Theory - A brief note on Selection Rules

Atoms are extremely complicated and have a ton of levels. When an atom interacts with light, either through absorption or emission, it will move from one state to another. However, the atom isn’t able to go from one random state to another. There are a few restrictions that allow us to figure out what possible states we can go to.

Before we talk about selection rules, remember that conservation of energy must always hold. A photon is a packet of light and has a fixed amount of energy. As we learned in previous sections, different energy levels also have a fixed energy. When an atom absorbs a photon, the transition between the energy levels of the atom should match (or be very close to) the energy of the photon. Similarly, when at atom emits a photon, the change in the energy between the initial and final energy level should match that of the photon. Thus, conservation of energy helps dictate a few of our transitions.

There are still a ton of internal states of the atom, especially if we include hyperfine states. We solve for transition probabilities and use selection rules to determine the probabilities of transitioning from one state to another.

First, recall that photons have different polarizations. When an atom interacts with a photon of a certain polarization, it may only be able to transition to certain states. This can functionally occur due to conservation of momentum and how the dipole moments of the atoms interact with the different polarized light.

At a high level, let’s consider a few cases. We are concerned with the selection rules that change the quantum number $\Delta m = \pm 1$ and $\Delta m = 0$. In the first case, the atom is usually interacting with circularly polarized light. The photon will thus have an angular momentum component (which we can imagine corresponds to nonzero $\hat{J}_z$ eigenvalue), thus altering the atoms $m$ number to account for conservation of angular momentum. If the photon had linearly polarized light, its angular momentum will be 0, thus leaving the magnetic quantum number $m$ untouched.

References

• Foot Chapter 2 (Hydrogen Selection Rules)

• Chem LibreText Link

• U. Delaware notes on Selection Rules