Ben Sauerwine
Practice for Qualifying Exams
Problem Source:
CMU August 2002 Qualifying Exam
This problem considers the hyperfine structure of the 1s level of hydrogen.
Splitting
of this level is responsible for the famous “21 cm” emission line in radio astronomy
studies of interstellar hydrogen clouds.
The hydrogen atom Hamiltonian can be
expressed as:
hf
f
W
W
H
H
+
+
=
0
where the non-relativistic Hamiltonian
R
e
m
P
H
2
2
0
2
−
=
v
Recall that the 1s level is the hydrogen atom orbital ground state, an eigenstates of
that is positive definite and spherically symmetric (with zero angular
momentum), and varies like
0
H
(
)
0
,
,
a
R
e
R
−
≈
φ
θ
ψ
.
The “fine structure” corrections are
of relativistic origins and may be expanded as
D
SO
mv
f
W
W
W
W
+
+
=
with
(
)
(
)
R
c
W
S
L
R
b
W
P
a
W
D
SO
mv
δ
v
v
v
⋅
=
=
3
4
with a, b, c constants.
The “hyperfine structure” corrections result from interaction
of the electron with the nuclear magnetic moment.
Expressed in terms of nuclear
spin
I
v
, electron spin
, and angular momentum
S
v
L
v
,
(
)
(
)
(
)
(
)
[
]
(
)
(
)
R
I
S
f
I
S
n
I
n
S
R
e
I
L
d
W
hf
v
v
v
v
v
v
v
v
v
δ
⋅
+
⋅
−
⋅
⋅
+
⋅
=
ˆ
ˆ
3
3
with d, e, and f constants and
the unit vector in the direction of
n
ˆ
R
v
.

This ** preview** has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*