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Display information for equation id:math.1661.52 on revision:1661

* Page found: Der harmonische Oszillator (eq math.1661.52)

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Der harmonische Oszillator Eq: math.1661.52

Hash: cd9ca545dda6a181a5a52c06ae0db154

TeX (original user input):

\begin{align}
& \left\langle  0 \right|{{a}^{n}}{{\left( {{a}^{+}} \right)}^{n}}\left| 0 \right\rangle =\left\langle  0 \right|{{a}^{n-1}}\left( {{\left( {{a}^{+}} \right)}^{n}}a+\left[ a,{{\left( {{a}^{+}} \right)}^{n}} \right] \right)\left| 0 \right\rangle =\left\langle  0 \right|{{a}^{n-1}}{{\left( {{a}^{+}} \right)}^{n}}a\left| 0 \right\rangle +n\left\langle  0 \right|{{a}^{n-1}}{{\left( {{a}^{+}} \right)}^{n-1}}\left| 0 \right\rangle  \\
& \left\langle  0 \right|{{a}^{n-1}}{{\left( {{a}^{+}} \right)}^{n}}a\left| 0 \right\rangle =0 \\
& \Rightarrow n\left\langle  0 \right|{{a}^{n-1}}{{\left( {{a}^{+}} \right)}^{n-1}}a\left| 0 \right\rangle =n\left( n-1 \right)\left\langle  0 \right|{{a}^{n-2}}{{\left( {{a}^{+}} \right)}^{n-2}}a\left| 0 \right\rangle \Rightarrow ...\Rightarrow  \\
\end{align}

TeX (checked):

{\begin{aligned}&\left\langle 0\right|{{a}^{n}}{{\left({{a}^{+}}\right)}^{n}}\left|0\right\rangle =\left\langle 0\right|{{a}^{n-1}}\left({{\left({{a}^{+}}\right)}^{n}}a+\left[a,{{\left({{a}^{+}}\right)}^{n}}\right]\right)\left|0\right\rangle =\left\langle 0\right|{{a}^{n-1}}{{\left({{a}^{+}}\right)}^{n}}a\left|0\right\rangle +n\left\langle 0\right|{{a}^{n-1}}{{\left({{a}^{+}}\right)}^{n-1}}\left|0\right\rangle \\&\left\langle 0\right|{{a}^{n-1}}{{\left({{a}^{+}}\right)}^{n}}a\left|0\right\rangle =0\\&\Rightarrow n\left\langle 0\right|{{a}^{n-1}}{{\left({{a}^{+}}\right)}^{n-1}}a\left|0\right\rangle =n\left(n-1\right)\left\langle 0\right|{{a}^{n-2}}{{\left({{a}^{+}}\right)}^{n-2}}a\left|0\right\rangle \Rightarrow ...\Rightarrow \\\end{aligned}}

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\begin{aligned} ⟨ 0 | a^{n} {(a^{+})}^{n} | 0 ⟩ = ⟨ 0 | a^{n - 1} ({(a^{+})}^{n} a + [a, {(a^{+})}^{n}]) | 0 ⟩ = ⟨ 0 | a^{n - 1} {(a^{+})}^{n} a | 0 ⟩ + n ⟨ 0 | a^{n - 1} {(a^{+})}^{n - 1} | 0 ⟩ \\ ⟨ 0 | a^{n - 1} {(a^{+})}^{n} a | 0 ⟩ = 0 \\ \Rightarrow n ⟨ 0 | a^{n - 1} {(a^{+})}^{n - 1} a | 0 ⟩ = n (n - 1) ⟨ 0 | a^{n - 2} {(a^{+})}^{n - 2} a | 0 ⟩ \Rightarrow . . . \Rightarrow \end{aligned}

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