QM question of the $z$ Matrix element in Angular Momentum Basis

I found a quite challenge quantum mechanics problem in a preparation sample test for a midterm.

Consider an electron moving in a central potential. Suppose that we know the matrix element of the $$z$$-position operator between two states: $$\langle j',m'|\,z\,|j,m\rangle$$ (i) Justify that $$m'=m$$ for this matrix element to be non-zero;

(ii) What are the constraints on $$j$$ and $$j'$$ for non-zero matrix elements?

(iii) Given a none-zero matrix element $$\langle j',m'|\,z\,|j,m\rangle$$ (with $$m=m'$$ and suitable constraints on $$j'$$ and $$j$$), give a general formula to compute the matrix elements: $$\langle j',m'''|\,x\,|j,m''\rangle$$

For the first question. I used $$[J_z,z]=0$$, then one can easily show that $$m'=m$$. However, for the second question, I stuck for a while by using the algebraic methods. I tried to use spherical harmonic wavefunction $$Y_m^j$$. However, if $$\hat{z}=\hat{r}\cos(\hat{\theta})$$, then one will eventually compute the following integral:

$$\int_{-1}^{1}P^m_j(x)P^{m'}_{j'}(x)xdx$$ Using the recursion formula for Legendre polynomial $$xP^m_l(x)=C_1P^m_{l-1}+C_2P^m_{l+1}$$, I conclude that $$|j-j'|=1$$. However, since this is an exam question, and, in the help sheet, there is no above relation offered. Hence, I would guess there should be an algebraic way to solve the problem.

For the (iii), again using the analytic method can do the problem, but the related formula is not given in the test. I also expect an algebraic way to do it.

Can someone give me a hint or some detailed calculation?

• I would also like this answered - I have a textbook that claims these follow from the Wigner-Eckart theorem but doesn't provide a description of how. – jacob1729 Nov 25 '18 at 23:31
• However the question doesn't actually ask you to prove these things so its possible the exam expects you to have memorised these 'selection rules' and that is why you find them hard to prove in the given time - you might not be meant to? – jacob1729 Nov 25 '18 at 23:32
• The Wigner-Eckart Theorem is provided indeed. I may play with it for a while. – Hamio Jiang Nov 25 '18 at 23:34
• I think you need to transform form the cartesian basis to the spherical basis, i.e.: $z \rightarrow \hat e^0$, so it is one of the basis vectors. – JEB Nov 25 '18 at 23:56
• Hint for (ii): try the commutator of J^2 with z. Part (iii) is essentially asking you to prove the Wigner-Eckart theorem, which can be found in a lot of textbooks although I'm not sure I could reproduce exactly what they're looking for here. – Will C Nov 26 '18 at 0:25