Physics 742 – Graduate Quantum Mechanics 2
Solution Set P
1.
[25] A hydrogen atom is in interstellar space in the 1s state, but not in the true
ground state (
F
= 0), but rather in the hyperfine excited state (
F
=1), specifically
in the state
1
2
, , ,
,
1,0,
,1,0
I
F
n l j F m
φ
=
=
.
It is going to transition to the true
ground state
1
2
, , ,
,
1,0,
,0,0
F
F
n l j F m
φ
=
=
via a magnetic dipole interaction.
(a) [5] Write out the initial and final states in terms of the explicit spin states of
the electron and proton
,
± ±
.
Find all nonzero components of the matrix
F
I
φ
φ
S
, where S is the electron spin operator.
The explicit form for these spin states can be found, for example, eq. (9.35).
We
therefore have
(
)
(
)
1
2
1
2
1,0
0,0
=
+ − + − +
=
+ − − − +
The spin operator acting on the initial state yields
(
)
(
)
(
)
(
)
(
)
(
)
1
1
1
2
2
8
1
1
1
2
2
8
1
1
1
1
2
2
2
8
10
,
10
,
10
00 .
x
x
y
y
z
z
S
S
i
i
S
σ
σ
σ
=
+ − + − +
=
− − + + +
=
+ − + − +
=
− − −
+ +
=
+ − + − +
=
+ − − − +
=
=
=
=
=
=
=
=
Therefore, the only nonzero matrix element is
S
z
, so
1
2
ˆ
F
I
φ
φ
=
S
k
=
(b) [8] Show that the rate for this transition for a wave going in a specific
direction with a definite polarization is given by
(
)
(
)
2
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 Spring '04
 Unknow
 Physics, mechanics, Atom, graduate quantum mechanics

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