Ben Sauerwine
Practice for Qualifying Exams
Thanks to Yossef and Shiang Yong for their input in this problem.
Problem Source:
CMU August 2001 Qualifying Exam
Electron in a spherical well
An electron is in a spherical well of radius a and depth
, i.e., the non-relativistic
Hamiltonian is
0
V
V
m
p
H
+
=
2
2
v
(1)
with m = the mass of an electron and
a
r
V
a
r
V
V
>
=
≤
−
=
0
0
(2)
In this problem, we take
1
2
=
π
h
.
The solutions of the Schrodinger equation are of
the form
( )
(
)
s
lm
Y
r
R
χ
φ
θ
,
, where
( )
r
R
is the radial wave equation,
(
)
φ
θ
,
lm
Y
is a
spherical harmonic, and
s
χ
is a non-relativistic spinor.
( )
r
R
is a solution to the
equation
(
)
(
)
0
1
2
1
2
2
2
=
⎥
⎦
⎤
⎢
⎣
⎡
+
−
−
+
⎥
⎦
⎤
⎢
⎣
⎡
R
r
l
l
V
E
m
dr
dR
r
dr
d
r
(3)
where E is the energy of the state.
Note that in spherical coordinates, the
operator has the form
2
∇
(
φ
θ
,
1
1
2
2
2
2
Ω
+
⎥
⎦
⎤
⎢
⎣
⎡
=
∇
r
dr
dR
r
dr
d
r
)
(4)
where
Ω
is an operator in the angular variables.
The differential equation satisfied
by spherical Bessel or Hankel functions is
( )
(
)
( )
0
1
1
1
2
2
2
=
⎥
⎦
⎤
⎢
⎣
⎡
+
−
+
⎥
⎦
⎤
⎢
⎣
⎡
x
F
x
l
l
dx
x
dF
x
dx
d
x
l
l
(5)

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