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Display information for equation id:math.1732.62 on revision:1732

* Page found: Zeitabhängige Störungsrechnung (eq math.1732.62)

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Hash: 370c9ef2f7063ef0c91830cf466a0e4f

TeX (original user input):

\begin{align}
& {{D}_{t}}(0)={{\left( \frac{t}{\hbar } \right)}^{2}} \\
& \begin{matrix}
\lim   \\
t\to \infty   \\
\end{matrix}\left( {{D}_{t}}(0) \right)=\infty  \\
& \int_{-\infty }^{\infty }{{{D}_{t}}(E)}=\int_{-\infty }^{\infty }{{}}dE\frac{4{{\sin }^{2}}\left( \frac{Et}{2\hbar } \right)}{{{E}^{2}}}=\frac{2t}{\hbar }\int_{-\infty }^{\infty }{{}}d\xi \frac{{{\sin }^{2}}\xi }{{{\xi }^{2}}} \\
& \int_{-\infty }^{\infty }{{}}d\xi \frac{{{\sin }^{2}}\xi }{{{\xi }^{2}}}=\pi  \\
& \Rightarrow \int_{-\infty }^{\infty }{{{D}_{t}}(E)}=\frac{2\pi }{\hbar }t \\
\end{align}

TeX (checked):

{\begin{aligned}&{{D}_{t}}(0)={{\left({\frac {t}{\hbar }}\right)}^{2}}\\&{\begin{matrix}\lim \\t\to \infty \\\end{matrix}}\left({{D}_{t}}(0)\right)=\infty \\&\int _{-\infty }^{\infty }{{{D}_{t}}(E)}=\int _{-\infty }^{\infty }{}dE{\frac {4{{\sin }^{2}}\left({\frac {Et}{2\hbar }}\right)}{{E}^{2}}}={\frac {2t}{\hbar }}\int _{-\infty }^{\infty }{}d\xi {\frac {{{\sin }^{2}}\xi }{{\xi }^{2}}}\\&\int _{-\infty }^{\infty }{}d\xi {\frac {{{\sin }^{2}}\xi }{{\xi }^{2}}}=\pi \\&\Rightarrow \int _{-\infty }^{\infty }{{{D}_{t}}(E)}={\frac {2\pi }{\hbar }}t\\\end{aligned}}

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Dt(0)=(t)2limt(Dt(0))=Dt(E)=dE4sin2(Et2)E2=2tdξsin2ξξ2dξsin2ξξ2=πDt(E)=2πt
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Calculated based on the variables occurring on the entire Zeitabhängige Störungsrechnung page

Identifiers

  • Dt
  • t
  • t
  • Dt
  • Dt
  • E
  • E
  • E
  • t
  • E
  • t
  • ξ
  • ξ
  • ξ
  • ξ
  • ξ
  • ξ
  • π
  • Dt
  • E
  • π
  • t

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