Homework 1

This homework is due on Tuesday January 16th. You should submit a PDF of your solutions on myCourses.

Do the following 4 questions on path integrals from Townsend: 8.1, 8.3, 8.4, 8.5.

In 8.1, make sure you get to the same answer as equation (6.76) in Townsend. In 8.3, comment on the form of your answer.

Hints:

• For 8.1, a useful formula is

\[\int_{-\infty}^\infty dx\ e^{-(ax^2+bx)} = \sqrt{\pi\over a} e^{b^2/4a}\]

(where \(\mathrm{Re}(a)>0\)).

• For 8.3: for a quadratic potential, the propagator is \(\propto e^{iS_c/\hbar}\), so you just need to determine \(S_c\), the action corresponding to the classical path \(x_c(t)\). You could start with the general solution for motion of an oscillator \(x(t) = A\sin\omega t+B\cos\omega t\).