Bilder

Bilder in der QM

Schrödinger-Bild

nur Zustände ${{\left|\psi \right\rangle }_{t}}$ zeitabhängig

Eigenvektoren $\left|n\right\rangle$ und Operatoren ${{\hat {A}}_{S}}$ sind nicht zeitabhängig

zeitentwicklung mit Zeitentwicklungsoperator ${\hat {U}}=\exp \left(-{\frac {\mathfrak {i}}{\hbar }}{{\hat {H}}_{s}}t\right)$:

${{\left|\psi \right\rangle }_{t}}={\hat {U}}{{\left|\psi \right\rangle }_{0}}$

${{\hat {A}}_{S}}$ definiert eine symmetrische quadratische From

geometrisch

Zustandsvektor wird um feste Hauptachsen mit Zeitentwicklungsooerator gedreht.

Schrödinger Gleichung

$E\left|\psi \right\rangle =H\left|\psi \right\rangle$

$E=i\hbar {{\partial }_{t}}$

Heisenberg-Bild

Zustände zeitunabhängig ${{\left|\psi \right\rangle }_{t}}={{\left|\psi \right\rangle }_{0}}$

Operatoren ${{\hat {A}}_{W}}\left(t\right)$ und Eigenvektoren ${{\left|n\right\rangle }_{t}}$ zeitabhängig.

transfomration von Operatoren ins Heisenbergbild

$\left\langle {{\hat {A}}_{S}}\right\rangle ={}_{t}{{\left\langle \psi |{{\hat {A}}_{S}}|\psi \right\rangle }_{t}}={}_{0}{{\left\langle \psi \left|\underbrace {{{\hat {U}}^{+}}{{\hat {A}}_{S}}{\hat {U}}} _{:={{\hat {A}}_{H}}}\right|\psi \right\rangle }_{0}}=\left\langle {{\hat {A}}_{H}}\right\rangle$

Hamilton Operator

${{\hat {H}}_{S}}={{\hat {H}}_{H}}$ folgt aus Bewegungsgleichung

Wechselwirkungsbild

${{\hat {H}}_{w}}={{\hat {H}}_{0,S}}+{{\hat {H}}_{1,S}}$

${{\hat {H}}_{1}}$ ist als Störung zu interpretieren

${{\hat {A}}_{W}}={\hat {U}}_{0}^{+}{{\hat {A}}_{S}}{{U}_{0}}$ mit ${{\hat {U}}_{0}}=\exp \left(-{\frac {i}{\hbar }}{{\hat {H}}_{0}}t\right)$

${{\breve {A}}_{W}}={{d}_{t}}{{\hat {A}}_{W}}={\frac {i}{\hbar }}\left[{\hat {H}},{\hat {A}}\right]$

${\begin{aligned}&\left\langle {{\hat {A}}_{S}}\right\rangle ={}_{t}{{\left\langle \underbrace {\psi |{{\hat {U}}_{0}}} _{\left\langle {{\psi }_{W}}\right|}\underbrace {{\hat {U}}_{0}^{+}{{\hat {A}}_{S}}{{\hat {U}}_{0}}} _{{\hat {A}}_{W}}{\hat {U}}_{0}^{+}|\psi \right\rangle }_{t}}={}_{W}{{\left\langle \psi \left|{{\hat {A}}_{W}}\right|\psi \right\rangle }_{W}}\\&{{d}_{t}}{{\left|\psi \right\rangle }_{W}}={\frac {i}{\hbar }}{{\hat {H}}_{0,S}}{\hat {U}}_{0}^{+}{{\left|\psi \right\rangle }_{t}}+{\hat {U}}_{0}^{+}{{\partial }_{t}}{{\left|\psi \right\rangle }_{t}}\\&{{\partial }_{t}}{{\left|\psi \right\rangle }_{t}}={\frac {1}{i\hbar }}{{\hat {H}}_{0,S}}{{\left|\psi \right\rangle }_{t}}={\frac {1}{i\hbar }}{{\hat {H}}_{0,S}}{{\hat {U}}_{0}}{{\left|\psi \right\rangle }_{W}}\\&\Rightarrow {{d}_{t}}{{\left|\psi \right\rangle }_{W}}={\frac {1}{\hbar i}}\left(-{{\hat {H}}_{0,S}}+\underbrace {{\hat {U}}_{0}^{+}{{\hat {H}}_{0,S}}{{\hat {U}}_{0}}} _{{{\hat {H}}_{W}}={{H}_{0,S}}+{{H}_{1,S}}}\right){{\left|\psi \right\rangle }_{W}}={\frac {1}{i\hbar }}\left({{\hat {H}}_{W}}\right){{\left|\psi \right\rangle }_{W}}\\&\Rightarrow i\hbar {{d}_{t}}{{\left|\psi \right\rangle }_{W}}={{\hat {H}}_{1,S}}{{\left|\psi \right\rangle }_{W}}\\\end{aligned}}$