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Display information for equation id:math.1661.45 on revision:1661
* Page found: Der harmonische Oszillator (eq math.1661.45)
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TeX (original user input):
\begin{align}
& \hat{H}{{\left( {{a}^{+}} \right)}^{n+1}}\left| 0 \right\rangle =\left( {{a}^{+}}\hat{H}+\hbar \omega {{a}^{+}} \right){{\left( {{a}^{+}} \right)}^{n}}\left| 0 \right\rangle ={{a}^{+}}\left( \hat{H}+\hbar \omega \right){{\left( {{a}^{+}} \right)}^{n}}\left| 0 \right\rangle \\
& \left( \hat{H}+\hbar \omega \right){{\left( {{a}^{+}} \right)}^{n}}\left| 0 \right\rangle =\left( \hbar \omega \left( n+\frac{1}{2} \right)+\hbar \omega \right){{\left( {{a}^{+}} \right)}^{n}}\left| 0 \right\rangle \\
& \Rightarrow \hat{H}{{\left( {{a}^{+}} \right)}^{n+1}}\left| 0 \right\rangle ={{a}^{+}}\left( \hat{H}+\hbar \omega \right){{\left( {{a}^{+}} \right)}^{n}}\left| 0 \right\rangle =\hbar \omega \left( n+1+\frac{1}{2} \right){{\left( {{a}^{+}} \right)}^{n+1}}\left| 0 \right\rangle \\
\end{align}
TeX (checked):
{\begin{aligned}&{\hat {H}}{{\left({{a}^{+}}\right)}^{n+1}}\left|0\right\rangle =\left({{a}^{+}}{\hat {H}}+\hbar \omega {{a}^{+}}\right){{\left({{a}^{+}}\right)}^{n}}\left|0\right\rangle ={{a}^{+}}\left({\hat {H}}+\hbar \omega \right){{\left({{a}^{+}}\right)}^{n}}\left|0\right\rangle \\&\left({\hat {H}}+\hbar \omega \right){{\left({{a}^{+}}\right)}^{n}}\left|0\right\rangle =\left(\hbar \omega \left(n+{\frac {1}{2}}\right)+\hbar \omega \right){{\left({{a}^{+}}\right)}^{n}}\left|0\right\rangle \\&\Rightarrow {\hat {H}}{{\left({{a}^{+}}\right)}^{n+1}}\left|0\right\rangle ={{a}^{+}}\left({\hat {H}}+\hbar \omega \right){{\left({{a}^{+}}\right)}^{n}}\left|0\right\rangle =\hbar \omega \left(n+1+{\frac {1}{2}}\right){{\left({{a}^{+}}\right)}^{n+1}}\left|0\right\rangle \\\end{aligned}}
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data-mjx-texclass="OPEN">|</mo><mn>0</mn><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>H</mi><mo>^</mo></mover></mrow></mrow><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>a</mi><mrow data-mjx-texclass="ORD"><mo>+</mo></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mn>0</mn><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo>=</mo><msup><mi>a</mi><mrow data-mjx-texclass="ORD"><mo>+</mo></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>H</mi><mo>^</mo></mover></mrow></mrow><mo>+</mo><mi data-mjx-alternate="1">ℏ</mi><mi>ω</mi><mo 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