Homework 2
(due Friday, September, 6)
1. Consider the wave function at t = 0 ψ(x, 0) = C sin(2πx) cos(πx) on the interval
0 ≤ x ≤ 1.
(a) What is the normalization constant, C?
(b) 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.)
(c) The energies of the eigenstates are En = h̄2 π 2 n2 /(2m) for a = 1. What is ψ(x, t)?
(d) Compute the expectation value of x as a function of time for this ψ(x, t).
(e) Compute the expectation value of p as a function of time for ψ(x, t).
(f) Compute the expectation value of x2 as a function of time for this ψ(x, t).
(g) Compute the expectation value of p2 as a function of time for this ψ(x, t).
(h) Compute the expectation value of the energy as a function of time.
(i) Check the uncertainty principle.
2. Let the wave function at t = 0 in an infinite square well potential (0 < x < 1) be
ψ(x, 0) = C sin(2πx) for 0 < x < 1/2 and ψ(x, 0) = 0 for 1/2 < x < 1.
(a) What is C so that ψ is normalized?
(b) Determine the coefficients cn so that
ψ(x, 0) =
∞
X
√
cn 2 sin(πnx).
(1)
n=1
(c) Modify the code leftside.m so to perform this sum. Print out a graph comparing
ψ(x, 0) to the sum of the first 100 terms. Note we are comparing ψ(x, 0) rather
than |ψ(x, 0)|2 .
(d) Modify the code movie.m to compute |ψ(x, t)|2 . Print out your code and describe
in words what happens to probability density, |ψ(x, t)|2 , as a function of time.