Question

A particle of mass m is confined to a one-dimensional region 0 ≤ x ≤ a as shown in Fig. At t = 0 its normalized wave function is 

\(\psi (x, t = 0) =\sqrt{8/5a}\left[1 + \cos (\frac{\pi x}a)\right]\sin (\pi x/a)\).

(a) What is the wave function at a later time t = t₀?

(b) What is the average energy of the system at t = 0 and at t = t₀?

(c) What is the probability that the particle is found in the left half of the box (i.e., in the region 0 ≤ x ≤ a/2) at t = t₀?

Answer 1

The time-independent Schrodinger equation for 0 < x < a is

\(\frac{h^2}{2m} \frac{d^2\psi}{dx^2} + E\psi = 0\).

It has solution \(\psi\)(x) = A sin kz, where k is given by k² = \(\frac{2mE}{h^2}\), satisfying \(\psi\)(O) = 0. The boundary condition $(a) = 0 then requires ka = nr.

Hence the normalized eigenfunctions are

(a) Thus

(b) The average energy of the system is

(c) The probability of finding the particle in 0 ≤ x ≤ a/2 at t = t₀ is