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Display information for equation id:math.1732.85 on revision:1732
* Page found: Zeitabhängige Störungsrechnung (eq math.1732.85)
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\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}
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data-mjx-texclass="CLOSE">⟩</mo></mrow><mo data-mjx-texclass="CLOSE">|</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi>t</mi><mi>δ</mi><mo stretchy="false">(</mo><msub><mi>E</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mo>−</mo><msub><mi>E</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>n</mi><mn>0</mn></mrow></mrow></msub><mo>−</mo><mi data-mjx-alternate="1">ℏ</mi><mi>ω</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>π</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mi data-mjx-alternate="1">ℏ</mi></mrow></mfrac></mrow><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><mi>n</mi><mo data-mjx-texclass="CLOSE">|</mo></mrow><msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>F</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mo>+</mo></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>n</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo data-mjx-texclass="CLOSE">|</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi>t</mi><mi>δ</mi><mo stretchy="false">(</mo><msub><mi>E</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mo>−</mo><msub><mi>E</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>n</mi><mn>0</mn></mrow></mrow></msub><mo>+</mo><mi data-mjx-alternate="1">ℏ</mi><mi>ω</mi><mo stretchy="false">)</mo><mo>+</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><mi>n</mi><mo data-mjx-texclass="CLOSE">|</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>F</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>n</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo>*</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><mi>n</mi><mo data-mjx-texclass="CLOSE">|</mo></mrow><msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>F</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mo>+</mo></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>n</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mo>−</mo><mi>i</mi><msup><mi mathvariant="normal">Ω</mi><mrow data-mjx-texclass="ORD"><mo>−</mo></mrow></msup><mi>t</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></msup><mo>−</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>i</mi><msup><mi mathvariant="normal">Ω</mi><mrow data-mjx-texclass="ORD"><mo>+</mo></mrow></msup><mi>t</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></msup><mo>−</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msup><mi data-mjx-alternate="1">ℏ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><msup><mi mathvariant="normal">Ω</mi><mrow data-mjx-texclass="ORD"><mo>+</mo></mrow></msup><msup><mi mathvariant="normal">Ω</mi><mrow data-mjx-texclass="ORD"><mo>−</mo></mrow></msup></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">}</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>+</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><mi>n</mi><mo data-mjx-texclass="CLOSE">|</mo></mrow><msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>F</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mo>+</mo></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>n</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo>*</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><mi>n</mi><mo data-mjx-texclass="CLOSE">|</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>F</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>n</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mo>−</mo><mi>i</mi><msup><mi mathvariant="normal">Ω</mi><mrow data-mjx-texclass="ORD"><mo>+</mo></mrow></msup><mi>t</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></msup><mo>−</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>i</mi><msup><mi mathvariant="normal">Ω</mi><mrow data-mjx-texclass="ORD"><mo>−</mo></mrow></msup><mi>t</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></msup><mo>−</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msup><mi data-mjx-alternate="1">ℏ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><msup><mi mathvariant="normal">Ω</mi><mrow data-mjx-texclass="ORD"><mo>+</mo></mrow></msup><msup><mi mathvariant="normal">Ω</mi><mrow data-mjx-texclass="ORD"><mo>−</mo></mrow></msup></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">}</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><mi>n</mi><mo data-mjx-texclass="CLOSE">|</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>F</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>n</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo>*</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><mi>n</mi><mo data-mjx-texclass="CLOSE">|</mo></mrow><msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>F</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mo>+</mo></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>n</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mi>:</mi><mo>=</mo><mi>A</mi><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>i</mi><mi>γ</mi></mrow></mrow></msup></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><mi>n</mi><mo data-mjx-texclass="CLOSE">|</mo></mrow><msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>F</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mo>+</mo></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>n</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo>*</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><mi>n</mi><mo data-mjx-texclass="CLOSE">|</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>F</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>n</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mi>:</mi><mo>=</mo><mi>A</mi><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>γ</mi></mrow></mrow></msup></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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