Problem 1:

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 2:

(a)  Given that the radial function for the ground state of hydrogen is R(r) = (2/a₀^3/2) exp(-r/a₀), find the value of radius at which the ground state radial probability reaches a maximum.

(b)  What is the probability of finding the ground state electron within the Bohr radius?

(c)  Using your calculation in parts (a) and (b), comment on the similarity and the difference of the hydrogen atom ground state in quantum mechanics compared to the Bohr model of the hydrogen atom (in terms of energy and radius), and give a short explanation of how the quantum mechanical model solves the uncertainty issue in the Bohr model.

Problem 3:

A particle with mass m is in ground state of 1D harmonic oscillator potential.  The potential energy function is U(x) = ½mω²x².
(a)  A position measurement is made.  What is the probability of finding the particle at the middle of the potential well with a position uncertainty Δx << (ħ/(mω))^½?
(b)  If the particle is found within Δx at the middle of the potential well, what is the probability that a subsequent energy measurement will still find the particle in the ground state?

Problem 4:

A negative K meson with mass m = 1000 electron masses is captured into a circular Bohr orbit around a lead nucleus (Z = 82).  Assume it starts with principal quantum number n = 10 and then cascades down through n = 9, 8, 7, ... etc.
(a)  What is the energy of the photon emitted in the n = 10 to n = 9 transition?
(b)  What is the approximate radius of the lead nucleus if no further quanta are observed after the n = 4 to n = 3 transition (because of nuclear absorption of the K meson)?