Elements of Modern Physics
98
Thus, to the leading order, all the energy levels are shifted by the same
amount.
Example 6
For the 3-dimensional harmonic oscillator, the Hamiltonian
H =
2
2
1
1
2
2
+ kr
m
p
(3.178)
is the sum of 1-dimensional Hamiltonians in the three directions. Hence the
wave function is a product of the three wave functions,
ψ (x, y, z) = ψ nx (x) ψ ny (y) ψ nz (z)
(3.179)
where ψ nx (x), etc. are given in Eq. (3.111), and the energy is
E =
3
2
x
y
z
n n n


+ + +
ω




(3.180)
These solutions can be written in terms of spherical coordinates so as to
exhibit the angular momentum content of the states. For example, the ground
state is
ψ (r, θ, φ) =
3/ 2
0
2 2
0
1/ 4
2
( , ) exp (–
/ 2)
Y
r
α
θ φ
α
π
E =
3
2
ω
(3.181)
while the state with n x = 1, n y = n z = 0 can be written as
ψ (r, θ, φ) =
5/ 2
1
1
22
1
1
1/ 4 1/ 2
2
[
( , )
( , )] exp(
/2)
3
Y
Y
r
r
−
α
θ φ −
θ φ
−α
π
E =
5
2
ω
(3.182)
PROBLEMS
1. A one-dimensional wave packet has the form ψ(x) = 0 for |x| > a, and
ψ(x) = (2a)
–1/2
for |x| ≤ a. What is the wave function in the momentum
space? Demonstrate the uncertainty principle for this wave packet.
2. For a particle coming from the left with energy E, and a potential changing
from V (x) = 0 for x < 0 to V (x) = – V 0 for x ≥ 0, obtain the transmission
and reflection coefficients T and R respectively. Show that R + T = 1.