I'm solving a question that required me to work with a Cartesian tensor, $A_i$, with Wigner-Eckart theorem.

The book was Sakurai Modern Quantum Mechanics with information given in related posts:

Ra. Decomposition of a Cartesian tensor

Rb. Irreducible form of Spherical tensor operators

Rc. Representing a reducible Cartesian tensor as a spherical tensor

The issue I have had was that the form of Wigner-Eckart theorem was given in spherical tensor. But $A_i$ was a cartesian tensor.

Qa. How to decompose $A_i$ into $T_{q}^{(k)} = Y_{l=k}^{m=q}(A_i)$? Is it a summation over index $m/q,l/k$? Does it have coefficients? And how to determine the rank of the tensor?

There was a projection theorem when $j=j'$, Eq 3.11.40 $$\langle \alpha', jm'|V_q |\alpha, jm\rangle = \frac{\langle \alpha', jm'|J\cdot V|\alpha, jm\rangle} {\hbar^2 j(j+1)} \langle jm' |J_q|jm\rangle$$

However, it was very confusing. From the reading, the book seemed to indicate that the subscript $V_q$ here indicate the spherical $m=q$.

Qb. Could projection theorem be used to deal with Cartesian tensor?


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