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Display information for equation id:math.1732.58 on revision:1732
* Page found: Zeitabhängige Störungsrechnung (eq math.1732.58)
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\begin{align}
& {{g}_{n}}^{(1)}(t)=-\frac{i}{\hbar }\int_{0}^{t}{d\tau }{{e}^{\left( i\frac{\left( {{E}_{n}}-{{E}_{n0}} \right)\tau }{\hbar } \right)}}\left\langle n \right|\hat{V}\left| {{n}_{0}} \right\rangle =-\left\langle n \right|\hat{V}\left| {{n}_{0}} \right\rangle \frac{{{e}^{\left( i\frac{\left( {{E}_{n}}-{{E}_{n0}} \right)t}{\hbar } \right)}}-1}{{{E}_{n}}-{{E}_{n0}}} \\
& {{\left| {{g}_{n}}^{(1)}(t) \right|}^{2}}={{\left| \left\langle n \right|\hat{V}\left| {{n}_{0}} \right\rangle \right|}^{2}}\left\{ \frac{{{e}^{\left( -i\frac{\left( {{E}_{n}}-{{E}_{n0}} \right)t}{\hbar } \right)}}-1}{{{E}_{n}}-{{E}_{n0}}} \right\}\left\{ \frac{{{e}^{\left( i\frac{\left( {{E}_{n}}-{{E}_{n0}} \right)t}{\hbar } \right)}}-1}{{{E}_{n}}-{{E}_{n0}}} \right\}:={{\left| \left\langle n \right|\hat{V}\left| {{n}_{0}} \right\rangle \right|}^{2}}\left\{ \frac{\left( {{e}^{\left( -i\Omega t \right)}}-1 \right)\left( {{e}^{\left( i\Omega t \right)}}-1 \right)}{{{\Omega }^{2}}{{\hbar }^{2}}} \right\} \\
& \Omega :=\frac{\left( {{E}_{n}}-{{E}_{n0}} \right)}{\hbar } \\
& \Rightarrow {{\left| {{g}_{n}}^{(1)}(t) \right|}^{2}}={{\left| \left\langle n \right|\hat{V}\left| {{n}_{0}} \right\rangle \right|}^{2}}\frac{2\left( 1-\cos \Omega t \right)}{{{\Omega }^{2}}{{\hbar }^{2}}}={{\left| \left\langle n \right|\hat{V}\left| {{n}_{0}} \right\rangle \right|}^{2}}\frac{4{{\sin }^{2}}\frac{\Omega }{2}t}{{{\Omega }^{2}}{{\hbar }^{2}}} \\
& \frac{4{{\sin }^{2}}\frac{\Omega }{2}t}{{{\Omega }^{2}}{{\hbar }^{2}}}:={{D}_{t}}\left( {{E}_{n}}-{{E}_{n0}} \right) \\
& \Rightarrow {{\left| {{g}_{n}}^{(1)}(t) \right|}^{2}}={{\left| \left\langle n \right|\hat{V}\left| {{n}_{0}} \right\rangle \right|}^{2}}{{D}_{t}}\left( {{E}_{n}}-{{E}_{n0}} \right) \\
\end{align}
TeX (checked):
{\begin{aligned}&{{g}_{n}}^{(1)}(t)=-{\frac {i}{\hbar }}\int _{0}^{t}{d\tau }{{e}^{\left(i{\frac {\left({{E}_{n}}-{{E}_{n0}}\right)\tau }{\hbar }}\right)}}\left\langle n\right|{\hat {V}}\left|{{n}_{0}}\right\rangle =-\left\langle n\right|{\hat {V}}\left|{{n}_{0}}\right\rangle {\frac {{{e}^{\left(i{\frac {\left({{E}_{n}}-{{E}_{n0}}\right)t}{\hbar }}\right)}}-1}{{{E}_{n}}-{{E}_{n0}}}}\\&{{\left|{{g}_{n}}^{(1)}(t)\right|}^{2}}={{\left|\left\langle n\right|{\hat {V}}\left|{{n}_{0}}\right\rangle \right|}^{2}}\left\{{\frac {{{e}^{\left(-i{\frac {\left({{E}_{n}}-{{E}_{n0}}\right)t}{\hbar }}\right)}}-1}{{{E}_{n}}-{{E}_{n0}}}}\right\}\left\{{\frac {{{e}^{\left(i{\frac {\left({{E}_{n}}-{{E}_{n0}}\right)t}{\hbar }}\right)}}-1}{{{E}_{n}}-{{E}_{n0}}}}\right\}:={{\left|\left\langle n\right|{\hat {V}}\left|{{n}_{0}}\right\rangle \right|}^{2}}\left\{{\frac {\left({{e}^{\left(-i\Omega t\right)}}-1\right)\left({{e}^{\left(i\Omega t\right)}}-1\right)}{{{\Omega }^{2}}{{\hbar }^{2}}}}\right\}\\&\Omega :={\frac {\left({{E}_{n}}-{{E}_{n0}}\right)}{\hbar }}\\&\Rightarrow {{\left|{{g}_{n}}^{(1)}(t)\right|}^{2}}={{\left|\left\langle n\right|{\hat {V}}\left|{{n}_{0}}\right\rangle \right|}^{2}}{\frac {2\left(1-\cos \Omega t\right)}{{{\Omega }^{2}}{{\hbar }^{2}}}}={{\left|\left\langle n\right|{\hat {V}}\left|{{n}_{0}}\right\rangle \right|}^{2}}{\frac {4{{\sin }^{2}}{\frac {\Omega }{2}}t}{{{\Omega }^{2}}{{\hbar }^{2}}}}\\&{\frac {4{{\sin }^{2}}{\frac {\Omega }{2}}t}{{{\Omega }^{2}}{{\hbar }^{2}}}}:={{D}_{t}}\left({{E}_{n}}-{{E}_{n0}}\right)\\&\Rightarrow {{\left|{{g}_{n}}^{(1)}(t)\right|}^{2}}={{\left|\left\langle n\right|{\hat {V}}\left|{{n}_{0}}\right\rangle \right|}^{2}}{{D}_{t}}\left({{E}_{n}}-{{E}_{n0}}\right)\\\end{aligned}}
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data-mjx-texclass="ORD"><mn>4</mn><msup><mi>sin</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">Ω</mi></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow><mi>t</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msup><mi mathvariant="normal">Ω</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><msup><mi data-mjx-alternate="1">ℏ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mrow></mfrac></mrow><mi>:</mi><mo>=</mo><msub><mi>D</mi><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</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 data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msup><msub><mi>g</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE">|</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>=</mo><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><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>V</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 data-mjx-texclass="CLOSE">|</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><msub><mi>D</mi><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</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 data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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