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Display information for equation id:math.1661.84 on revision:1661

* Page found: Der harmonische Oszillator (eq math.1661.84)

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Der harmonische Oszillator Eq: math.1661.84

Hash: 6fe3479b20c45bba01b561e01ede8862

TeX (original user input):

\begin{align}
& {{\phi }_{1}}(\xi )={{a}^{+}}{{\phi }_{0}}(\xi )=\frac{1}{i\sqrt{2}}\left( \xi -\frac{d}{d\xi } \right){{\phi }_{0}}(\xi )=-\frac{1}{i\sqrt{2}}{{e}^{\left( \frac{{{\xi }^{2}}}{2} \right)}}\frac{d}{d\xi }\left( {{e}^{\left( -\frac{{{\xi }^{2}}}{2} \right)}}{{\phi }_{0}}(\xi ) \right) \\
& \Rightarrow {{\phi }_{1}}(\xi )=-\frac{1}{i\sqrt{2}}{{e}^{\left( \frac{{{\xi }^{2}}}{2} \right)}}\frac{d}{d\xi }\left( {{A}_{0}}{{e}^{\left( -{{\xi }^{2}} \right)}} \right) \\
& {{A}_{0}}={{\left( \frac{m\omega }{\hbar \pi } \right)}^{\frac{1}{4}}} \\
\end{align}

TeX (checked):

{\begin{aligned}&{{\phi }_{1}}(\xi )={{a}^{+}}{{\phi }_{0}}(\xi )={\frac {1}{i{\sqrt {2}}}}\left(\xi -{\frac {d}{d\xi }}\right){{\phi }_{0}}(\xi )=-{\frac {1}{i{\sqrt {2}}}}{{e}^{\left({\frac {{\xi }^{2}}{2}}\right)}}{\frac {d}{d\xi }}\left({{e}^{\left(-{\frac {{\xi }^{2}}{2}}\right)}}{{\phi }_{0}}(\xi )\right)\\&\Rightarrow {{\phi }_{1}}(\xi )=-{\frac {1}{i{\sqrt {2}}}}{{e}^{\left({\frac {{\xi }^{2}}{2}}\right)}}{\frac {d}{d\xi }}\left({{A}_{0}}{{e}^{\left(-{{\xi }^{2}}\right)}}\right)\\&{{A}_{0}}={{\left({\frac {m\omega }{\hbar \pi }}\right)}^{\frac {1}{4}}}\\\end{aligned}}

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MathML (5.169 KB / 561 B) :

\begin{aligned} ϕ_{1} (ξ) = a^{+} ϕ_{0} (ξ) = \frac{1}{i \sqrt{2}} (ξ - \frac{d}{d ξ}) ϕ_{0} (ξ) = - \frac{1}{i \sqrt{2}} e^{(\frac{ξ^{2}}{2})} \frac{d}{d ξ} (e^{(- \frac{ξ^{2}}{2})} ϕ_{0} (ξ)) \\ \Rightarrow ϕ_{1} (ξ) = - \frac{1}{i \sqrt{2}} e^{(\frac{ξ^{2}}{2})} \frac{d}{d ξ} (A_{0} e^{(- ξ^{2})}) \\ A_{0} = {(\frac{m ω}{ℏ π})}^{\frac{1}{4}} \end{aligned}

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Identifiers

$ϕ_{1}$

$ξ$

$a$

$ϕ_{0}$

$ξ$

$i$

$ξ$

$d$

$d$

$ξ$

$ϕ_{0}$

$ξ$

$i$

$e$

$ξ$

$d$

$d$

$ξ$

$e$

$ξ$

$ϕ_{0}$

$ξ$

$ϕ_{1}$

$ξ$

$i$

$e$

$ξ$

$d$

$d$

$ξ$

$A_{0}$

$e$

$ξ$

$A_{0}$

$m$

$ω$

$π$

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