1. Calculate perturbatively the eigenstates and energies of the following potential:
[math]\displaystyle{ V (x) = \epsilon (x-a)x }[/math] for [math]\displaystyle{ 0\lt x\lt a }[/math]
[math]\displaystyle{ V (x) = \infty }[/math] elsewhere.
2. a) Evaluate the correction to the bound states of hydrogen from the fact that the proton has a finite size. Assume that the charge is uniformly distributed in the proton, so that
[math]\displaystyle{ V(r)=-{e^2\over r} }[/math] outside the proton
[math]\displaystyle{ V(r)=-{e^2\over 2 r_0} \left(3-{r^2\over r_0^2}\right) }[/math] inside the proton
with [math]\displaystyle{ r_0\approx }[/math] 1 fm the proton radius.
b) Same question for an electron bound to a uranium nucleus.
3. Calculate the corrections to the harmonic oscillator energies due to a potential [math]\displaystyle{ V(x)=\epsilon x }[/math].
Hint: express [math]\displaystyle{ x }[/math] in terms of creation and annihilation operators.
4. Stark effect: calculate the separation of levels for [math]\displaystyle{ n=2 }[/math] in the hydrogen atom due to the presence of a constant electric field in the [math]\displaystyle{ z }[/math] direction, i.e. corresponding to a potential
[math]\displaystyle{ V(z) = −ezE }[/math],
where [math]\displaystyle{ E }[/math] is the magnitude of the electric field.
IFPA wiki
