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* Page found: Spin- Operatoren und Zustände (eq math.1693.25)

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\begin{align}

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

& =\left\langle  \downarrow  \right|{{{\hat{\bar{\sigma }}}}_{1}}^{2}+{{{\hat{\bar{\sigma }}}}_{2}}^{2}+i\left[ {{{\hat{\bar{\sigma }}}}_{1}},{{{\hat{\bar{\sigma }}}}_{2}} \right]\left| \downarrow  \right\rangle  \\

& \left[ {{{\hat{\bar{\sigma }}}}_{1}},{{{\hat{\bar{\sigma }}}}_{2}} \right]=2i{{{\hat{\bar{\sigma }}}}_{3}} \\

& {{{\hat{\bar{\sigma }}}}_{1}}^{2}+{{{\hat{\bar{\sigma }}}}_{2}}^{2}={{{\hat{\bar{\sigma }}}}^{2}}-{{{\hat{\bar{\sigma }}}}_{3}}^{2} \\

& \Rightarrow \alpha *\alpha =\left\langle  \downarrow  \right|{{{\hat{\bar{\sigma }}}}^{2}}-{{{\hat{\bar{\sigma }}}}_{3}}^{2}-2{{{\hat{\bar{\sigma }}}}_{3}}\left| \downarrow  \right\rangle =\left\langle  \downarrow  \right|3-1+2\left| \downarrow  \right\rangle =4 \\

& \Rightarrow \left| \alpha  \right|=2 \\

\end{align}

TeX (checked): 

{\begin{aligned}&\alpha *\alpha =\left\langle \downarrow \right|{{\hat {\bar {\sigma }}}_{+}}^{+}{{\hat {\bar {\sigma }}}_{+}}\left|\downarrow \right\rangle =\left\langle \downarrow \right|\left({{\hat {\bar {\sigma }}}_{1}}-i{{\hat {\bar {\sigma }}}_{2}}\right)\left({{\hat {\bar {\sigma }}}_{1}}+i{{\hat {\bar {\sigma }}}_{2}}\right)\left|\downarrow \right\rangle \\&=\left\langle \downarrow \right|{{\hat {\bar {\sigma }}}_{1}}^{2}+{{\hat {\bar {\sigma }}}_{2}}^{2}+i\left[{{\hat {\bar {\sigma }}}_{1}},{{\hat {\bar {\sigma }}}_{2}}\right]\left|\downarrow \right\rangle \\&\left[{{\hat {\bar {\sigma }}}_{1}},{{\hat {\bar {\sigma }}}_{2}}\right]=2i{{\hat {\bar {\sigma }}}_{3}}\\&{{\hat {\bar {\sigma }}}_{1}}^{2}+{{\hat {\bar {\sigma }}}_{2}}^{2}={{\hat {\bar {\sigma }}}^{2}}-{{\hat {\bar {\sigma }}}_{3}}^{2}\\&\Rightarrow \alpha *\alpha =\left\langle \downarrow \right|{{\hat {\bar {\sigma }}}^{2}}-{{\hat {\bar {\sigma }}}_{3}}^{2}-2{{\hat {\bar {\sigma }}}_{3}}\left|\downarrow \right\rangle =\left\langle \downarrow \right|3-1+2\left|\downarrow \right\rangle =4\\&\Rightarrow \left|\alpha \right|=2\\\end{aligned}}

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α*α=⟨↓|σ¯^++σ¯^+|↓⟩=⟨↓|(σ¯^1−iσ¯^2)(σ¯^1+iσ¯^2)|↓⟩=⟨↓|σ¯^12+σ¯^22+i[σ¯^1,σ¯^2]|↓⟩[σ¯^1,σ¯^2]=2iσ¯^3σ¯^12+σ¯^22=σ¯^2−σ¯^32⇒α*α=⟨↓|σ¯^2−σ¯^32−2σ¯^3|↓⟩=⟨↓|3−1+2|↓⟩=4⇒|α|=2
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Calculated based on the variables occurring on the entire Spin- Operatoren und Zustände page

Identifiers

  • α
  • α
  • σ¯^+
  • σ¯^+
  • σ¯^1
  • i
  • σ¯^2
  • σ¯^1
  • i
  • σ¯^2
  • σ¯^1
  • σ¯^2
  • i
  • σ¯^1
  • σ¯^2
  • σ¯^1
  • σ¯^2
  • i
  • σ¯^3
  • σ¯^1
  • σ¯^2
  • σ¯^
  • σ¯^3
  • α
  • α
  • σ¯^
  • σ¯^3
  • σ¯^3
  • α

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