Square Well

A particle of mass m is confined to a space 0 < x < a in one dimension by infinitely high walls at x = 0, a. At t = 0 the particle is initially in the left half of the well with constant probability

(i)

a) Find the time-dependent wave function ψ (x, t).

b) What is the probability that the particle is in the nth eigenstate?

c) Write an expression for the average value of the particle energy.

SOLUTION

a) The most general solution is

(1)

(2)

(3)

We evaluate the coefficients by using the initial condition at t = 0:

(4)

The term cos (nπ/2) is either 1, 0, or –1, depending on the value of n. The answer to (a) is to use the above expression for a_n in (S.5.11.3). The answer to part (b) is that the probability of being in the nth eigenstate is |a_n|². The answer to part (c) is that the average value of the energy is

(5)

This latter series does not converge. It takes an infinite amount of energy to form the initial wave function.