Problem 1:

The potential energy of the nuclei of a diatomic molecule as a function of their separation r is given by
U(r) = -2D[a₀/r - a₀²/r²].
Here D is a constant with units of energy.
Approximate this potential energy function near its minimum by a harmonic oscillator potential energy function and determine the vibrational energy levels of the molecule with zero angular momentum.

Problem 2:

Find the rotational energy levels of a diatomic molecule with atoms of mass m₁ and m₂.  Take r₁ and r₂ to be the distances from atoms 1 and 2 to the center of mass.

Problem 3:

The operators a and a^† when acting on the energy eigenstates of the harmonic oscillator, denoted by |n>, have the property
a^†|n> = (n + 1)^½|n + 1>,  a|n> = n^½|n - 1>.
We have x = (a + a^†)/(2α),  p = -i(a - a^†)/(2β), where α =√(mω/(2ħ)),  β =1/√(2mωħ).
Find the mean value and root mean square deviation of p², when the oscillator is in the energy eigenstate |n>.

Problem 4:

An exotic atom consists of a Helium nucleus (Z = 2) and an electron and an antiproton p(bar) both in n = 2 states.  Take the mass of the p(bar) to be 2000 electron masses and that of the helium nucleus to be 8000 m_e.  For an electron in the n = 1 state of hydrogen E = -13.6 eV.
(a)  How much energy is required to remove the electron from this atom?
(b)  How much energy is required to remove the p(bar) from this atom?
(c)  Assume both the p(bar) and the electron are in 2p states.  Then each can de-excite to their ground state.  It is observed that radiation always accompanies those transitions when the electron jumps first, but when the p(bar) jumps first there is often no photon emitted.  Explain!

Problem 5:

An electron in the hydrogen atom occupies the combined position and spin state

R₂₁(r)[√⅓ Y₁₀(θ,φ)χ₊ + √⅔ Y₁₁(θ,φ)χ_-].

(a)  If you measure L², what value(s) might you get, and with what probability(ies)?
(b)  If you measure L_z, what value(s) might you get, and with what probability(ies)?
(c)  If you measure S², what value(s) might you get, and with what probability(ies)?
(d)  If you measure S_z, what value(s) might you get, and with what probability(ies)?
(e)  If you measured the position of the electron, what is the probability density for finding the electron at r, θ, φ in terms of the variables given above.
(f)  If you measured both S_z and the distance of the electron from the proton, what is the probability per unit length for finding the particle with spin up a distance r from the proton in terms of the variables given above?

Useful integral:  ∫₀^πsinθ dθ∫₀^2πdφ |Y_lm(θ,φ))|² = 1.