ρ ˙ = L ρ = − i ℏ [ H , ρ ] {\displaystyle {\dot {\rho }}={\mathcal {L}}\rho =-{\frac {i}{\color {Gray}\hbar }}\left[{H,\rho }\right]} mit
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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 ) | = H ^ ⟨ Ψ ( t ) | , H ^ = H ^ + ⇒ ∂ t ⟨ Ψ ( t ) | = i H ^ ⟨ Ψ ( t ) | {\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|={\hat {H}}\left\langle \Psi \left(t\right)\right|,\,{\hat {H}}={{\hat {H}}^{+}}\Rightarrow {{\partial }_{t}}\left\langle \Psi \left(t\right)\right|={\mathfrak {i}}{\hat {H}}\left\langle \Psi \left(t\right)\right|\\\end{aligned}}}
Dichtematrix
ρ ^ = | Ψ ( t ) ⟩ ⟨ Ψ ( t ) | {\displaystyle {\hat {\rho }}=\left|\Psi \left(t\right)\right\rangle \left\langle \Psi \left(t\right)\right|}
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