Homework Problems

1. The interaction term for Electric Quadrupole transitions correspond to a linear combination of spherical harmonics, $Y_{2m}$, and are parity even. Find the selection rules for E2 transitions.

2. Magnetic dipole transitions are due to an axial vector operator and hence are proportional to the $Y_{1m}$ but do not change parity (unlike a vector operator). What are the M1 selection rules?

3. Draw the energy level diagram for hydrogen up to $n=3$. Show the allowed E1 transitions. Use another color to show the allowed E2 and M1 transitions.

4. Calculate the decay rate for the $3p\rightarrow 1s$ transition.

5. Calculate the decay rate for the $3d\rightarrow 2p$ transition in hydrogen.

6. Assume that we prepare Hydrogen atoms in the $\psi_{n\ell m}=\psi_{211}$ state. We set up an experiment with the atoms at the origin and detectors sensitive to the polariztion along each of the 3 coordinate axes. What is the probability that a photon with its wave vector pointing along the axis will be Left Circularly Polarized?

7. Photons from the $3p\rightarrow 1s$ transition are observed coming from the sun. Quantitatively compare the natural line width to the widths from Doppler broadening and collision broadening expected for radiation from the sun's surface.