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* Page found: Kugelsymmetrische Potentiale (eq math.1673.3)

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\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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[L̂3,r̂12]=[L̂3,r̂1]r̂1+r̂1[L̂3,r̂1]=iℏr̂2r̂1+r̂1iℏr̂2=2iℏr̂2r̂1[L̂3,r̂22]=[L̂3,r̂2]r̂2+r̂2[L̂3,r̂2]=−iℏr̂1r̂2−r̂2iℏr̂1=−2iℏr̂2r̂[L̂3,r̂32]=[L̂3,r̂3]r̂3+r̂3[L̂3,r̂3]=0
<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><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo>,</mo><msup><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><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 stretchy="false">=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo>,</mo><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo stretchy="false">+</mo><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo>,</mo><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo stretchy="false">=</mo><mi>i</mi><mi alternate="1">ℏ</mi><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo stretchy="false">+</mo><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mi>i</mi><mi alternate="1">ℏ</mi><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo stretchy="false">=</mo><mn>2</mn><mi>i</mi><mi alternate="1">ℏ</mi><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mtd></mtr><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><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo>,</mo><msup><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><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 stretchy="false">=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo>,</mo><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo stretchy="false">+</mo><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo>,</mo><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo stretchy="false">=</mo><mo stretchy="false">−</mo><mi>i</mi><mi alternate="1">ℏ</mi><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo stretchy="false">−</mo><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mi>i</mi><mi alternate="1">ℏ</mi><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo stretchy="false">=</mo><mo stretchy="false">−</mo><mn>2</mn><mi>i</mi><mi alternate="1">ℏ</mi><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover></mtd></mtr><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><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo>,</mo><msup><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><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 stretchy="false">=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo>,</mo><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo stretchy="false">+</mo><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo>,</mo><msub><mover><mi>r</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo stretchy="false">=</mo><mn>0</mn></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

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Identifiers

  • L̂3
  • r̂1
  • L̂3
  • r̂1
  • r̂1
  • r̂1
  • L̂3
  • r̂1
  • i
  • r̂2
  • r̂1
  • r̂1
  • i
  • r̂2
  • i
  • r̂2
  • r̂1
  • L̂3
  • r̂2
  • L̂3
  • r̂2
  • r̂2
  • r̂2
  • L̂3
  • r̂2
  • i
  • r̂1
  • r̂2
  • r̂2
  • i
  • r̂1
  • i
  • r̂2
  • r̂
  • L̂3
  • r̂3
  • L̂3
  • r̂3
  • r̂3
  • r̂3
  • L̂3
  • r̂3

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