#dipole operator

Okay, so for a transition between energy eigenstates, there needs to be an exchange of a photon with the correct energy. I'm assuming you know that.

Since photons are waves of the electromagnetic field, they impose an electric moment on charged particles, which in a vast majority of cases can be modeled as a simple dipole moment.

Now here's where the quantum mechanics starts: The dipole moment is expressed as a linear operator, which when applied to the wave function of a particular state gives you back the eigenvalue for the dipole moment of that state. However, since we want to describe the transition between states, and the operator only applies to the ket of the wave function, the bra and the ket which it is nestled in between are of the different states, aka the starting and the final electron state.

The operator applies to the starting state ket, which can then be completed on the left with the final state bra, and then integrated over to obtain the transition dipole moment integral.

This integral will tell you the expectation value for the transition. For the selection rules, you don't actually have to precisely calculate this integral, you just have to find out whether or not it is zero, because if it is, that means you have an impossible transition on your hands.

Depending on your representation, the dipole operator as well as the wave functions will look different, as will the space you integrate over.

So first, find which representation (spatial, spherical, momentum space, etc.) you are working with, how the dipole operator looks for that representation, and then pick the two states you want to see if a dipole transition exists between them.

The tricky part is usually to get the wave function representation right, and then to leverage the symmetries of that function to determine if the value of the integral is zero or not. The representation that i find most common for tasks like this is this one, which separates the wave function into a radial and two angular parts. It is also already conveniently expressed in terms of 3 quantum numbers, that being the main quantum number n, the orbital angular momentum number l, and the magnetic quantum number m.

I am afraid you'll have to learn the quirks and symmetries of the generalized Laguerre polynomials as well as the spherical harmonics, in order to make statements about the transition dipole moment integral. However, once you get a feel for their symmetries and remember in what special cases integrals vanish (like integrating an odd function over a symmetric interval, etc.) you will be able to derive the selection rules.