G25.2666: Quantum Chemistry and Dynamics

Problem set #7
Due: May 3, 2005

1.
The time-dependent Schrödinger equation for a particle moving in a potential subject to an electromagnetic field is

Show that the Schrödinger equation is invariant under a gauge transformation

2.
Suppose a harmonic oscillator with frequency and mass has a charge and subject to a time-dependent perturbation for a time interval . Here is the position operator and is the magnitude of an applied electric field.
a.
Calculate the transition probability for a transition between the and levels of the oscillator within first-order perturbation theory.
b.
Calculate the transition probability for a transition between the and levels of the oscillator within the lowest non-vanishing order in perturbation theory. What is the lowest order needed to obtain a non-trivial result?

3.
As a simple model for photoelectron spectroscopy, consider a one-dimensional particle moving in a potential , where is the position operator.
a.
Show that the Hamiltonian admits a single bound state with energy with associated eigenfunction

Hint: Try integrating the eigenvalue equation between and and show that as , the derivative of the eigenfunction has a discontinuity at .

b.
For positive energies , there will be two stationary wave functions corresponding to a particle incident from the left or from the right. These eigenfunctions are given by

Show that these eigenfunctions satisfy the orthonormalization condition

c.
Assuming the particle carries a charge and is subject to a perturbation of the form

where is the amplitude of an applied electric field, calculate the transition rate using the Fermi Golden Rule expression for transitions from the bound state to a state with energy .

4.
Problem 13.2, page 215 from Introduction to Quantum Mechanics in Chemistry.

5.
Problem 13.6, page 217 from Introduction to Quantum Mechanics in Chemistry.