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\begin{document}

Generalized angular momentum is defined by the relation:

\( [J_{i},J_{j}]=i\hbar \epsilon _{ijk}J_{k} \), and \( [J^{2},J_{i}]=0 \)

In non-orthonormal coordinates:\( [\overrightarrow{a}*\overrightarrow{J},\overrightarrow{b}*\overrightarrow{J}]=i\hbar (\overrightarrow{a}x\overrightarrow{b})*\overrightarrow{J} \)

from this, you can derive the orbital angular momentum spectrum.

Define \( J_{+}=J_{x}+iJ_{y} \), \( J_{-}=J_{x}-iJ_{y}=J^{\dagger }_{+} \)

This means: \( [J_{z},J_{+}]=\hbar J_{+} \),\( [J_{z},J_{-}]=-\hbar J_{-} \), \( [J_{+},J_{-}]=2\hbar J_{z} \)

\( J^{2}=\frac{1}{2}(J_{+}J_{-}+J_{-}J_{+})+J^{2}_{z} \)still commutes with everything.

CSCO is \( J^{2},J_{z} \)\( \Rightarrow J^{2}|lm>=l\hbar ^{2}|lm> \) and \( J_{z}|lm>=m\hbar |lm> \)

\( l\geq 0 \)\( <p|J^{2}|p>=<p|\overrightarrow{J^{\dagger }}*\overrightarrow{J}|p> \)

\( l,m \) are related.

\( <lm|J^{2}-J^{2}_{z}|lm>=l-m^{2}=<lm|J_{+}J_{+}^{\dagger }+J^{\dagger }_{+}J_{+}|lm>\Rightarrow l\geq m^{2} \)

\( J_{z}J_{+}|lm>=(J_{+}J_{z}+\hbar J_{+})|lm>=(m+1)\hbar J_{+}|lm> \)

Similarly, \( J_{z}J_{-}|lm>=(m-1)\hbar J_{-}|lm> \), \( J^{2}J_{\pm }|lm>=l\hbar ^{2}J\pm |lm> \)

\( \Rightarrow J_{\pm }|lm>=C_{lm\pm }\hbar ^{2}|lm\pm 1> \)

It must be that:

\( \exists j \)s.t. \( J_{+}|lj>=0 \)

\( J_{-}J_{+}|lj>\Rightarrow l-j^{2}-j=0 \)

Similarly, for a minimal j, \( J_{+}J_{-}|j'm>\Rightarrow l-j'^{2}+j' \)\( \Rightarrow m\in [-j,j] \)

\( j \) is an integer because if it is not, the ladder operators will yield
contradictions.

\( \Rightarrow j=0,1/2,1,3/2,... \) and \( l=0,3/4,2,15/4 \)

m must be an integer because otherwise \( Y_{lm} \)is multivalued.

\( <\overrightarrow{r}|lm>=Y_{lm}(\overrightarrow{r}) \)

\end{document}