Tensors and Groups in Quantum

A rank-k tensor \(\hat T_k^q\) is defined as

\[\begin{split}\left[\hat J_z, \hat T_k^1 \right] &= q\hbar \hat T_k^q \\ \left[ \hat J_{\pm}, \hat T_k^q \right] & = \sqrt{(k\mp q)(k\pm q + 1)}\hbar \hat T_{k}^{q\pm 1} .\end{split}\]

Wigner-Eckart Theorem

Wigner-Eckart theorem is

\[\bra{n'j'm'}\hat T_k^q \ket{njm} = \bra{n'j'}\vert \hat T_k \vert \ket{nj} \braket{j'm';kj}{kq;jm},\]

where \(j,j'\) are the angular momentum quantum numbers and \(n, n'\) are quantum numbers which are not related to angular momentum.

It seems that tensor \(\hat T_k^q\) is a source of angular momentum. The maximum angular momentum it can provide is \(k\).