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Display information for equation id:math.1651.45 on revision:1651

* Page found: Der harmonische Oszillator (eq math.1651.45)

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Der harmonische Oszillator Eq: math.1661.45

Hash: 7cf944ac7427204500df210831957a9f

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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\begin{matrix} \hat{H} {(a^{+})}^{n + 1} | 0 ⟩ = (a^{+} \hat{H} + ℏ ω a^{+}) {(a^{+})}^{n} | 0 ⟩ = a^{+} (\hat{H} + ℏ ω) {(a^{+})}^{n} | 0 ⟩ \\ (\hat{H} + ℏ ω) {(a^{+})}^{n} | 0 ⟩ = (ℏ ω (n + \frac{1}{2}) + ℏ ω) {(a^{+})}^{n} | 0 ⟩ \\ \Rightarrow \hat{H} {(a^{+})}^{n + 1} | 0 ⟩ = a^{+} (\hat{H} + ℏ ω) {(a^{+})}^{n} | 0 ⟩ = ℏ ω (n + 1 + \frac{1}{2}) {(a^{+})}^{n + 1} | 0 ⟩ \end{matrix}

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Identifiers

$\hat{H}$

$a$

$n$

$a$

$\hat{H}$

$ω$

$a$

$a$

$n$

$a$

$\hat{H}$

$ω$

$a$

$n$

$\hat{H}$

$ω$

$a$

$n$

$ω$

$n$

$ω$

$a$

$n$

$\hat{H}$

$a$

$n$

$a$

$\hat{H}$

$ω$

$a$

$n$

$ω$

$n$

$a$

$n$

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