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Display information for equation id:math.1783.25 on revision:1783

* Page found: Streuamplitude und Streuquerschnitt (eq math.1783.25)

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Streuamplitude und Streuquerschnitt Eq: math.1783.25

Hash: 92a9ab91615a03331449c7ed5cd36f84

TeX (original user input):

\begin{align}
  & {{\left( {{{\bar{j}}}_{s}} \right)}_{r}}=\frac{\hbar }{2im}\left( {{\Psi }_{S}}*\frac{\partial }{\partial r}{{\Psi }_{S}}-{{\Psi }_{S}}\frac{\partial }{\partial r}{{\Psi }_{S}}* \right)=\frac{\hbar }{2im}{{\left| f({{{\bar{e}}}_{r}}) \right|}^{2}}\left( \frac{{{e}^{-ikr}}}{r}\frac{\partial }{\partial r}\frac{{{e}^{ikr}}}{r}-\frac{{{e}^{ikr}}}{r}\frac{\partial }{\partial r}\frac{{{e}^{-ikr}}}{r} \right) \\ 
 & \Rightarrow {{\left( {{{\bar{j}}}_{s}} \right)}_{r}}=\frac{\hbar }{2im}{{\left| f({{{\bar{e}}}_{r}}) \right|}^{2}}\left( \frac{{{e}^{-ikr}}}{r}\left( \frac{ik}{r}-\frac{1}{{{r}^{2}}} \right){{e}^{ikr}}-\frac{{{e}^{ikr}}}{r}\left( -\frac{ik}{r}-\frac{1}{{{r}^{2}}} \right){{e}^{-ikr}} \right)=\frac{\hbar \bar{k}}{m{{r}^{2}}}{{\left| f({{{\bar{e}}}_{r}}) \right|}^{2}} \\ 
\end{align}

TeX (checked):

{\begin{aligned}&{{\left({{\bar {j}}_{s}}\right)}_{r}}={\frac {\hbar }{2im}}\left({{\Psi }_{S}}*{\frac {\partial }{\partial r}}{{\Psi }_{S}}-{{\Psi }_{S}}{\frac {\partial }{\partial r}}{{\Psi }_{S}}*\right)={\frac {\hbar }{2im}}{{\left|f({{\bar {e}}_{r}})\right|}^{2}}\left({\frac {{e}^{-ikr}}{r}}{\frac {\partial }{\partial r}}{\frac {{e}^{ikr}}{r}}-{\frac {{e}^{ikr}}{r}}{\frac {\partial }{\partial r}}{\frac {{e}^{-ikr}}{r}}\right)\\&\Rightarrow {{\left({{\bar {j}}_{s}}\right)}_{r}}={\frac {\hbar }{2im}}{{\left|f({{\bar {e}}_{r}})\right|}^{2}}\left({\frac {{e}^{-ikr}}{r}}\left({\frac {ik}{r}}-{\frac {1}{{r}^{2}}}\right){{e}^{ikr}}-{\frac {{e}^{ikr}}{r}}\left(-{\frac {ik}{r}}-{\frac {1}{{r}^{2}}}\right){{e}^{-ikr}}\right)={\frac {\hbar {\bar {k}}}{m{{r}^{2}}}}{{\left|f({{\bar {e}}_{r}})\right|}^{2}}\\\end{aligned}}

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\begin{aligned} {({\bar{j}}_{s})}_{r} = \frac{ℏ}{2 i m} (Ψ_{S} * \frac{\partial}{\partial r} Ψ_{S} - Ψ_{S} \frac{\partial}{\partial r} Ψ_{S} *) = \frac{ℏ}{2 i m} {| f ({\bar{e}}_{r}) |}^{2} (\frac{e^{- i k r}}{r} \frac{\partial}{\partial r} \frac{e^{i k r}}{r} - \frac{e^{i k r}}{r} \frac{\partial}{\partial r} \frac{e^{- i k r}}{r}) \\ \Rightarrow {({\bar{j}}_{s})}_{r} = \frac{ℏ}{2 i m} {| f ({\bar{e}}_{r}) |}^{2} (\frac{e^{- i k r}}{r} (\frac{i k}{r} - \frac{1}{r^{2}}) e^{i k r} - \frac{e^{i k r}}{r} (- \frac{i k}{r} - \frac{1}{r^{2}}) e^{- i k r}) = \frac{ℏ \bar{k}}{m r^{2}} {| f ({\bar{e}}_{r}) |}^{2} \end{aligned}

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Identifiers

${\bar{j}}_{s} r$

$i$

$m$

$Ψ_{S}$

$r$

$Ψ_{S}$

$Ψ_{S}$

$r$

$Ψ_{S}$

$i$

$m$

$f$

${\bar{e}}_{r}$

$e$

$i$

$k$

$r$

$r$

$r$

$e$

$i$

$k$

$r$

$r$

$e$

$i$

$k$

$r$

$r$

$r$

$e$

$i$

$k$

$r$

$r$

${\bar{j}}_{s} r$

$i$

$m$

$f$

${\bar{e}}_{r}$

$e$

$i$

$k$

$r$

$r$

$i$

$k$

$r$

$r$

$e$

$i$

$k$

$r$

$e$

$i$

$k$

$r$

$r$

$i$

$k$

$r$

$r$

$e$

$i$

$k$

$r$

$\bar{k}$

$m$

$r$

$f$

${\bar{e}}_{r}$

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