1
Problem set 5  Free space and sharp potentials
Due: Oct. 19, 2011, in class
Posted on Oct. 12, 2011
1.
A particle in a parabolic potential is set free
. This is similar to the 3rd problem in problem set 2. Imagine
a parabolic potential described by the frequency
ω
=
q
k
m
. If we prepare the system in the

n
= 1
i
state, it will
stay in this state as long as the system is still described by the Hamiltonian of the harmonic oscillator.
(a) What is the functional form of this wavefunction at a given time
t
?
(b) Now assume that at
t
= 0 the potential suddenly drops to zero,
V
(
x
) = 0. Find the time evolution of the
wavefunction following the steps below:
i. Express the wavefunction at time
t
= 0 as a combination of the stationary states of the free particle
Schr¨
odinger equation.
ii. What is the appropriate time evolution of each of the above stationary states?
iii. Add up the stationary states with the corresponding phases to find the total, time dependent wave
function.
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 Fall '10
 TamiPeregBarnea
 mechanics, Work, Fundamental physics concepts, group velocity, stationary states

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