Heisenberg Equation of Motion

$\frac{d}{dt}{⟨A⟩}_{\psi }={⟨\frac{1}{i\hslash }[A,H]⟩}_{\psi }$ describes the time evolution of Measurement uncertainty in the Heisenberg picture. This result ties to the Liouville–von Neumann equation from the Schrodinger picture, which applies to a Density Operator $\hat{\rho }$, where

• $H$ is the Hamiltonian
• $[\cdot ,\cdot ]$ is the Commutator
• ${⟨A⟩}_{\psi }$ is the Expectation of a Measurement under Operator $A$ given we’re in the Quantum state $\psi$

Proof

We will make use of Schrodinger Equation and the fact that $H$ is a Self-Adjoint Operator.

$$\begin{array}{rll}\frac{d}{dt}{⟨A⟩}_{\psi }& =\frac{d}{dt}⟨\psi ,A\psi ⟩& \\ & =⟨\frac{d}{dt}\psi ,A\psi ⟩+⟨\psi ,A\frac{d}{dt}\psi ⟩& \text{As dtdψ=iℏ1Hψ}\\ & =⟨\frac{1}{i\hslash }H\psi ,A\psi ⟩+⟨\psi ,A\frac{1}{i\hslash }H\psi ⟩& \\ & =\frac{1}{i\hslash }\left(-⟨H\psi ,A\psi ⟩+⟨\psi ,AH\psi ⟩\right)& \text{As H is self-adjoint, i.e ⟨Hψ∣ϕ⟩=⟨ψ∣Hϕ⟩}\\ & =\frac{1}{i\hslash }\left(-⟨\psi ,HA\psi ⟩+⟨\psi ,AH\psi ⟩\right)& \\ & =\frac{1}{i\hslash }⟨\psi ,(-HA+AH)\psi ⟩& \\ & =\frac{1}{i\hslash }{⟨AH-HA⟩}_{\psi }& \\ & ={⟨\frac{1}{i\hslash }[A,H]⟩}_{\psi }& \end{array}$$