Question

Consider the wave function at t = 0, ψ(x, 0) = C sin(3πx/2) cos(πx/2) on the interval 0 ≤ x ≤ 1.

(1) What is the normalization constant, C?
(2) Express ψ(x,0) as a linear combination of the eigenstates of the infinite square well on the interval, 0 < x < 1. (You will only need two terms.)
(3) The energies of the eigenstates are En = h̄²π²n²/(2m) for a = 1. What is ψ(x, t)?
(4) Compute the expectation value of x as a function of time for this ψ(x, t)
(5) Compute the expectation value of p as a function of time for ψ(x, t)
(6) Compare how 〈p〉 and 〈x〉 are related to what one would expect classically?
(7) Compute the expectation value of the energy as a function of time.

Answer 1

1) The normalisation condition is

This gives

Hence the normalization constant is

2) Using the trigonometric identity

one can write

And noting that the eigenstates of infinite square well potential for a = 1are

we can write the given wave function as a linear combination of the eigenstates of the infinite square well as follows

3) The wave function as a function of time is

where

and

4) Note that

Hence

where

Therefore

The first term gives

Similarly

and

Therefore the expectation value of x as a function of time is