Subject: Quantum Mechanics
Edition: 2nd
Author: Amit Goswami
Problems:
Chapter 9. Systems of Two Degrees of Freedom
#1. Consider the problem of a two-dimensional harmonic oscillator in Cartesian coordinates. The Hamiltonian is given as
H = {(Px)^2}/(2m) + {(Py)^2}/(2m) +(1/2)k(x^2 + y^2)
a) Write H and Lz in terms of the creation and annihilation operators in two dimensions, ax, ax dagger, ay, ay dagger. Show, using the commutation relation of the a and a dagger operators, that Lz commutes with H.
b) Consider the two eigenstates |U 01> and |U 10> belonging to the first excited state of the oscillator (N = 1). Are these eigenstates also of Lz? Demonstrate your answer using the form of Lz you derived in part a) in terms of a, a dagger operators. Reconcile your answer with the conclusion reached in part a).
#2. Consider a particle confined to a circular box. Show that the Schrodinger equation is separable in cylindrical coordinates Rho, Phi and find and solve the equations for the radial and the angular motion. (Hint: The radial equation is the Bessel equation and its solutions are called Bessel functions; see any book on special functions of mathematical physics.)
#6. Consider two coupled oscillators. The Hamiltonian is given as