Homework 4
(due Wednesday, September, 17)
The first problem in this homework set is very nearly identical to the first problem in
the harmonic oscillator homework from last year. Thus, I have disabled the solution on-line
from Fall 2011 temporarily. All of the equation numbers below are from 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].
q
(b) The characteristic length scale of the harmonic oscillator is l = h̄/mω. For
√
n = 0, 1, 2 plot lψn vs. x/l. Print out your plot and your Matlab/Octave code.
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
1
ψ(x) = √ (ψn (x) + iψn+1 (x))
2
(1)
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
Ψ(x, 0) = cos(θ)ψ0 (x) + sin(θ)eiϕ ψ1 (x),
(2)
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:
dhxi
hpi
=
dt
m
dhpi
= −khxi.
dt
(3)
(4)