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Display information for equation id:math.1824.8 on revision:1824

* Page found: Das Wasserstoffatom (relativistsich) (eq math.1824.8)

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TeX (original user input): 

\begin{align}

& {{\left( \hbar Q \right)}^{2}}=\beta \left( \tilde{\bar{\sigma }}\bar{L}+\hbar  \right)\beta \left( \tilde{\bar{\sigma }}\bar{L}+\hbar  \right)={{\beta }^{2}}{{\left( \tilde{\bar{\sigma }}\bar{L}+\hbar  \right)}^{2}} \\

& \left[ \beta ,\tilde{\bar{\sigma }} \right]=0=\left( \begin{matrix}

\left[ 1,\tilde{\bar{\sigma }} \right] & {}  \\

{} & -\left[ 1,\tilde{\bar{\sigma }} \right]  \\

\end{matrix} \right) \\

& {{\beta }^{2}}=1 \\

& \Rightarrow {{\left( \hbar Q \right)}^{2}}=\left( \tilde{\bar{\sigma }}\bar{L} \right)\left( \tilde{\bar{\sigma }}\bar{L} \right)+2\hbar \left( \tilde{\bar{\sigma }}\bar{L} \right)+{{\hbar }^{2}} \\

& \left( \tilde{\bar{\sigma }}\bar{L} \right)\left( \tilde{\bar{\sigma }}\bar{L} \right)={{L}^{2}}+i\tilde{\bar{\sigma }}\left( \bar{L}\times \bar{L} \right) \\

& \left( \bar{L}\times \bar{L} \right)=i\hbar \bar{L} \\

& \Rightarrow \left( \tilde{\bar{\sigma }}\bar{L} \right)\left( \tilde{\bar{\sigma }}\bar{L} \right)={{L}^{2}}-\hbar \tilde{\bar{\sigma }}(\bar{L}) \\

\end{align}

TeX (checked): 

{\begin{aligned}&{{\left(\hbar Q\right)}^{2}}=\beta \left({\tilde {\bar {\sigma }}}{\bar {L}}+\hbar \right)\beta \left({\tilde {\bar {\sigma }}}{\bar {L}}+\hbar \right)={{\beta }^{2}}{{\left({\tilde {\bar {\sigma }}}{\bar {L}}+\hbar \right)}^{2}}\\&\left[\beta ,{\tilde {\bar {\sigma }}}\right]=0=\left({\begin{matrix}\left[1,{\tilde {\bar {\sigma }}}\right]&{}\\{}&-\left[1,{\tilde {\bar {\sigma }}}\right]\\\end{matrix}}\right)\\&{{\beta }^{2}}=1\\&\Rightarrow {{\left(\hbar Q\right)}^{2}}=\left({\tilde {\bar {\sigma }}}{\bar {L}}\right)\left({\tilde {\bar {\sigma }}}{\bar {L}}\right)+2\hbar \left({\tilde {\bar {\sigma }}}{\bar {L}}\right)+{{\hbar }^{2}}\\&\left({\tilde {\bar {\sigma }}}{\bar {L}}\right)\left({\tilde {\bar {\sigma }}}{\bar {L}}\right)={{L}^{2}}+i{\tilde {\bar {\sigma }}}\left({\bar {L}}\times {\bar {L}}\right)\\&\left({\bar {L}}\times {\bar {L}}\right)=i\hbar {\bar {L}}\\&\Rightarrow \left({\tilde {\bar {\sigma }}}{\bar {L}}\right)\left({\tilde {\bar {\sigma }}}{\bar {L}}\right)={{L}^{2}}-\hbar {\tilde {\bar {\sigma }}}({\bar {L}})\\\end{aligned}}

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(ℏQ)2=β(σ¯~L¯+ℏ)β(σ¯~L¯+ℏ)=β2(σ¯~L¯+ℏ)2[β,σ¯~]=0=([1,σ¯~]−[1,σ¯~])β2=1⇒(ℏQ)2=(σ¯~L¯)(σ¯~L¯)+2ℏ(σ¯~L¯)+ℏ2(σ¯~L¯)(σ¯~L¯)=L2+iσ¯~(L¯×L¯)(L¯×L¯)=iℏL¯⇒(σ¯~L¯)(σ¯~L¯)=L2−ℏσ¯~(L¯)
<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"><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi alternate="1">ℏ</mi><mi>Q</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">=</mo><mi>β</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo>~</mo></mover><mover><mi>L</mi><mo>¯</mo></mover><mo stretchy="false">+</mo><mi alternate="1">ℏ</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>β</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo>~</mo></mover><mover><mi>L</mi><mo>¯</mo></mover><mo stretchy="false">+</mo><mi alternate="1">ℏ</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">=</mo><msup><mi>β</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo>~</mo></mover><mover><mi>L</mi><mo>¯</mo></mover><mo stretchy="false">+</mo><mi alternate="1">ℏ</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></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><mi>β</mi><mo>,</mo><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo>~</mo></mover><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo stretchy="false">=</mo><mn>0</mn><mo stretchy="false">=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mo data-mjx-texclass="OPEN"></mo><mtable><mtr><mtd><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mn>1</mn><mo>,</mo><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo>~</mo></mover><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">−</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mn>1</mn><mo>,</mo><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo>~</mo></mover><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true" data-mjx-texclass="CLOSE"></mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><msup><mi>β</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">=</mo><mn>1</mn></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi alternate="1">ℏ</mi><mi>Q</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo>~</mo></mover><mover><mi>L</mi><mo>¯</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo>~</mo></mover><mover><mi>L</mi><mo>¯</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">+</mo><mn>2</mn><mi alternate="1">ℏ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo>~</mo></mover><mover><mi>L</mi><mo>¯</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">+</mo><msup><mi alternate="1">ℏ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></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><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo>~</mo></mover><mover><mi>L</mi><mo>¯</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo>~</mo></mover><mover><mi>L</mi><mo>¯</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">=</mo><msup><mi>L</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">+</mo><mi>i</mi><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo>~</mo></mover><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>L</mi><mo>¯</mo></mover><mo stretchy="false">×</mo><mover><mi>L</mi><mo>¯</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow></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><mover><mi>L</mi><mo>¯</mo></mover><mo stretchy="false">×</mo><mover><mi>L</mi><mo>¯</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">=</mo><mi>i</mi><mi alternate="1">ℏ</mi><mover><mi>L</mi><mo>¯</mo></mover></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo>~</mo></mover><mover><mi>L</mi><mo>¯</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo>~</mo></mover><mover><mi>L</mi><mo>¯</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">=</mo><msup><mi>L</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">−</mo><mi alternate="1">ℏ</mi><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo>~</mo></mover><mo stretchy="false">(</mo><mover><mi>L</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

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Calculated based on the variables occurring on the entire Das Wasserstoffatom (relativistsich) page

Identifiers

  • Q
  • β
  • σ¯~
  • L¯
  • β
  • σ¯~
  • L¯
  • β
  • σ¯~
  • L¯
  • β
  • σ¯~
  • σ¯~
  • σ¯~
  • β
  • Q
  • σ¯~
  • L¯
  • σ¯~
  • L¯
  • σ¯~
  • L¯
  • σ¯~
  • L¯
  • σ¯~
  • L¯
  • L
  • i
  • σ¯~
  • L¯
  • L¯
  • L¯
  • L¯
  • i
  • L¯
  • σ¯~
  • L¯
  • σ¯~
  • L¯
  • L
  • σ¯~
  • L¯

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