Problem 1:

The Schroedinger equation for a particle in an isotropic harmonic oscillator potential in coordinate-space representation is

-[ħ²/(2m)]∇²ψ(r,t) + ½mω²r²ψ(r,t) = iħ(∂/∂t)ψ(r,t).

(a)  What is the corresponding Schroedinger equation in momentum-space representation?
(b)  Write down the ground state wave function ψ(r,t) for the particle.

Problem 2:

(a)  Find the energy eigenfunctions and energy levels for a spinless particle confined to a two-dimensional rectangular box, with |x| < a and |y| < b.
Do not just write down your answer, but derive it.  Justify each step.
(b)  Make a diagram or table showing the 5 lowest energy levels and the degeneracies for b = a and for b = 2a.

Problem 3:

The Rydberg constant, R_H = 109737.568525/cm is one of the most accurately known fundamental constants.
(a)  Find the wave number of the Balmer alpha line (n = 3 to n' = 2) in atomic hydrogen.  Neglect fine structure.
(b)  Is the Balmer alpha line in atomic deuterium shifted towards the blue or towards the red compared to normal hydrogen?
(c)  Calculate the shift in wave number between deuterium and hydrogen.

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)?