NIU Physics PhD Candidacy Exam – Spring 2010 – Quantum Mechanics
DO ONLY THREE OUT OF FOUR QUESTIONS
Problem 1. In many systems, the Hamiltonian is invariant under rotations. An example is the
hydrogen atom where the potential V(r) in the Hamiltonian
H=p2
2m+V(r),
depends only on the distance to the origin.
An infinitesimally small rotation along the z-axis of the wavefunction is given by
Rz,dϕψ(x, y , z) = ψ(x−ydϕ, y +xdϕ, z),
(a) Show that this rotation can be expressed in terms of the angular momentum component
Lz. [10 points]
(b) Starting from the expression of Lzin Cartesian coordinates, show that Lzcan be related to
the derivative with respect to ϕin spherical coordinates. Derive the ϕ-dependent part of the
wavefunction corresponding to an eigenstate of Lz. [10 points]
(c) Show that if Rz,dϕ commutes with the Hamiltonian, then there exist eigenfunctions of Hthat
are also eigenfunctions of Rz,dϕ . [10 points]
(d) Using the fact that Liwith i=x, y, z commute with the Hamiltonian, show that L2commutes
with the Hamiltonian. [10 points]
Problem 2. To an harmonic oscillator Hamiltonian
H= ¯hωa†a,
we add a term
H′=λ(a†+a).
This problem is known as the displaced harmonic oscillator. It can be diagonalized exactly by
adding a constant (let us call it ∆) to the step operators.
(a) Express the constant ∆ in terms of ¯hω and λ. [8 points]
(b) The energies are shifted by a constant energy. Express that energy in terms of ¯hω and λ. [8
points]
(c) Express the new eigenstates |˜niin terms of the displaced oscillator operator ˜a†. [8 points]
(d) Calculate the matrix elements h˜n|0i.[8 points]
(e) An harmonic oscillator is in the ground state of H. At a certain time, the Hamiltonian
suddenly changes to H+H′. Plot the probability and change in energy for the final states |˜ni
for ˜n= 0,···,5 for ∆ = 2. [8 points]
