Homework 3
(due Wednesday, September, 21)
All the problems in this homework set are about the harmonic oscillator, specifically
computing properties of the harmonic oscillator in quantum mechanics using the raising and
lowering operators. You should not have to do any integrals to solve these problems. All
equation numbers cited in the questions refer to Griffiths’ book.
1. Wave function: We computed the ground state of the harmonic oscillator in class
[2.59].
(a) Starting from this ground state, use the raising and lowering operators [2.47] to
compute ψ1 and ψ2 with [2.67].
(b) The characteristic length scale of the harmonic oscillator is l =
√
n = 0, 1, 2 plot lψn vs. x/l.
q
h̄/mω. For
2. Expectation values: The position and momentum operators may be expressed in
terms of raising and lowering operators [2.69], and one knows the effect of the raising and lowering operators on the harmonic oscillator eigenstates [2.66]. For the nth
eigenstate, ψn , compute the following expectation values
(a) hxi
(b) hx2 i
(c) σx
(d) hpi
(e) hp2 i
(f) σp
(g) Check the uncertainty principle.
(h) Expectation value of the kinetic energy, p2 /2m.
(i) Expectation value of the potential energy, mω 2 x2 /2.
(j) Expectation value of the total energy.
3. Time dependence: Let the wave function at t = 0 be given by
1 Ψ(x, 0) = √ ψ0 (x) + eiϕ ψ1 (x) ,
2
(1)
where ϕ is real and ψ0 and ψ1 are the harmonic oscillator ground state and first excited
state, respectively.
(a) What is Ψ(x, t)?
(b) For Ψ(x, t) compute the expectation value of x as a function of time.
(c) Also for Ψ(x, t) compute the expectation value of p as a function of time.
(d) Show that the results from (b) and (c) satisfy the classical differential equations:
hpi
dhxi
=
dt
m
dhpi
= −khxi.
dt
(2)
(3)
(e) Repeat (b) and (c) for
1 Ψ(x, 0) = √ ψ0 (x) + eiϕ ψ2 (x) .
2
(4)