Ph135c. Solution set #1, 4/9/10
1.
We can approximate this potential by an infinite radial square well (taking care to shift the energies
by
V
0
at the end):
V
(
r
) =
0
r < a
∞
r > a
We look for a solution of Schrodinger’s equation in the form:
ψ
k‘m
(
r, θ, φ
) =
u
k‘
(
r
)
r
Y
‘
m
(
θ, φ
)
Then we find that
u
k‘
(
r
) must satisfy (unless otherwise noted, we set
~
= 1):

1
2
m
d
2
u
k‘
dr
2
+
‘
(
‘
+ 1)
2
mr
2
u
k‘
=
k
2
2
m
u
k‘
where we’ve written the energy as
E
=
k
2
2
m
. The solutions can be expressed in terms of spherical
Bessel functions
j
‘
(
x
):
u
k‘
(
r
)
∝
rj
‘
(
kr
)
The allowed energies are determined by the condition
u
k‘
(
a
) = 0.
If we denote the
n
th zero of
j
‘
(
x
) by
x
n‘
(that is,
j
‘
(
x
n‘
) = 0 and 0
< x
1
‘
< x
2
‘
< ...
), then:
E
n‘
=
x
n‘
2
2
ma
2
One can look up the zeros of the spherical Bessel functions in a reference table, and the lowest ten
values are:
x
1
‘
x
2
‘
x
3
‘
‘
= 0
:
3
.
14
6
.
28
9
.
42
‘
= 1
:
4
.
49
7
.
72
‘
= 2
:
5
.
76
9
.
10
‘
= 3
:
7
.
00
‘
= 4
:
8
.
18
‘
= 5
:
9
.
39
The corresponding energy levels are plotted in figure 1. The degeneracies are determined by the
angular momentum via
d
(
‘
) = 2
‘
+ 1. That is, the
s
levels have 1 state, the
p
levels have 3 states, and
so on. If we have identical spin
1
2
fermions, there are 2 total states for each wavefunction, coming from
the two independent spins.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 Smith
 Physics, Energy, Kinetic Energy, Fundamental physics concepts, Noether's theorem, total spin

Click to edit the document details