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

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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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(r¯⋅p¯)[(r¯⋅p¯)+ℏi]Ψ(r,ϑ,ϕ)=−ℏ2r∂∂r(r∂∂r+1)Ψ(r,ϑ,ϕ)=−ℏ2r[∂∂r(r∂Ψ∂r)+∂Ψ∂r]=−ℏ2r[(r∂2Ψ∂r2)+2∂Ψ∂r]=−ℏ2r∂2∂r2(rΨ)
<math xmlns="http://www.w3.org/1998/Math/MathML" class="mwe-math-element mwe-math-element-inline"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mtable displaystyle="true"><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">⋅</mo><mover><mi>p</mi><mo>¯</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">⋅</mo><mover><mi>p</mi><mo>¯</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi alternate="1">ℏ</mi></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><mi>Ψ</mi><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>ϑ</mi><mo>,</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">−</mo><msup><mi alternate="1">ℏ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi>r</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>r</mi></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>r</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>r</mi></mrow></mrow></mfrac></mrow><mo stretchy="false">+</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>Ψ</mi><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>ϑ</mi><mo>,</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">=</mo><mo stretchy="false">−</mo><msup><mi alternate="1">ℏ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi>r</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>r</mi></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>r</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>Ψ</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>r</mi></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>Ψ</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>r</mi></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo stretchy="false">=</mo><mo stretchy="false">−</mo><msup><mi alternate="1">ℏ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi>r</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>r</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msup><mi>∂</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi>Ψ</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msup><mi>r</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">+</mo><mn>2</mn><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>Ψ</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>r</mi></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo stretchy="false">=</mo><mo stretchy="false">−</mo><msup><mi alternate="1">ℏ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi>r</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>∂</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msup><mi>r</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>r</mi><mi>Ψ</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

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Calculated based on the variables occurring on the entire Kugelsymmetrische Potentiale page

Identifiers

  • r¯
  • p¯
  • r¯
  • 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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