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* Page found: Kugelsymmetrische Potentiale (eq math.1677.34)

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Kugelsymmetrische Potentiale Eq: math.1677.34

Hash: eb7985616797051e240f1b039edae357

TeX (original user input):

\begin{align}

& \left( \bar{r}\cdot \bar{p} \right)\left[ \left( \bar{r}\cdot \bar{p} \right)+\frac{\hbar }{i} \right]\Psi (r,\vartheta ,\phi )=-{{\hbar }^{2}}r\frac{\partial }{\partial r}\left( r\frac{\partial }{\partial r}+1 \right)\Psi (r,\vartheta ,\phi ) \\

& =-{{\hbar }^{2}}r\left[ \frac{\partial }{\partial r}\left( r\frac{\partial \Psi }{\partial r} \right)+\frac{\partial \Psi }{\partial r} \right]=-{{\hbar }^{2}}r\left[ \left( r\frac{{{\partial }^{2}}\Psi }{\partial {{r}^{2}}} \right)+2\frac{\partial \Psi }{\partial r} \right]=-{{\hbar }^{2}}r\frac{{{\partial }^{2}}}{\partial {{r}^{2}}}\left( r\Psi  \right) \\

\end{align}

TeX (checked):

{\begin{aligned}&\left({\bar {r}}\cdot {\bar {p}}\right)\left[\left({\bar {r}}\cdot {\bar {p}}\right)+{\frac {\hbar }{i}}\right]\Psi (r,\vartheta ,\phi )=-{{\hbar }^{2}}r{\frac {\partial }{\partial r}}\left(r{\frac {\partial }{\partial r}}+1\right)\Psi (r,\vartheta ,\phi )\\&=-{{\hbar }^{2}}r\left[{\frac {\partial }{\partial r}}\left(r{\frac {\partial \Psi }{\partial r}}\right)+{\frac {\partial \Psi }{\partial r}}\right]=-{{\hbar }^{2}}r\left[\left(r{\frac {{{\partial }^{2}}\Psi }{\partial {{r}^{2}}}}\right)+2{\frac {\partial \Psi }{\partial r}}\right]=-{{\hbar }^{2}}r{\frac {{\partial }^{2}}{\partial {{r}^{2}}}}\left(r\Psi \right)\\\end{aligned}}

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\begin{aligned} (\bar{r} \cdot \bar{p}) [(\bar{r} \cdot \bar{p}) + \frac{ℏ}{i}] Ψ (r, ϑ, ϕ) = - ℏ^{2} r \frac{\partial}{\partial r} (r \frac{\partial}{\partial r} + 1) Ψ (r, ϑ, ϕ) \\ = - ℏ^{2} r [\frac{\partial}{\partial r} (r \frac{\partial Ψ}{\partial r}) + \frac{\partial Ψ}{\partial r}] = - ℏ^{2} r [(r \frac{\partial^{2} Ψ}{\partial r^{2}}) + 2 \frac{\partial Ψ}{\partial r}] = - ℏ^{2} r \frac{\partial^{2}}{\partial r^{2}} (r Ψ) \end{aligned}

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Identifiers

$\bar{r}$

$\bar{p}$

$\bar{r}$

$\bar{p}$

$i$

$Ψ$

$r$

$ϑ$

$ϕ$

$r$

$r$

$r$

$r$

$Ψ$

$r$

$ϑ$

$ϕ$

$r$

$r$

$r$

$Ψ$

$r$

$Ψ$

$r$

$r$

$r$

$Ψ$

$r$

$Ψ$

$r$

$r$

$r$

$r$

$Ψ$

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