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TeX (original user input): 

\begin{align}

& {{{\hat{L}}}^{2}}={{\varepsilon }_{jkl}}{{\varepsilon }_{jmn}}{{x}_{k}}{{p}_{l}}{{x}_{m}}{{p}_{n}}=\left( {{\delta }_{km}}{{\delta }_{\ln }}-{{\delta }_{kn}}{{\delta }_{lm}} \right){{x}_{k}}{{p}_{l}}{{x}_{m}}{{p}_{n}}= \\

& ={{x}_{m}}{{p}_{n}}{{x}_{m}}{{p}_{n}}-{{x}_{n}}{{p}_{m}}{{x}_{m}}{{p}_{n}} \\

& {{p}_{n}}{{x}_{m}}={{x}_{m}}{{p}_{n}}-i\hbar {{\delta }_{mn}} \\

& {{x}_{n}}{{p}_{m}}={{p}_{m}}{{x}_{n}}+i\hbar {{\delta }_{mn}} \\

& \Rightarrow {{{\hat{L}}}^{2}}={{x}_{m}}{{x}_{m}}{{p}_{n}}{{p}_{n}}-{{p}_{m}}{{x}_{n}}{{x}_{m}}{{p}_{n}}-2i\hbar {{x}_{m}}{{p}_{m}} \\

& {{p}_{m}}{{x}_{n}}{{x}_{m}}{{p}_{n}}={{p}_{m}}{{x}_{m}}{{x}_{n}}{{p}_{n}} \\

& {{p}_{m}}{{x}_{m}}={{x}_{m}}{{p}_{m}}-i\hbar {{\delta }_{mm}} \\

& {{\delta }_{mm}}=3 \\

& \Rightarrow {{{\hat{L}}}^{2}}={{x}_{m}}{{x}_{m}}{{p}_{n}}{{p}_{n}}-{{p}_{m}}{{x}_{n}}{{x}_{m}}{{p}_{n}}-2i\hbar {{x}_{m}}{{p}_{m}}={{x}_{m}}^{2}{{p}_{n}}^{2}-{{x}_{m}}{{p}_{m}}{{x}_{n}}{{p}_{n}}+3i\hbar {{x}_{n}}{{p}_{n}}-2i\hbar {{x}_{m}}{{p}_{m}} \\ 

& \Rightarrow {{{\hat{L}}}^{2}}={{x}_{m}}^{2}{{p}_{n}}^{2}-\left( {{x}_{m}}{{p}_{m}} \right)\left( {{x}_{n}}{{p}_{n}} \right)+i\hbar {{x}_{m}}{{p}_{m}} \\

& {{{\hat{L}}}^{2}}={{r}^{2}}{{p}^{2}}-{{\left( \bar{r}\cdot \bar{p} \right)}^{2}}+i\hbar \left( \bar{r}\cdot \bar{p} \right) \\

\end{align}

TeX (checked): 

{\begin{aligned}&{{\hat {L}}^{2}}={{\varepsilon }_{jkl}}{{\varepsilon }_{jmn}}{{x}_{k}}{{p}_{l}}{{x}_{m}}{{p}_{n}}=\left({{\delta }_{km}}{{\delta }_{\ln }}-{{\delta }_{kn}}{{\delta }_{lm}}\right){{x}_{k}}{{p}_{l}}{{x}_{m}}{{p}_{n}}=\\&={{x}_{m}}{{p}_{n}}{{x}_{m}}{{p}_{n}}-{{x}_{n}}{{p}_{m}}{{x}_{m}}{{p}_{n}}\\&{{p}_{n}}{{x}_{m}}={{x}_{m}}{{p}_{n}}-i\hbar {{\delta }_{mn}}\\&{{x}_{n}}{{p}_{m}}={{p}_{m}}{{x}_{n}}+i\hbar {{\delta }_{mn}}\\&\Rightarrow {{\hat {L}}^{2}}={{x}_{m}}{{x}_{m}}{{p}_{n}}{{p}_{n}}-{{p}_{m}}{{x}_{n}}{{x}_{m}}{{p}_{n}}-2i\hbar {{x}_{m}}{{p}_{m}}\\&{{p}_{m}}{{x}_{n}}{{x}_{m}}{{p}_{n}}={{p}_{m}}{{x}_{m}}{{x}_{n}}{{p}_{n}}\\&{{p}_{m}}{{x}_{m}}={{x}_{m}}{{p}_{m}}-i\hbar {{\delta }_{mm}}\\&{{\delta }_{mm}}=3\\&\Rightarrow {{\hat {L}}^{2}}={{x}_{m}}{{x}_{m}}{{p}_{n}}{{p}_{n}}-{{p}_{m}}{{x}_{n}}{{x}_{m}}{{p}_{n}}-2i\hbar {{x}_{m}}{{p}_{m}}={{x}_{m}}^{2}{{p}_{n}}^{2}-{{x}_{m}}{{p}_{m}}{{x}_{n}}{{p}_{n}}+3i\hbar {{x}_{n}}{{p}_{n}}-2i\hbar {{x}_{m}}{{p}_{m}}\\&\Rightarrow {{\hat {L}}^{2}}={{x}_{m}}^{2}{{p}_{n}}^{2}-\left({{x}_{m}}{{p}_{m}}\right)\left({{x}_{n}}{{p}_{n}}\right)+i\hbar {{x}_{m}}{{p}_{m}}\\&{{\hat {L}}^{2}}={{r}^{2}}{{p}^{2}}-{{\left({\bar {r}}\cdot {\bar {p}}\right)}^{2}}+i\hbar \left({\bar {r}}\cdot {\bar {p}}\right)\\\end{aligned}}

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L^2=εjklεjmnxkplxmpn=(δkmδln−δknδlm)xkplxmpn==xmpnxmpn−xnpmxmpnpnxm=xmpn−iℏδmnxnpm=pmxn+iℏδmn⇒L^2=xmxmpnpn−pmxnxmpn−2iℏxmpmpmxnxmpn=pmxmxnpnpmxm=xmpm−iℏδmmδmm=3⇒L^2=xmxmpnpn−pmxnxmpn−2iℏxmpm=xm2pn2−xmpmxnpn+3iℏxnpn−2iℏxmpm⇒L^2=xm2pn2−(xmpm)(xnpn)+iℏxmpmL^2=r2p2−(r¯⋅p¯)2+iℏ(r¯⋅p¯)
<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd></mtd><mtd><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><mi>&#x03B5;</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>j</mi><mi>k</mi><mi>l</mi></mrow></mrow></msub><msub><mi>&#x03B5;</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>j</mi><mi>m</mi><mi>n</mi></mrow></mrow></msub><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mi>&#x03B4;</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>m</mi></mrow></mrow></msub><msub><mi>&#x03B4;</mi><mrow data-mjx-texclass="ORD"><mi>ln</mi></mrow></msub><mo>&#x2212;</mo><msub><mi>&#x03B4;</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>n</mi></mrow></mrow></msub><msub><mi>&#x03B4;</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>l</mi><mi>m</mi></mrow></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>l</mi></mrow></msub><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><msub><mi>p</mi><mrow 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data-mjx-texclass="OPEN">(</mo><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mi>i</mi><mi data-mjx-alternate="1">&#x210F;</mi><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub></mtd></mtr><mtr><mtd></mtd><mtd><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><msup><mi>r</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><msup><mi>p</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>&#x2212;</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo>&#x22C5;</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>+</mo><mi>i</mi><mi data-mjx-alternate="1">&#x210F;</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo>&#x22C5;</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>

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Identifiers

  • L^
  • εjkl
  • εjmn
  • xk
  • pl
  • xm
  • pn
  • δkm
  • δ
  • δkn
  • δlm
  • xk
  • pl
  • xm
  • pn
  • xm
  • pn
  • xm
  • pn
  • xn
  • pm
  • xm
  • pn
  • pn
  • xm
  • xm
  • pn
  • i
  • δmn
  • xn
  • pm
  • pm
  • xn
  • i
  • δmn
  • L^
  • xm
  • xm
  • pn
  • pn
  • pm
  • xn
  • xm
  • pn
  • i
  • xm
  • pm
  • pm
  • xn
  • xm
  • pn
  • pm
  • xm
  • xn
  • pn
  • pm
  • xm
  • xm
  • pm
  • i
  • δmm
  • δmm
  • L^
  • xm
  • xm
  • pn
  • pn
  • pm
  • xn
  • xm
  • pn
  • i
  • xm
  • pm
  • xm
  • pn
  • xm
  • pm
  • xn
  • pn
  • i
  • xn
  • pn
  • i
  • xm
  • pm
  • L^
  • xm
  • pn
  • xm
  • pm
  • xn
  • pn
  • i
  • xm
  • pm
  • L^
  • r
  • p
  • r¯
  • p¯
  • i
  • r¯
  • p¯

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