ρ ˙ = L ρ = − i ℏ [ H , ρ ] {\displaystyle {\dot {\rho }}={\mathcal {L}}\rho =-{\frac {i}{\color {Gray}\hbar }}\left[{H,\rho }\right]} mit
[1]
Schrödingergleichung
i ∂ t Ψ ( t ) = H ^ Ψ ( t ) {\displaystyle {{\mathfrak {i}}{\partial }_{t}}\Psi (t)={\hat {H}}\Psi (t)}
Dirac Notation
Ket: | i ∂ t Ψ ( t ) ⟩ = | H ^ Ψ ( t ) ⟩ i ∂ t | Ψ ( t ) ⟩ = H ^ | Ψ ( t ) ⟩ ⇒ ∂ t | Ψ ( t ) ⟩ = − i H ^ | Ψ ( t ) ⟩ {\displaystyle {\begin{aligned}&\left|{\mathfrak {i}}{{\partial }_{t}}\Psi \left(t\right)\right\rangle =\left|{\hat {H}}\Psi \left(t\right)\right\rangle \\&{\mathfrak {i}}{{\partial }_{t}}\left|\Psi \left(t\right)\right\rangle ={\hat {H}}\left|\Psi \left(t\right)\right\rangle \Rightarrow {{\partial }_{t}}\left|\Psi \left(t\right)\right\rangle =-{\mathfrak {i}}{\hat {H}}\left|\Psi \left(t\right)\right\rangle \\\end{aligned}}}
Bra:
⟨ i ∂ t Ψ ( t ) | = ⟨ H ^ Ψ ( t ) | - i ∂ t ⟨ Ψ ( t ) | = ⟨ Ψ ( t ) | H ^ , ( H ^ = H ^ + ) ⇒ ∂ t ⟨ Ψ ( t ) | = i ⟨ Ψ ( t ) | H ^ {\displaystyle {\begin{aligned}&\left\langle {\mathfrak {i}}{{\partial }_{t}}\Psi \left(t\right)\right|=\left\langle {\hat {H}}\Psi \left(t\right)\right|\\&{\text{-}}{\mathfrak {i}}{{\partial }_{t}}\left\langle \Psi \left(t\right)\right|=\left\langle \Psi \left(t\right)\right|{\hat {H}},\,\left({\hat {H}}={{\hat {H}}^{+}}\right)\Rightarrow {{\partial }_{t}}\left\langle \Psi \left(t\right)\right|={\mathfrak {i}}\left\langle \Psi \left(t\right)\right|{\hat {H}}\\\end{aligned}}}
Dichtematrix
ρ = | Ψ ( t ) ⟩ ⟨ Ψ ( t ) | {\displaystyle \rho =\left|\Psi \left(t\right)\right\rangle \left\langle \Psi \left(t\right)\right|}
einsetzen:
ρ ˙ = ∂ t ( | Ψ ( t ) ⟩ ⟨ Ψ ( t ) | ) = ( ∂ t | Ψ ( t ) ⟩ ) ⟨ Ψ ( t ) | + | Ψ ( t ) ⟩ ( ∂ t ⟨ Ψ ( t ) | ) = − i H ^ | Ψ ( t ) ⟩ ⟨ Ψ ( t ) | + | Ψ ( t ) ⟩ ⟨ Ψ ( t ) | i H ^ = − i ( H ^ ρ − ρ H ^ ) ≡ − i [ H ^ , ρ ] = i [ ρ , H ^ ] {\displaystyle {\begin{aligned}&{\dot {\rho }}={{\partial }_{t}}\left(\left|\Psi \left(t\right)\right\rangle \left\langle \Psi \left(t\right)\right|\right)\\&=\left({{\partial }_{t}}\left|\Psi \left(t\right)\right\rangle \right)\left\langle \Psi \left(t\right)\right|+\left|\Psi \left(t\right)\right\rangle \left({{\partial }_{t}}\left\langle \Psi \left(t\right)\right|\right)\\&=-{\mathfrak {i}}{\hat {H}}\left|\Psi \left(t\right)\right\rangle \left\langle \Psi \left(t\right)\right|+\left|\Psi \left(t\right)\right\rangle \left\langle \Psi \left(t\right)\right|{\mathfrak {i}}{\hat {H}}\\&=-{\mathfrak {i}}\left({\hat {H}}\rho -\rho {\hat {H}}\right)\equiv -{\mathfrak {i}}\left[{\hat {H}},\rho \right]={\mathfrak {i}}\left[\rho ,{\hat {H}}\right]\end{aligned}}}
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