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Display information for equation id:math.1676.50 on revision:1676
* Page found: Kugelsymmetrische Potentiale (eq math.1676.50)
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Hash: 25dd81bfa20a5f91dd44be31f13bd7a8
TeX (original user input):
\begin{align}
& -\frac{{{\hbar }^{2}}}{2m}\frac{Y}{r}\frac{{{d}^{2}}}{d{{r}^{2}}}\left( rR \right)+\frac{R}{2m{{r}^{2}}}\left( {{L}^{2}}Y \right)+Y\left( V(r)-E \right)R=0 \\
& {{L}^{2}}Y={{\hbar }^{2}}l(l+1)Y \\
& \Rightarrow -\frac{{{\hbar }^{2}}}{2m}\frac{Y}{r}\frac{{{d}^{2}}}{d{{r}^{2}}}\left( rR \right)+\frac{R}{2m{{r}^{2}}}\left( {{\hbar }^{2}}l(l+1)Y \right)+Y\left( V(r)-E \right)R=0 \\
& \Rightarrow -\frac{{{\hbar }^{2}}}{2m}\frac{{{d}^{2}}}{d{{r}^{2}}}\left( rR \right)+\left( \frac{{{\hbar }^{2}}l(l+1)}{2m{{r}^{2}}}+V(r)-E \right)\left( rR \right)=0 \\
\end{align}
TeX (checked):
{\begin{aligned}&-{\frac {{\hbar }^{2}}{2m}}{\frac {Y}{r}}{\frac {{d}^{2}}{d{{r}^{2}}}}\left(rR\right)+{\frac {R}{2m{{r}^{2}}}}\left({{L}^{2}}Y\right)+Y\left(V(r)-E\right)R=0\\&{{L}^{2}}Y={{\hbar }^{2}}l(l+1)Y\\&\Rightarrow -{\frac {{\hbar }^{2}}{2m}}{\frac {Y}{r}}{\frac {{d}^{2}}{d{{r}^{2}}}}\left(rR\right)+{\frac {R}{2m{{r}^{2}}}}\left({{\hbar }^{2}}l(l+1)Y\right)+Y\left(V(r)-E\right)R=0\\&\Rightarrow -{\frac {{\hbar }^{2}}{2m}}{\frac {{d}^{2}}{d{{r}^{2}}}}\left(rR\right)+\left({\frac {{{\hbar }^{2}}l(l+1)}{2m{{r}^{2}}}}+V(r)-E\right)\left(rR\right)=0\\\end{aligned}}
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<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd></mtd><mtd><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi data-mjx-alternate="1">ℏ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>m</mi></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>Y</mi></mrow><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>d</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>R</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>R</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>m</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><msup><mi>L</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi>Y</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mi>Y</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>−</mo><mi>E</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>R</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd><mtd><msup><mi>L</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi>Y</mi><mo>=</mo><msup><mi data-mjx-alternate="1">ℏ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi>l</mi><mo stretchy="false">(</mo><mi>l</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Y</mi></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi data-mjx-alternate="1">ℏ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>m</mi></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>Y</mi></mrow><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>d</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>R</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>R</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>m</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><msup><mi data-mjx-alternate="1">ℏ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi>l</mi><mo stretchy="false">(</mo><mi>l</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Y</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mi>Y</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>−</mo><mi>E</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>R</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi data-mjx-alternate="1">ℏ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>m</mi></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>d</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>R</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msup><mi data-mjx-alternate="1">ℏ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi>l</mi><mo stretchy="false">(</mo><mi>l</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>m</mi><msup><mi>r</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mrow></mfrac></mrow><mo>+</mo><mi>V</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>−</mo><mi>E</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>r</mi><mi>R</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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