Liouville-von-Neumann-Gleichung

$\dot{ρ} = ℒ ρ = - \frac{i}{ℏ} [H, ρ]$ mit

Herleitung

𝔦 \partial_{t} Ψ (t) = \hat{H} Ψ (t)

Dirac Notation

Ket:

\begin{matrix} | 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{matrix}

Bra:

\begin{matrix} ⟨ 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{matrix}

ρ = | Ψ (t) ⟩ ⟨ Ψ (t) |

einsetzen:

\begin{matrix} \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{matrix}