Schrodinger Equation - Introduction to Quantum Mechanics - Exam, Exams of Quantum Mechanics

Download Schrodinger Equation - Introduction to Quantum Mechanics - Exam and more Exams Quantum Mechanics in PDF only on Docsity! NIU Physics PhD Candidacy Exam – Fall 2010 – Quantum Mechanics DO ONLY THREE OUT OF FOUR QUESTIONS Problem 1. We consider a spinless particle with mass m and charge q that is confined to move on a circle of radius R centered around the origin in the x-y plane. (a) Write down the Schrödinger equation for this particle and solve it to find the eigenenergies and corresponding normalized eigenfunctions. Are there degeneracies? [ 10 points ] (b) This system is perturbed by an electric field E pointing along the x axis. To lowest nonvan- ishing order in perturbation theory, find the corrections to the eigenenergies of the system. [10 points] (c) What are the corrections to the eigenfunctions due to the field E in lowest nonvanishing order? [10 points] (d) Next we consider instead of the electric field E the effect of a magnetic field B pointing along the z axis. Evaluate to lowest nonvanishing order in perturbation theory the corrections to the eigenenergies of the system. [10 points] Problem 2. Let us consider two spins S and S ′ with S = S ′ = 1 2 . The z components of the spin are Sz = ±12 and S ′z = ±12 . We can define a basis set as |SSz, S ′S ′z〉 (or simplified |Sz, S ′z〉). The spins interact with each other via the interaction H = TS · S′, (1) where T is a coupling constant. S and S′ work on the spins S and S ′, respectively. (a) Rewrite the interaction in terms of Sz, S ′ z and step up and down operators S± and S ′ ± . [10 points] (b) Find the eigenvalues of H when the spins are parallel. [10 points] (c) Find the eigenvalues of H for Sz + S ′ z = 0. [13 points] (d) Give a physical interpretation of the eigenenergies and eigenstates of H.[7 points]