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Display information for equation id:math.1656.15 on revision:1656
* Page found: Der harmonische Oszillator (eq math.1656.15)
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TeX (original user input):
\begin{align}
& \left[ a,{{\left( {{a}^{+}} \right)}^{n+1}} \right]=a{{\left( {{a}^{+}} \right)}^{n+1}}-{{\left( {{a}^{+}} \right)}^{n+1}}a=a{{\left( {{a}^{+}} \right)}^{n+1}}-{{\left( {{a}^{+}} \right)}^{n}}a{{a}^{+}}+{{\left( {{a}^{+}} \right)}^{n}}a{{a}^{+}}-{{\left( {{a}^{+}} \right)}^{n+1}}a \\
& \Rightarrow \left[ a,{{\left( {{a}^{+}} \right)}^{n+1}} \right]=\left[ a,{{\left( {{a}^{+}} \right)}^{n}} \right]{{a}^{+}}+{{\left( {{a}^{+}} \right)}^{n}}\left[ a,{{a}^{+}} \right] \\
& \left[ a,{{\left( {{a}^{+}} \right)}^{n}} \right]=n{{\left( {{a}^{+}} \right)}^{n-1}} \\
& \Rightarrow \left[ a,{{\left( {{a}^{+}} \right)}^{n+1}} \right]=n{{\left( {{a}^{+}} \right)}^{n-1}}{{a}^{+}}+{{\left( {{a}^{+}} \right)}^{n}}=\left( n+1 \right){{\left( {{a}^{+}} \right)}^{n}} \\
\end{align}
TeX (checked):
{\begin{aligned}&\left[a,{{\left({{a}^{+}}\right)}^{n+1}}\right]=a{{\left({{a}^{+}}\right)}^{n+1}}-{{\left({{a}^{+}}\right)}^{n+1}}a=a{{\left({{a}^{+}}\right)}^{n+1}}-{{\left({{a}^{+}}\right)}^{n}}a{{a}^{+}}+{{\left({{a}^{+}}\right)}^{n}}a{{a}^{+}}-{{\left({{a}^{+}}\right)}^{n+1}}a\\&\Rightarrow \left[a,{{\left({{a}^{+}}\right)}^{n+1}}\right]=\left[a,{{\left({{a}^{+}}\right)}^{n}}\right]{{a}^{+}}+{{\left({{a}^{+}}\right)}^{n}}\left[a,{{a}^{+}}\right]\\&\left[a,{{\left({{a}^{+}}\right)}^{n}}\right]=n{{\left({{a}^{+}}\right)}^{n-1}}\\&\Rightarrow \left[a,{{\left({{a}^{+}}\right)}^{n+1}}\right]=n{{\left({{a}^{+}}\right)}^{n-1}}{{a}^{+}}+{{\left({{a}^{+}}\right)}^{n}}=\left(n+1\right){{\left({{a}^{+}}\right)}^{n}}\\\end{aligned}}
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