Homework 2
(due Wednesday, September, 14)
1. A particle in an infinite square well (0 < x < a) has wave function
Ψ(x, 0) = αψ1 (x) + βeiϕ ψ2 (x)
(1)
at t = 0, where α, β, and ϕ are real.
(a) For Ψ to be normalized what is the constraint on α and β?
(b) What is Ψ(x, t)? Express your answer in terms of ω ≡ π 2 h̄/2ma2 .
(c) What is |Ψ(x, t)|2 ?
(d) Compute the expectation value of x: hxi. What is the effect on changing ϕ on
hxi? What is the effect on changing α and β on hxi?
(e) Compute the expectation value of p: hpi.
(f) Compute the expectation value of the energy: hHi.
2. For this problem
set a = 1. The wave function in an infinite square well at t = 0 is
√
equal to 1/ 2∆ for 0.5 − ∆ ≤ x ≤ 0.5 + ∆ and zero elsewhere. (We assume that ∆ is
less than one half.)
(a) Determine the coefficients cn so that
Ψ(x, 0) =
∞
X
cn ψn (x).
(2)
n=1
Why are some of the cn equal to zero?
(b) Using Matlab or equivalent make plots of
Ψ(x, 0) =
NX
max
cn ψn (x)
(3)
n=1
for Nmax = 4, 8, 16, 32. You can include all four plots on the same graph if you
include a legend for the plot. Take ∆ = 0.1.
(c) Using the same ∆ = 0.1 and ω ≡ π 2h̄/2ma2 , plot |Ψ(x, t)|2 for ωt = π/2 and
ωt = π keeping Nmax = 100 terms in the series. Note that in this question you
are plotting the magnitude of |Ψ(x, t)|2 , while in the previous you are plotting
Ψ(x, t).