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Display information for equation id:math.1667.20 on revision:1667

* Page found: Drehimpuls- Eigenzustände (eq math.1667.20)

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'''\begin{align}

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

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

& \to \left[ {{{\hat{L}}}_{+}},{{{\hat{L}}}_{-}} \right]=2\hbar {{{\hat{L}}}_{3}} \\

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

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

\end{align}

TeX (checked): 

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

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L^+L^−=(L^1+iL^2)(L^1−iL^2)=L^12+L^22−i[L^1,L^2]=L^2−L^32+ℏL^3L^−L^+=L^12+L^22+i[L^1,L^2]=L^2−L^32−ℏL^3→[L^+,L^−]=2ℏL^3[L^2,L^+]=0[L^2,L^−]=0
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data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo>+</mo><mi>i</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>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><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>1</mn></mrow></msub><mo>&#x2212;</mo><mi>i</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>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><msup><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>1</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>+</mo><msup><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>2</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>&#x2212;</mo><mi>i</mi><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>1</mn></mrow></msub><mo>,</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>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><msup><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>2</mn></mrow></msup><mo>&#x2212;</mo><msup><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><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>+</mo><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><mtd><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"><mo>&#x2212;</mo></mrow></msub><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"><mo>+</mo></mrow></msub><mo>=</mo><msup><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>1</mn></mrow></msub><mrow 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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><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>&#x2212;</mo><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><mtd><mo accent="false">&#x2192;</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"><mo>+</mo></mrow></msub><mo>,</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"><mo>&#x2212;</mo></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mn>2</mn><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><mtd><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msup><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>2</mn></mrow></msup><mo>,</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"><mo>+</mo></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msup><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>2</mn></mrow></msup><mo>,</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"><mo>&#x2212;</mo></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 Drehimpuls- Eigenzustände page

Identifiers

  • L^+
  • L^−
  • L^1
  • i
  • L^2
  • L^1
  • i
  • L^2
  • L^1
  • L^2
  • i
  • L^1
  • L^2
  • L^
  • L^3
  • L^3
  • L^−
  • L^+
  • L^1
  • L^2
  • i
  • L^1
  • L^2
  • L^
  • L^3
  • L^3
  • L^+
  • L^−
  • L^3
  • L^
  • L^+
  • L^
  • L^−

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