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* Page found: Spin- Operatoren und Zustände (eq math.1692.28)

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TeX (original user input): 

\begin{align}

& \left( {{{\hat{\bar{\sigma }}}}_{1}}+i{{{\hat{\bar{\sigma }}}}_{2}} \right)\left| \downarrow  \right\rangle =\left( {{{\hat{\bar{\sigma }}}}_{1}}+{{{\hat{\bar{\sigma }}}}_{1}} \right)\left| \downarrow  \right\rangle =2\left| \uparrow  \right\rangle  \\

& \left( {{{\hat{\bar{\sigma }}}}_{1}}-i{{{\hat{\bar{\sigma }}}}_{2}} \right)\left| \uparrow  \right\rangle =\left( {{{\hat{\bar{\sigma }}}}_{1}}+{{{\hat{\bar{\sigma }}}}_{1}} \right)\left| \uparrow  \right\rangle =2\left| \downarrow  \right\rangle  \\

& \Rightarrow {{{\hat{\bar{\sigma }}}}_{1}}\left| \downarrow  \right\rangle =\left| \uparrow  \right\rangle  \\

& {{{\hat{\bar{\sigma }}}}_{1}}\left| \uparrow  \right\rangle =\left| \downarrow  \right\rangle  \\

\end{align}

TeX (checked): 

{\begin{aligned}&\left({{\hat {\bar {\sigma }}}_{1}}+i{{\hat {\bar {\sigma }}}_{2}}\right)\left|\downarrow \right\rangle =\left({{\hat {\bar {\sigma }}}_{1}}+{{\hat {\bar {\sigma }}}_{1}}\right)\left|\downarrow \right\rangle =2\left|\uparrow \right\rangle \\&\left({{\hat {\bar {\sigma }}}_{1}}-i{{\hat {\bar {\sigma }}}_{2}}\right)\left|\uparrow \right\rangle =\left({{\hat {\bar {\sigma }}}_{1}}+{{\hat {\bar {\sigma }}}_{1}}\right)\left|\uparrow \right\rangle =2\left|\downarrow \right\rangle \\&\Rightarrow {{\hat {\bar {\sigma }}}_{1}}\left|\downarrow \right\rangle =\left|\uparrow \right\rangle \\&{{\hat {\bar {\sigma }}}_{1}}\left|\uparrow \right\rangle =\left|\downarrow \right\rangle \\\end{aligned}}

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(σ¯̂1+iσ¯̂2)|↓⟩=(σ¯̂1+σ¯̂1)|↓⟩=2|↑⟩(σ¯̂1−iσ¯̂2)|↑⟩=(σ¯̂1+σ¯̂1)|↑⟩=2|↓⟩⇒σ¯̂1|↓⟩=|↑⟩σ¯̂1|↑⟩=|↓⟩
<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><mover><mi>σ</mi><mo>¯</mo></mover><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo stretchy="false">+</mo><mi>i</mi><msub><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo stretchy="false">̂</mo></mover><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><mo stretchy="false">↓</mo><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo stretchy="false">=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo stretchy="false">+</mo><msub><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mo stretchy="false">↓</mo><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo stretchy="false">=</mo><mn>2</mn><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mo stretchy="false">↑</mo><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><msub><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo stretchy="false">−</mo><mi>i</mi><msub><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo stretchy="false">̂</mo></mover><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><mo stretchy="false">↑</mo><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo stretchy="false">=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo stretchy="false">+</mo><msub><mover><mover><mi>σ</mi><mo>¯</mo></mover><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mo stretchy="false">↑</mo><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo stretchy="false">=</mo><mn>2</mn><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mo stretchy="false">↓</mo><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><msub><mover><mover><mi>σ</mi><mo>¯</mo></mover><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><mo stretchy="false">↓</mo><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo stretchy="false">=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mo stretchy="false">↑</mo><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><msub><mover><mover><mi>σ</mi><mo>¯</mo></mover><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><mo stretchy="false">↑</mo><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo stretchy="false">=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mo stretchy="false">↓</mo><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></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 Spin- Operatoren und Zustände page

Identifiers

  • σ¯̂1
  • i
  • σ¯̂2
  • σ¯̂1
  • σ¯̂1
  • σ¯̂1
  • i
  • σ¯̂2
  • σ¯̂1
  • σ¯̂1
  • σ¯̂1
  • σ¯̂1

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