Selection Rules (Time Dependent Perturbation Theory)

[Diracobama2181]

Homework Statement

Suppose that the system is a single spinless particle of mass $$M$$ and charge $$e$$ in a central Coulomb potential and the quantum numbers of the initial state are labeled $$|n, l, m>$$ in the usual way. It is subject to perturbation with magnetic field $$ B(t) = B_0e ^{−λt}$$ pointing in the x direction, which adds a term $$eL_xB(t)/(2Mc)$$ to the Hamiltonian. Find the quantum number selection rules for allowed transitions to a different state.

Relevant Equations

$$[L_x,X]=0$$

I suppose my question is, since X commutes for H, does this mean that the selection rules are $$<n',l',m'|X|n,l,m>=0$$ unless $$l'=l\pm 1$$ and $$m'=m\pm 1$$, as specified in Shankar?

 

[Nate810]

No, that's not correct. The selection rules for the matrix element of any operator $X$ are given by $$\langle n',l',m'|X|n,l,m\rangle=0$$ unless $$l'-l=\Delta l$$ and $$m'-m=\Delta m$$ where $\Delta l$ and $\Delta m$ are constants depending on the particular operator $X$. In the case of the position operator, $\Delta l=\pm 1$ and $\Delta m=\pm 1$.