# Exercises JRC1

From IFPA wiki

1. Calculate perturbatively the eigenstates and energies of the following potential:

$V (x) = \epsilon (x-a)x$ for $0\lt x\lt a$

$V (x) = \infty$ 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

$V(r)=-{e^2\over r}$ outside the proton

$V(r)=-{e^2\over 2 r_0} \left(3-{r^2\over r_0^2}\right)$ inside the proton

with $r_0\approx$ 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 $V(x)=\epsilon x$.

Hint: express $x$ in terms of creation and annihilation operators.

4. Stark effect: calculate the separation of levels for $n=2$ in the hydrogen atom due to the presence of a constant electric field in the $z$ direction, i.e. corresponding to a potential

$V(z) = −ezE$,

where $E$ is the magnitude of the electric field.

IFPA Wiki maintained by Atri B.