PHY4605–Introduction to Quantum Mechanics II
Spring 2004
Problem Set 2
Jan. 14, 2004
Due: Jan. 24, 2004
Reading: PH notes
1. Gauge invariance. Griffiths problem 4.61.
2. Harmonic Oscillator in B-field.
Consider a 3D simple harmonic oscillator, H = p2 /2m + mω 2 r2 /2, with eigenstates given in an angular momentum basis ψn,`,m .
(a) Given that the eigenvalues of this problem may be written h̄ω(2n+`+3/2),
verify they are in a 1-1 correspondence with the eigenvalues for the 3D SHO
you can write down in a Cartesian basis (recall these follow directly from
the theorem that if H = H1 + H2 + H3 , E = E1 + E2 + E3 if the Hα
commute), i.e. check that the eigenvalues and their degeneracies are the
same in the two representations. Do not try to find the functions ψn,`,m
explicitly.
Suppose the oscillating particle has charge e, and is now placed in a weak,
spatially homogeneous magnetic field B.
(b) Show the form of the Hamiltonian is
H = H0 +
e
B·L
2mc
(1)
where H0 is the Hamiltonian for the 3D SHO in zero field, and L is the
orbital angular momentum operator for the system. [Hint: for convenience
take A = (1/2)r × B, and recall ∇ · (φA) = A · ∇φ + (∇ · A)φ for scalar
field φ and vector field A.]
(c) Sketch the spectrum for the three lowest energy states as a function of B,
i.e. show how the three lowest lying eigenstates in B = 0 split (if they do)
upon application of a field.
1