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* Page found: Drehimpuls- Eigenzustände (eq math.1667.4)

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\begin{align}
  & \left[ {{{\hat{L}}}_{1}},{{{\hat{L}}}_{2}} \right]=\left[ \left( {{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}-{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}} \right),\left( {{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}-{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}} \right) \right]={{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}-{{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}-{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}+{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}}{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}-{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}{{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}+{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}}+{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}-{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}} \\ 
 & ={{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}-{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}}+{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}}{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}-{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}{{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}={{{\hat{r}}}_{2}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{1}}-{{{\hat{r}}}_{2}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{3}}{{{\hat{p}}}_{1}}+{{{\hat{r}}}_{1}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{3}}{{{\hat{p}}}_{2}}-{{{\hat{r}}}_{1}}{{{\hat{p}}}_{3}}{{{\hat{r}}}_{3}}{{{\hat{p}}}_{2}} \\ 
 & ={{{\hat{r}}}_{2}}\left[ {{{\hat{p}}}_{3}},{{{\hat{r}}}_{3}} \right]{{{\hat{p}}}_{1}}+{{{\hat{r}}}_{1}}\left[ {{{\hat{r}}}_{3}},{{{\hat{p}}}_{3}} \right]{{{\hat{p}}}_{2}}=\frac{\hbar }{i}{{{\hat{r}}}_{2}}{{{\hat{p}}}_{1}}-\frac{\hbar }{i}{{{\hat{r}}}_{1}}{{{\hat{p}}}_{2}}=i\hbar {{{\hat{L}}}_{3}} \\ 
\end{align}

TeX (checked): 

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

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[L^1,L^2]=[(r^2p^3−r^3p^2),(r^3p^1−r^1p^3)]=r^2p^3r^3p^1−r^2p^3r^1p^3−r^3p^2r^3p^1+r^3p^2r^1p^3−r^3p^1r^2p^3+r^3p^1r^3p^2+r^1p^3r^2p^3−r^1p^3r^3p^2=r^2p^3r^3p^1−r^1p^3r^3p^2+r^3p^2r^1p^3−r^3p^1r^2p^3=r^2p^3r^3p^1−r^2r^3p^3p^1+r^1r^3p^3p^2−r^1p^3r^3p^2=r^2[p^3,r^3]p^1+r^1[r^3,p^3]p^2=ℏir^2p^1−ℏir^1p^2=iℏL^3
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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>3</mn></mrow></msub><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</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><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><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</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>1</mn></mrow></msub><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</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>3</mn></mrow></msub><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub></mtd></mtr><mtr><mtd></mtd><mtd><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>p</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>p</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>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>p</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>p</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi data-mjx-alternate="1">&#x210F;</mi></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></mfrac></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><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo>&#x2212;</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi data-mjx-alternate="1">&#x210F;</mi></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></mfrac></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><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><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>L</mi><mo>^</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>

Translations to Computer Algebra Systems

Translation to Maple

In Maple:

Translation to Mathematica

In Mathematica:

Similar pages

Calculated based on the variables occurring on the entire Drehimpuls- Eigenzustände page

Identifiers

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

MathML observations

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Abgerufen von „https://wiki.physikerwelt.de/wiki/Spezial:Formelinformation“