Homework Problems

1. A hydrogen atom is placed in an electric field which is uniform in space and turns on at $t=0$ then decays exponentially. That is, $\vec{E}(t)=0$ for $t<0$ and $\vec{E}(t)=\vec{E}_0 e^{-\gamma t}$ for $t>0$. What is the probability that, as $t\rightarrow\infty$, the hydrogen atom has made a transition to the $2p$ state?

2. A one dimensional harmonic oscillator is in its ground state. It is subjected to the additional potential $W=-e\xi x$ for a a time interval $\tau$. Calculate the probability to make a transition to the first excited state (in first order). Now calculate the probability to make a transition to the second excited state. You will need to calculate to second order.

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. 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.