Wigner Eckart theorem / Electric dipole

[anony]

Homework Statement

The eletric dipole of the atom D = qR is a vector op, ie transforms according to j = 1 rep of SU(2). Use wigner eckart theorem show

<1, 0, 0|D|1, 0, 0> = 0

(<n',l',m'|D|n, l, m> = 0

Homework Equations

http://ocw.mit.edu/NR/rdonlyres/Chemistry/5-73Fall-2005/BEE27FD3-DE39-468F-B7B7-2EB89C58B89D/0/sec9.pdf

Last equation on page 3, and that q + m = m' for the CG coefficients to be non zero

The Attempt at a Solution

Okay, so converting R to a spherical tensor T, T would be of rank 1 so k = 1, and q = -1, 0, 1. I'm incredibly confused as I've been reading from plenty of different sources. In my latest read I found that maybe q = -1, 0, 1 in the case where l = 1 and hence m = -1, 0, 1 (which they have labelled q), and so in the case where m = 0 (ie. here) then q = 0 only. But I'm relatively sure that its the former.

Anyway, for q = -1, 1, q + m = +-1 + 0 = hence for q = +-1, the CG coefficients are zero. But for q = 0, q + m = 0 + 0 = m' hence this should be a non zero coefficient? Then I tried to check the other requirement:
|j - k| <= j' <= j + k

But I don't know what j'/j is? j is meant to be the total angular momentum right? We are told j = 1 in the problem, but then what's j'? or is j' =1? arghhhh!

Help would be MUCH appreciated! Thanks!

EDIT:
so, in some problems I see they've had j -> l, j' -> l'?

In that case,
|0-1| <= 0 <= 1 => 1 <= 0 not true, so that they are all zero coefficients and no need to faff about with q+m = m'?

Ah, I should have mentioned this is a spinless atom. so j = l + s right? and since l = 0, j = j' = 0, and hence its as I showed in my last edit right?

 

[Jhero]

Thank you for your post regarding the electric dipole of the atom. To answer your question, let's first review the Wigner-Eckart theorem. This theorem states that for a vector operator, the matrix elements <j',m'|D|j,m> are proportional to the Clebsch-Gordan coefficient <j',m'|j,m> and the reduced matrix element <j'||D||j>, where j' and j are the total angular momenta of the initial and final states, and m' and m are their respective magnetic quantum numbers.

In the case of a spinless atom, the total angular momentum j is equal to the orbital angular momentum l. Therefore, j' and j are both equal to l, and the Clebsch-Gordan coefficient <l',m'|l,m> will be non-zero only when l' = l and m' = m. This means that the only allowed matrix element is <l,m|D|l,m>, and any other combination will be equal to zero.

In your problem, j = j' = 1, which corresponds to l = l' = 1. Therefore, the only non-zero matrix element will be <1,0|D|1,0>, and all other combinations will be equal to zero. This confirms the result that you were asked to prove, <1,0,0|D|1,0,0> = 0.

I hope this helps clarify your understanding of the Wigner-Eckart theorem and its application to the electric dipole of the atom. If you have any further questions, please don't hesitate to ask.