# Radial and angular parts of matrix elements for alkali atoms - calculating Quadrupole matrix elements

I am trying to calculate the quadrupole matrix element for an alkali ($$^{87}$$Rb specifically) and have come to the expression for the matrix element being as follows

$$\left=\int dr R_f(r) r^4 R_i(r) \int d\Omega Y^{m_1*}_{\ell_1} Y^{q}_{k} Y^{m_2}_{\ell_2}$$

The radial part of the matrix element is easy to understand, but I am stuck on the angular part. I have the expression:

$$\int d\Omega Y^{m_1*}_{\ell_1} Y^{m_2}_{\ell_2} Y^{m_3}_{\ell_3} = \sqrt{\frac{(2\ell_2 + 1)(2\ell_3+1)}{(2\ell_1+1)}}\left<\ell_1 0|\ell_2 0 \ell_3 0\right>\left<\ell_2 m_2 \ell_3 m_3|\ell_1 m_1\right>$$

which I understand.

My difficulty comes from the fact that for a real state defined by $$\left|n \ell j\right>$$ I get confused on how to fill in the values of $$\ell$$ and $$m$$ in the spherical harmonics. In all textbooks they only define the state using $$\left|nlm\right>$$ in which case this is trivial.

My understanding is that $$m_\ell$$ is not a good quantum number in this case because of the LS coupling present?

Perhaps it would be clearer if you wrote out in more detaild what $$|i\rangle$$ and $$|f\rangle$$ are. You say that magnetic quantum number is not 'good' for your case, yet it appears on the right-hand side of your first expression.

I must say I am not exactly clear what you are trying to do, but I assume that your atom has filled shells apart from one electron in a non-filled shell, and you are looking at how this electron will interact with an electromagnetic field via quadrupole interaction. Well, even if the magnetic number eigenstates are not the eigenstates of the Hamiltonian, they should still provide a suitable basis to decompose your electron wavefunction. That decomposition will not be fixed in time, true, but it will not prevent you from computing the matrix element for the quadrupole.

In some sense, this is quite reasonable. You have an electron in some oriented state that interacts with an oriented external field (hence quadrupole transition). However the orientation of the electron state is not fully probed by your interaction (since quadrupole electric operator does not couple to spin), hence you can expect to see some uncertainty in your interaction-dependent observations.

A more formal way to proceed here would be:

$$\langle f |Q^k_q | i \rangle=\langle f|\left[ \int d^3 r \int d^3r'\:|\mathbf{r}\rangle\langle\mathbf{r}|Q^k_q|\mathbf{r}' \rangle\langle\mathbf{r}'|\right]| i \rangle=\int d^3 r \int d^3r'\,\langle f | \mathbf{r}\rangle \langle \mathbf{r}' | i\rangle\,\langle\mathbf{r}|Q^k_q|\mathbf{r}' \rangle$$

You then use the fact that quadrupole operator is diagonal in the position basis, i.e.:

$$\langle\mathbf{r}|Q^k_q|\mathbf{r}' \rangle=\delta\left(\mathbf{r}-\mathbf{r}'\right)Q^k_q\left(\mathbf{r}\right)$$

So:

$$\langle f |Q^k_q | i \rangle=\int d^3 r \,\langle f | \mathbf{r}\rangle \langle \mathbf{r} | i\rangle\,Q^k_q\left(\mathbf{r}\right)$$

Thus the question is, what is your electron wavefunction in position basis:

$$\langle \mathbf{r} | i\rangle=?$$

And note that your XYZ axis here (for purposes of specifying the wavefunction) are not arbitrary, you would have already fixed them in writing out the quadrupole operator. Otherwise you should really start from a tensorial expression for the quadrupole operator and the electron charge-current density operator (as a four-vector).