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Display information for equation id:math.1661.85 on revision:1661

* Page found: Der harmonische Oszillator (eq math.1661.85)

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Der harmonische Oszillator Eq: math.1661.85

Hash: 5f9278a67673087294c29166f7a5bc83

TeX (original user input):

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

TeX (checked):

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

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

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Identifiers

$ϕ_{n}$

$ξ$

$a$

$n$

$n$

$ϕ_{0}$

$ξ$

$i$

$n$

$n$

$n$

$ξ$

$d$

$d$

$ξ$

$n$

$ϕ_{0}$

$ξ$

$i$

$n$

$A_{0}$

$n$

$n$

$n$

$e$

$ξ$

$d$

$n$

$d$

$ξ$

$n$

$e$

$ξ$

$A_{0}$

$n$

$n$

$A_{n}$

$A_{0}$

$m$

$ω$

$π$

$n$

$e$

$ξ$

$d$

$n$

$d$

$ξ$

$n$

$e$

$ξ$

$H_{n}$

$ξ$

$e$

$ξ$

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