Physics 401 Homework 4---Due October 14
This homework assignments concerns the quantum mechanic harmonic oscillator. The wave
functions n
ψ
refers to the nth eigenfunction of this system.
1. Use ladder operators to show that:
a.
ω
ψψ
m
n
nn x2
)1(
1ˆ+
+=h
b.
ω
ψψ
m
n
nn x2
1ˆh
=
−
c. 0
ˆ=
kj x
ψψ
unless 1
±
=
jk
d. 2
)1(
1ˆ+
+=nm
nn ip
ω
ψψ
h
e. 2
1ˆnm
nn ip
ω
ψψ
h
−=
−
f. 0
ˆ=
kj p
ψψ
unless 1
±
=
jk
2. Suppose that at t=0
(
)
)()()()( 321
3
1xxxx
ψψψψ
++= , show that
(
)
titititi exexexetx
ωωωω
ψψψψ
3
3
2
21
2/
3
1)()()(),( −−−− ++=
3. Use the results of 1. and 2. to compute x and pas a function of time for the state in
2. Show that your result follows the classical equations of motion for a Harmonic
oscillator.
4. Use Ehrenfest’s theorem to show x and p always follows the classical harmonic
oscillator equations of motion.
Griffths: 2.10