- 523

- 0

## Main Question or Discussion Point

Given a rank-k spherically symmetric tensor operator [itex]\hat{T}^{(k)}_q[/itex] (in other words a family of 2k + 1 operators satisfying [itex][J_z,T_q^{(k)}] = q T_{q}^{(k)}[/itex] and [itex]J_{\pm},T_q^{(k)}] = \sqrt{(k\pm q + 11)(k \mp q)}T_{q\pm 1}^{(k)}[/itex] for all k.

We have the Wigner-Eckart thorem

[itex]\langle j',m' |T^{(k)}_q|j,m \rangle = \frac{1}{\sqrt{2j+1}}\langle jk; mq | jk; j'm' \rangle\langle j' || T^{(k)} || j \rangle[/itex]

where the ``double bar'' is independent of m, m' and q.

I want to calculate the double bar for the electric dipole operator (proportional to the position operator). I'm expecting the answer to be proportional to [itex]\sqrt{2j+1}[/itex].

The first thing to answer is whether the theorem applies, ie is the position operator an irreducible spherical tensor operator. Secondly, how would I go about computing the double bar in this case?

We have the Wigner-Eckart thorem

[itex]\langle j',m' |T^{(k)}_q|j,m \rangle = \frac{1}{\sqrt{2j+1}}\langle jk; mq | jk; j'm' \rangle\langle j' || T^{(k)} || j \rangle[/itex]

where the ``double bar'' is independent of m, m' and q.

I want to calculate the double bar for the electric dipole operator (proportional to the position operator). I'm expecting the answer to be proportional to [itex]\sqrt{2j+1}[/itex].

The first thing to answer is whether the theorem applies, ie is the position operator an irreducible spherical tensor operator. Secondly, how would I go about computing the double bar in this case?