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Exercises JRC1

From IFPA wiki

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

[math] V (x) = \epsilon (x-a)x[/math] for [math]0\lt x\lt a[/math]

[math] 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]V(r)=-{e^2\over r}[/math] outside the proton

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

with [math]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]V(x)=\epsilon x[/math].

Hint: express [math]x[/math] in terms of creation and annihilation operators.

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

[math] V(z) = −ezE [/math],

where [math]E[/math] is the magnitude of the electric field.

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