# Selection rules and type of photon?

Let us consider the following matrix element: $$\langle n',m',l'|x| n, m, l \rangle$$ For the corresponding radiative transition we have the selection rule that $\Delta m=\pm 1$. But will the photon emitted be circularly polarized i.e.: $$|\psi \rangle=|\pm m=1 \rangle$$ or linearly polarized (i.e. a combination of two circularly polarized waves) i.e.: $$|\psi \rangle=a(| m=1 \rangle+| m=-1 \rangle)$$

Given two states $|n,l,m⟩$ and $|n',l',m'⟩$, the following radiative transition matrix elements are nonzero: $$\left\langle n,l\pm1,m+1 \middle| x+iy \middle| n,l,m \right\rangle,$$ $$\left\langle n,l\pm1,m-1 \middle| x-iy \middle| n,l,m \right\rangle,$$ and $$\left\langle n,l\pm1,m \middle| z\middle| n,l,m \right\rangle,$$ and that's it (for transitions between states of definite angular momentum about the $z$ axis; you can also have transitions to e.g. $|n,l+1,m+1⟩+|n,l+1,m-1⟩$, using a linearly polarized photon in the $x$ direction and propagating in the $y,z$ plane, or to similar superposition states).

The first two correspond to absorption or emission of a circularly polarized photon propagating along the $z$ axis, with selection rule $\Delta m=±1$ on the atom, while the third one corresponds to absorption or emission of a linearly polarized photon polarized along the $z$ axis and propagating on any axis in the $x,y$ plane, with selection rule $\Delta m=0$ on the atom.

• No worries, I will take a look in Cohen-Tannoudji. Just to clarify, when we do have a transition to the state e.g. $|n,l+1,m+1⟩+|n,l+1,m-1⟩$ the state of the photon would be $a(|m-1\rangle +|m+1\rangle)$, which is a linearly polarized photon?? May 31, 2016 at 18:53
• @Quantumspaghettification Yes, so long as the photon is propagating along the $z$ axis (since otherwise the re-use of $m$ for $m_z$ is misleading). May 31, 2016 at 18:57
I assume you are talking of the transition dipole element which is typically defined as $$\langle n',m',l'|qx| n, m, l \rangle$$ with q being the charge of the dipole and x its position.