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==Harmonische zeitabhängige Störung== :<math>{{\hat{H}}^{1}}(t)=\hat{F}{{e}^{-i\omega t}}+{{\hat{F}}^{+}}{{e}^{i\omega t}}</math> hermitesch! Es folgt: :<math>\begin{align} & {{g}_{n}}(t)=-\frac{i}{\hbar }\int_{0}^{t}{d\tau }{{e}^{\left( i\frac{\left( {{E}_{n}}-{{E}_{n0}}-\hbar \omega \right)\tau }{\hbar } \right)}}\left\langle n \right|\hat{F}\left| {{n}_{0}} \right\rangle -\frac{i}{\hbar }\int_{0}^{t}{d\tau }{{e}^{\left( i\frac{\left( {{E}_{n}}-{{E}_{n0}}+\hbar \omega \right)\tau }{\hbar } \right)}}\left\langle n \right|{{{\hat{F}}}^{+}}\left| {{n}_{0}} \right\rangle \\ & \Rightarrow {{g}_{n}}(t)=-\left\langle n \right|\hat{F}\left| {{n}_{0}} \right\rangle \left\{ \frac{{{e}^{\left( i\frac{\left( {{E}_{n}}-{{E}_{n0}}-\hbar \omega \right)t}{\hbar } \right)-1}}}{{{E}_{n}}-{{E}_{n0}}-\hbar \omega } \right\}-\left\langle n \right|{{{\hat{F}}}^{+}}\left| {{n}_{0}} \right\rangle \left\{ \frac{{{e}^{\left( i\frac{\left( {{E}_{n}}-{{E}_{n0}}+\hbar \omega \right)t}{\hbar } \right)-1}}}{{{E}_{n}}-{{E}_{n0}}+\hbar \omega } \right\} \\ \end{align}</math> Somit folgt für die Übergangswahrscheinlichkeit von <math>\left| {{n}_{0}} \right\rangle </math> auf <math>\left| n \right\rangle </math> : :<math>\begin{align} & {{\left| {{\left\langle n | \Psi \right\rangle }_{t}} \right|}^{2}}={{\left| {{g}_{n}} \right|}^{2}}=\frac{2\pi }{\hbar }{{\left| \left\langle n \right|\hat{F}\left| {{n}_{0}} \right\rangle \right|}^{2}}t\delta ({{E}_{n}}-{{E}_{n0}}-\hbar \omega ) \\ & +\frac{2\pi }{\hbar }{{\left| \left\langle n \right|{{{\hat{F}}}^{+}}\left| {{n}_{0}} \right\rangle \right|}^{2}}t\delta ({{E}_{n}}-{{E}_{n0}}+\hbar \omega )+\left\langle n \right|\hat{F}\left| {{n}_{0}} \right\rangle *\left\langle n \right|{{{\hat{F}}}^{+}}\left| {{n}_{0}} \right\rangle \left\{ \frac{{{e}^{\left( -i\frac{\left( {{E}_{n}}-{{E}_{n0}}-\hbar \omega \right)t}{\hbar } \right)}}-1}{{{E}_{n}}-{{E}_{n0}}-\hbar \omega } \right\}\left\{ \frac{{{e}^{\left( i\frac{\left( {{E}_{n}}-{{E}_{n0}}+\hbar \omega \right)t}{\hbar } \right)}}-1}{{{E}_{n}}-{{E}_{n0}}+\hbar \omega } \right\} \\ & +\left\langle n \right|{{{\hat{F}}}^{+}}\left| {{n}_{0}} \right\rangle *\left\langle n \right|\hat{F}\left| {{n}_{0}} \right\rangle \left\{ \frac{{{e}^{\left( -i\frac{\left( {{E}_{n}}-{{E}_{n0}}+\hbar \omega \right)t}{\hbar } \right)}}-1}{{{E}_{n}}-{{E}_{n0}}+\hbar \omega } \right\}\left\{ \frac{{{e}^{\left( i\frac{\left( {{E}_{n}}-{{E}_{n0}}-\hbar \omega \right)t}{\hbar } \right)}}-1}{{{E}_{n}}-{{E}_{n0}}-\hbar \omega } \right\} \\ & {{\Omega }^{\pm }}:=\Omega \pm \omega =\frac{\left( {{E}_{n}}-{{E}_{n0}}\pm \hbar \omega \right)}{\hbar } \\ & \Rightarrow {{\left| {{\left\langle n | \Psi \right\rangle }_{t}} \right|}^{2}}={{\left| {{g}_{n}} \right|}^{2}}=\frac{2\pi }{\hbar }{{\left| \left\langle n \right|\hat{F}\left| {{n}_{0}} \right\rangle \right|}^{2}}t\delta ({{E}_{n}}-{{E}_{n0}}-\hbar \omega ) \\ & +\frac{2\pi }{\hbar }{{\left| \left\langle n \right|{{{\hat{F}}}^{+}}\left| {{n}_{0}} \right\rangle \right|}^{2}}t\delta ({{E}_{n}}-{{E}_{n0}}+\hbar \omega )+\left\langle n \right|\hat{F}\left| {{n}_{0}} \right\rangle *\left\langle n \right|{{{\hat{F}}}^{+}}\left| {{n}_{0}} \right\rangle \left\{ \frac{\left( {{e}^{\left( -i{{\Omega }^{-}}t \right)}}-1 \right)\left( {{e}^{\left( i{{\Omega }^{+}}t \right)}}-1 \right)}{{{\hbar }^{2}}{{\Omega }^{+}}{{\Omega }^{-}}} \right\} \\ & +\left\langle n \right|{{{\hat{F}}}^{+}}\left| {{n}_{0}} \right\rangle *\left\langle n \right|\hat{F}\left| {{n}_{0}} \right\rangle \left\{ \frac{\left( {{e}^{\left( -i{{\Omega }^{+}}t \right)}}-1 \right)\left( {{e}^{\left( i{{\Omega }^{-}}t \right)}}-1 \right)}{{{\hbar }^{2}}{{\Omega }^{+}}{{\Omega }^{-}}} \right\} \\ & \left\langle n \right|\hat{F}\left| {{n}_{0}} \right\rangle *\left\langle n \right|{{{\hat{F}}}^{+}}\left| {{n}_{0}} \right\rangle :=A{{e}^{-i\gamma }} \\ & \left\langle n \right|{{{\hat{F}}}^{+}}\left| {{n}_{0}} \right\rangle *\left\langle n \right|\hat{F}\left| {{n}_{0}} \right\rangle :=A{{e}^{i\gamma }} \\ \end{align}</math> :<math>\begin{align} & \Rightarrow {{\left| {{\left\langle n | \Psi \right\rangle }_{t}} \right|}^{2}}={{\left| {{g}_{n}} \right|}^{2}}=\frac{2\pi }{\hbar }{{\left| \left\langle n \right|\hat{F}\left| {{n}_{0}} \right\rangle \right|}^{2}}t\delta ({{E}_{n}}-{{E}_{n0}}-\hbar \omega ) \\ & +\frac{2\pi }{\hbar }{{\left| \left\langle n \right|{{{\hat{F}}}^{+}}\left| {{n}_{0}} \right\rangle \right|}^{2}}t\delta ({{E}_{n}}-{{E}_{n0}}+\hbar \omega )+A{{e}^{-i\gamma }}\left\{ \frac{\left( {{e}^{\left( -i{{\Omega }^{-}}t \right)}}-1 \right)\left( {{e}^{\left( i{{\Omega }^{+}}t \right)}}-1 \right)}{{{\hbar }^{2}}{{\Omega }^{+}}{{\Omega }^{-}}} \right\} \\ & +A{{e}^{i\gamma }}\left\{ \frac{\left( {{e}^{\left( -i{{\Omega }^{+}}t \right)}}-1 \right)\left( {{e}^{\left( i{{\Omega }^{-}}t \right)}}-1 \right)}{{{\hbar }^{2}}{{\Omega }^{+}}{{\Omega }^{-}}} \right\} \\ \end{align}</math> Weiter gilt :<math>A{{e}^{-i\gamma }}\left\{ \frac{\left( {{e}^{\left( -i{{\Omega }^{-}}t \right)}}-1 \right)\left( {{e}^{\left( i{{\Omega }^{+}}t \right)}}-1 \right)}{{{\hbar }^{2}}{{\Omega }^{+}}{{\Omega }^{-}}} \right\}+A{{e}^{i\gamma }}\left\{ \frac{\left( {{e}^{\left( -i{{\Omega }^{+}}t \right)}}-1 \right)\left( {{e}^{\left( i{{\Omega }^{-}}t \right)}}-1 \right)}{{{\hbar }^{2}}{{\Omega }^{+}}{{\Omega }^{-}}} \right\}=\frac{4A}{{{\hbar }^{2}}{{\Omega }^{+}}{{\Omega }^{-}}}\cos \left( \omega t-\gamma \right)\left[ \cos \left( \omega t \right)-\cos \left( \Omega t \right) \right]</math> Für <math>\omega \ne 0,\Omega \ne 0</math> sind diese Terme jedoch rein oszillierend. Für <math>t\to \infty </math> sind diese jedoch vernachlässigbar gegen Terme <math>\tilde{\ }t\delta ({{E}_{n}}-{{E}_{n0}}\pm \hbar \omega )=t\delta (\hbar {{\Omega }^{\pm }})</math> Somit folgt für <math>t\to \infty </math> : :<math>{{\left| {{\left\langle n | \Psi \right\rangle }_{t}} \right|}^{2}}={{\left| {{g}_{n}} \right|}^{2}}=\frac{2\pi }{\hbar }{{\left| \left\langle n \right|\hat{F}\left| {{n}_{0}} \right\rangle \right|}^{2}}t\delta ({{E}_{n}}-{{E}_{n0}}-\hbar \omega )+\frac{2\pi }{\hbar }{{\left| \left\langle n \right|{{{\hat{F}}}^{+}}\left| {{n}_{0}} \right\rangle \right|}^{2}}t\delta ({{E}_{n}}-{{E}_{n0}}+\hbar \omega )</math> Für Zeit gegen unendlich kann man dann leicht die Übergangswahrscheinlichkeit zwischen <math>\left| {{n}_{0}} \right\rangle </math> und <math>\left| n \right\rangle </math> pro Zeiteinheit, durch Ableitung nach der Zeit erhalten: :<math>{{W}_{nn0}}=\frac{d}{dt}{{\left| {{\left\langle n | \Psi \right\rangle }_{t}} \right|}^{2}}=\frac{2\pi }{\hbar }{{\left| \left\langle n \right|\hat{F}\left| {{n}_{0}} \right\rangle \right|}^{2}}\delta ({{E}_{n}}-{{E}_{n0}}-\hbar \omega )+\frac{2\pi }{\hbar }{{\left| \left\langle {{n}_{0}} \right|\hat{F}\left| n \right\rangle \right|}^{2}}\delta ({{E}_{n}}-{{E}_{n0}}+\hbar \omega )</math> Die Terme lassen sich identifizieren: :<math>\frac{2\pi }{\hbar }{{\left| \left\langle n \right|\hat{F}\left| {{n}_{0}} \right\rangle \right|}^{2}}\delta ({{E}_{n}}-{{E}_{n0}}-\hbar \omega )</math> steht für die Absorption eines Quants der Energie <math>\hbar \omega </math> bei gleichzeitiger Anregung des Übergangs von <math>\left| {{n}_{0}} \right\rangle </math> auf<math>\left| n \right\rangle </math>, was einem Energiesprung von <math>{{E}_{n}}-{{E}_{n0}}</math> entspricht. Das Quant wird also von Niveau <math>\left| {{n}_{0}} \right\rangle </math> auf <math>\left| n \right\rangle </math> gehievt :<math>\frac{2\pi }{\hbar }{{\left| \left\langle {{n}_{0}} \right|{{{\hat{F}}}^{+}}\left| n \right\rangle \right|}^{2}}\delta ({{E}_{n}}-{{E}_{n0}}+\hbar \omega )</math> steht für die Emission eines Quants der Energie <math>\hbar \omega </math> bei gleichzeitiger Anregung des Übergangs von <math>\left| n \right\rangle </math> auf<math>\left| {{n}_{0}} \right\rangle </math>, was einer Energieabgabe von <math>{{E}_{n0}}-{{E}_{n}}</math> entspricht. Das Quant fällt dabei vom diesmal höheren Niveau <math>\left| {{n}_{0}} \right\rangle </math> auf das Niveau <math>\left| n \right\rangle </math> herunter.
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