Liouville-von-Neumann-Gleichung

Herleitung

i \partial_{t} Ψ (t) = \hat{H} Ψ (t)

Dirac Notation

Ket:

\begin{aligned} | i \partial_{t} Ψ (t) ⟩ = | \hat{H} Ψ (t) ⟩ \\ i \partial_{t} | Ψ (t) ⟩ = \hat{H} | Ψ (t) ⟩ \Rightarrow \partial_{t} | Ψ (t) ⟩ = - i \hat{H} | Ψ (t) ⟩ \end{aligned}

Bra:

\begin{aligned} ⟨ i \partial_{t} Ψ (t) | = ⟨ \hat{H} Ψ (t) | \\ - i \partial_{t} ⟨ Ψ (t) | = ⟨ Ψ (t) | \hat{H}, (\hat{H} = {\hat{H}}^{+}) \Rightarrow \partial_{t} ⟨ Ψ (t) | = i ⟨ Ψ (t) | \hat{H} \end{aligned}

ρ = | Ψ (t) ⟩ ⟨ Ψ (t) |

einsetzen:

\begin{aligned} \dot{ρ} = \partial_{t} (| Ψ (t) ⟩ ⟨ Ψ (t) |) \\ = (\partial_{t} | Ψ (t) ⟩) ⟨ Ψ (t) | + | Ψ (t) ⟩ (\partial_{t} ⟨ Ψ (t) |) \\ = - i \hat{H} | Ψ (t) ⟩ ⟨ Ψ (t) | + | Ψ (t) ⟩ ⟨ Ψ (t) | i \hat{H} \\ = - i (\hat{H} ρ - ρ \hat{H}) \equiv - i [\hat{H}, ρ] = i [ρ, \hat{H}] \end{aligned}