Liouville–von Neumann equation

$\frac{d{\rho }_t}{dt}=\frac{1}{i\hslash }[H,{\rho }_t]$

where:

• ${\rho }_t=|{\psi }_t⟩⟨{\psi }_t|$ is a Density Operator evolving according to the Schrodinger Equation $i\hslash \frac{d}{dt}|{\psi }_t⟩=H|{\psi }_t⟩$
• $[\cdot ,\cdot ]$ is the Commutator

Proof

$$\begin{array}{rll}\frac{d}{dt}{\rho }_t& =\frac{d}{dt}(|{\psi }_t⟩⟨{\psi }_t|)& \text{By the product rule}\\ & =\frac{d|{\psi }_t⟩}{dt}⟨{\psi }_t|+|{\psi }_t⟩\frac{d⟨{\psi }_t|}{dt}& \text{Using Schrodinger’s Eqn and it’s dual}\\ & =\frac{1}{i\hslash }H|{\psi }_t⟩⟨{\psi }_t|-\frac{1}{i\hslash }|{\psi }_t⟩⟨{\psi }_t|H& \text{As ρt=∣ψt⟩⟨ψt∣}\\ & =\frac{1}{i\hslash }H{\rho }_t-\frac{1}{i\hslash }{\rho }_{t}H& \text{By the definition of commutator}\\ & =\frac{1}{i\hslash }[H,{\rho }_t]& \end{array}$$