The two sets of matrices:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]{G_1} = i\hbar \left( {\begin{array}{*{20}{c}}

0 & 0 & 0 \\

0 & 0 & { - 1} \\

0 & 1 & 0 \\

\end{array}} \right){\rm{ }}{G_2} = i\hbar \left( {\begin{array}{*{20}{c}}

0 & 0 & 1 \\

0 & 0 & 0 \\

{ - 1} & 0 & 0 \\

\end{array}} \right){\rm{ }}{G_3} = i\hbar \left( {\begin{array}{*{20}{c}}

0 & { - 1} & 0 \\

1 & 0 & 0 \\

0 & 0 & 0 \\

\end{array}} \right)[/tex]

and

[tex]{J_1} = \frac{\hbar }{{\sqrt 2 }}\left( {\begin{array}{*{20}{c}}

0 & 1 & 0 \\

1 & 0 & 1 \\

0 & 1 & 0 \\

\end{array}} \right){\rm{ }}{J_2} = \frac{{i\hbar }}{{\sqrt 2 }}\left( {\begin{array}{*{20}{c}}

0 & { - 1} & 0 \\

1 & 0 & { - 1} \\

0 & 1 & 0 \\

\end{array}} \right){\rm{ }}{J_3} = \hbar \left( {\begin{array}{*{20}{c}}

1 & 0 & 0 \\

0 & 0 & 0 \\

0 & 0 & { - 1} \\

\end{array}} \right)[/tex]

They both satisfy the common commutation relation

[tex][{L_i},{L_j}] = i\hbar {\varepsilon _{ijk}}{L_k}[/tex] with L substituted by G or J

So both can be used to describe angular momentum. G is called the Cartesian basis representation and J is called spherical basis representation. This is absolutely new to me, so I want to acquire some general knowledge about it. For example, why are G and J named like that (Cartesian and spherical....), how are they related and so on. And any source of reference will be helpful.

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# Representation of angular momentum matrix in Cartesian and spherical basis

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