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Display information for equation id:math.1737.11 on revision:1737

* Page found: Induzierte Emission und Absorption von Lichtquanten in Atomen (eq math.1737.11)

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\begin{align}
& -\frac{e}{m}\left\langle  n \right|{{e}^{i\bar{k}\bar{r}}}{{{\bar{A}}}_{0}}\hat{\bar{p}}\left| {{n}_{0}} \right\rangle \cong -\frac{i}{\hbar }\frac{em}{2m}{{{\bar{A}}}_{0}}\left\langle  n \right|{{{\hat{H}}}_{0}}\hat{\bar{r}}-\hat{\bar{r}}{{{\hat{H}}}_{0}}\left| {{n}_{0}} \right\rangle =-\frac{i}{2\hbar }({{E}_{n}}-{{E}_{n0}}){{{\bar{A}}}_{0}}e\left\langle  n \right|\hat{\bar{r}}\left| {{n}_{0}} \right\rangle  \\
& {{{\bar{A}}}_{0}}=-\frac{{{{\bar{E}}}_{0}}}{\omega } \\
& e\left\langle  n \right|\hat{\bar{r}}\left| {{n}_{0}} \right\rangle :={{{\bar{d}}}_{nn0}} \\
\end{align}

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- e m ⁢ ⟨ n | e i ⁢ k ¯ ⁢ r ¯ ⁢ A ¯ 0 ⁢ p ¯ ^ | n 0 ⟩ ≅ - i ℏ ⁢ e ⁢ m 2 ⁢ m ⁢ A ¯ 0 ⁢ ⟨ n | H ^ 0 ⁢ r ¯ ^ - r ¯ ^ ⁢ H ^ 0 | n 0 ⟩ = - i 2 ⁢ ℏ ⁢ ( E n - E n ⁢ 0 ) ⁢ A ¯ 0 ⁢ e ⁢ ⟨ n | r ¯ ^ | n 0 ⟩ A ¯ 0 = - E ¯ 0 ω e ⁢ ⟨ n | r ¯ ^ | n 0 ⟩ := d ¯ n ⁢ n ⁢ 0 missing-subexpression 𝑒 𝑚 quantum-operator-product 𝑛 superscript 𝑒 𝑖 ¯ 𝑘 ¯ 𝑟 subscript ¯ 𝐴 0 ^ ¯ 𝑝 subscript 𝑛 0 𝑖 Planck-constant-over-2-pi 𝑒 𝑚 2 𝑚 subscript ¯ 𝐴 0 quantum-operator-product 𝑛 subscript ^ 𝐻 0 ^ ¯ 𝑟 ^ ¯ 𝑟 subscript ^ 𝐻 0 subscript 𝑛 0 𝑖 2 Planck-constant-over-2-pi subscript 𝐸 𝑛 subscript 𝐸 𝑛 0 subscript ¯ 𝐴 0 𝑒 quantum-operator-product 𝑛 ^ ¯ 𝑟 subscript 𝑛 0 missing-subexpression subscript ¯ 𝐴 0 subscript ¯ 𝐸 0 𝜔 missing-subexpression assign 𝑒 quantum-operator-product 𝑛 ^ ¯ 𝑟 subscript 𝑛 0 subscript ¯ 𝑑 𝑛 𝑛 0 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle-\frac{e}{m}\left% \langle n\right|{{e}^{i\bar{k}\bar{r}}}{{{\bar{A}}}_{0}}\hat{\bar{p}}\left|{{n% }_{0}}\right\rangle\cong-\frac{i}{\hbar}\frac{em}{2m}{{{\bar{A}}}_{0}}\left% \langle n\right|{{{\hat{H}}}_{0}}\hat{\bar{r}}-\hat{\bar{r}}{{{\hat{H}}}_{0}}% \left|{{n}_{0}}\right\rangle=-\frac{i}{2\hbar}({{E}_{n}}-{{E}_{n0}}){{{\bar{A}% }}_{0}}e\left\langle n\right|\hat{\bar{r}}\left|{{n}_{0}}\right\rangle\\ &\displaystyle{{{\bar{A}}}_{0}}=-\frac{{{{\bar{E}}}_{0}}}{\omega}\\ &\displaystyle e\left\langle n\right|\hat{\bar{r}}\left|{{n}_{0}}\right\rangle% :={{{\bar{d}}}_{nn0}}\\ \end{aligned}}}

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−em⟨n|eik¯r¯A¯0p¯^|n0⟩≅−iℏem2mA¯0⟨n|H^0r¯^−r¯^H^0|n0⟩=−i2ℏ(En−En0)A¯0e⟨n|r¯^|n0⟩A¯0=−E¯0ωe⟨n|r¯^|n0⟩:=d¯nn0

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