Question

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

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.

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

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