) | \psi} = \braket{\psi|A_H(t)|\psi}. \end{align*}

Since time evolution $U$ is unitary, this formulation is easy to work with. Here are some nice properties that follow immediately.

\begin{align*} C_S = A_S B_S \quad &\implies \quad C_H = U\adj A_S B_S U = A_H B_H \\ [A_S,B_C] = C_S \quad &\implies \quad [A_H,B_H] = U\adj [A_S,B_S] U = U\adj C_S U = C_H \\ [H_S(t_1),H_S(t_2)] = 0 \quad &\implies \quad H_H(t) = H_S(t) \\ [A_S,H_S(t)] = 0 \quad&\implies\quad A_H(t) = A_S \end{align*}

To derive the time evolution of a Heisenberg operator, start with

\begin{align*} i\hbar \frac{\partial }{\partial t} U(t,t_) &= H_S(t) U(t,t_0) \tag{1} \\ i\hbar \frac{\partial }{\partial t} U\adj(t,t_0) &= -U\adj(t,t_0) H_S(t) \tag{2} \\ i\hbar \frac{\d }{\d t} A_H(t) &= i\hbar \frac{\partial }{\partial t} \left( U\adj(t,t_0) A_S U(t,t_0) \right) \\ &= i\hbar \left( (\partial_t U\adj) A_S U + U\adj (\partial_t A_S) U + U\adj A_S \partial_t U \right) \\ &= - U\adj H_S A_S U + U\adj A_S H_S U + i\hbar U\adj (\partial_t A_S) U \tag{from 1, 2} \\ &= U\adj [A_S,H_S] U + i\hbar (\partial_t A_S)_H (t) \\ &= [A_H,H_H] + i\hbar (\partial_t A_S)_H(t) \end{align*}

This is Heisenberg’s equation of motion.