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* Page found: Kugelsymmetrische Potentiale (eq math.1677.3)

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TeX (original user input): 

\begin{align}

& \left[ {{{\hat{L}}}_{3}},{{{\hat{r}}}_{1}}^{2} \right]=\left[ {{{\hat{L}}}_{3}},{{{\hat{r}}}_{1}} \right]{{{\hat{r}}}_{1}}+{{{\hat{r}}}_{1}}\left[ {{{\hat{L}}}_{3}},{{{\hat{r}}}_{1}} \right]=i\hbar {{{\hat{r}}}_{2}}{{{\hat{r}}}_{1}}+{{{\hat{r}}}_{1}}i\hbar {{{\hat{r}}}_{2}}=2i\hbar {{{\hat{r}}}_{2}}{{{\hat{r}}}_{1}} \\

& \left[ {{{\hat{L}}}_{3}},{{{\hat{r}}}_{2}}^{2} \right]=\left[ {{{\hat{L}}}_{3}},{{{\hat{r}}}_{2}} \right]{{{\hat{r}}}_{2}}+{{{\hat{r}}}_{2}}\left[ {{{\hat{L}}}_{3}},{{{\hat{r}}}_{2}} \right]=-i\hbar {{{\hat{r}}}_{1}}{{{\hat{r}}}_{2}}-{{{\hat{r}}}_{2}}i\hbar {{{\hat{r}}}_{1}}=-2i\hbar {{{\hat{r}}}_{2}}\hat{r} \\

& \left[ {{{\hat{L}}}_{3}},{{{\hat{r}}}_{3}}^{2} \right]=\left[ {{{\hat{L}}}_{3}},{{{\hat{r}}}_{3}} \right]{{{\hat{r}}}_{3}}+{{{\hat{r}}}_{3}}\left[ {{{\hat{L}}}_{3}},{{{\hat{r}}}_{3}} \right]=0 \\

\end{align}

TeX (checked): 

{\begin{aligned}&\left[{{\hat {L}}_{3}},{{\hat {r}}_{1}}^{2}\right]=\left[{{\hat {L}}_{3}},{{\hat {r}}_{1}}\right]{{\hat {r}}_{1}}+{{\hat {r}}_{1}}\left[{{\hat {L}}_{3}},{{\hat {r}}_{1}}\right]=i\hbar {{\hat {r}}_{2}}{{\hat {r}}_{1}}+{{\hat {r}}_{1}}i\hbar {{\hat {r}}_{2}}=2i\hbar {{\hat {r}}_{2}}{{\hat {r}}_{1}}\\&\left[{{\hat {L}}_{3}},{{\hat {r}}_{2}}^{2}\right]=\left[{{\hat {L}}_{3}},{{\hat {r}}_{2}}\right]{{\hat {r}}_{2}}+{{\hat {r}}_{2}}\left[{{\hat {L}}_{3}},{{\hat {r}}_{2}}\right]=-i\hbar {{\hat {r}}_{1}}{{\hat {r}}_{2}}-{{\hat {r}}_{2}}i\hbar {{\hat {r}}_{1}}=-2i\hbar {{\hat {r}}_{2}}{\hat {r}}\\&\left[{{\hat {L}}_{3}},{{\hat {r}}_{3}}^{2}\right]=\left[{{\hat {L}}_{3}},{{\hat {r}}_{3}}\right]{{\hat {r}}_{3}}+{{\hat {r}}_{3}}\left[{{\hat {L}}_{3}},{{\hat {r}}_{3}}\right]=0\\\end{aligned}}

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[L^3,r^12]=[L^3,r^1]r^1+r^1[L^3,r^1]=iℏr^2r^1+r^1iℏr^2=2iℏr^2r^1[L^3,r^22]=[L^3,r^2]r^2+r^2[L^3,r^2]=−iℏr^1r^2−r^2iℏr^1=−2iℏr^2r^[L^3,r^32]=[L^3,r^3]r^3+r^3[L^3,r^3]=0
<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><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>L</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo>,</mo><msup><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>L</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>L</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mi>i</mi><mi data-mjx-alternate="1">&#x210F;</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mi>i</mi><mi data-mjx-alternate="1">&#x210F;</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo>=</mo><mn>2</mn><mi>i</mi><mi data-mjx-alternate="1">&#x210F;</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>L</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo>,</mo><msup><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>L</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>L</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mo>&#x2212;</mo><mi>i</mi><mi data-mjx-alternate="1">&#x210F;</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo>&#x2212;</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mi>i</mi><mi data-mjx-alternate="1">&#x210F;</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo>=</mo><mo>&#x2212;</mo><mn>2</mn><mi>i</mi><mi data-mjx-alternate="1">&#x210F;</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>L</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo>,</mo><msup><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>L</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo>+</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>L</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><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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Calculated based on the variables occurring on the entire Kugelsymmetrische Potentiale page

Identifiers

  • L^3
  • r^1
  • L^3
  • r^1
  • r^1
  • r^1
  • L^3
  • r^1
  • i
  • r^2
  • r^1
  • r^1
  • i
  • r^2
  • i
  • r^2
  • r^1
  • L^3
  • r^2
  • L^3
  • r^2
  • r^2
  • r^2
  • L^3
  • r^2
  • i
  • r^1
  • r^2
  • r^2
  • i
  • r^1
  • i
  • r^2
  • r^
  • L^3
  • r^3
  • L^3
  • r^3
  • r^3
  • r^3
  • L^3
  • r^3

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