Physics 512
Winter 2003
Homework Set #12 – Solutions
1. The differential cross section for the ejection of an electron with momentum
k
f
by
an incident photon of momentum
k
(
ω
=
c

k

) and polarization ˆ (the photoelectric
effect) may be written as
dσ
d
Ω
=
α

k
f

2
πm
¯
hω
L
3

f

e
ik
·
r
p
·
ˆ

i

2
where the matrix element refers to the initial (bound) and final (free plane wave)
electron states. Derive this expression using the quantum theory of radiation (instead
of the classical treatment shown in class).
We begin by computing the transition rate for this process to occur.
For the rate,
Fermi’s golden rule for harmonic perturbations gives
w
=
2
π
¯
h

f
; 0

V

i
;
{
n
(
α
)
k
= 1
} 
2
ρ
f
(
E
f
)
Here we have used a notation where
i
and
f
denote the initial and final electron state,
while the remaining pieces are a shorthand label for the photons in the Fock space. For
absorption of a photon, we have
V
=
e
mc
A
(+)
·
p
=
e
mc
√
2
π
¯
hc
2
1
L
3
/
2
k,α
1
√
ω
a
α
(
k
)
p
·
ˆ
(
α
)
e
ik
·
r
Evaluating the matrix element in the photon Fock space, we see that the lowering
operator annihilates the initial photon, leaving the Fock vacuum (which is normalized
so that
0

0
= 1
). This selects out only one particular term in the sum (corresponding
to the single initial state photon). As a result
f
; 0

V

i
;
{
n
(
α
)
k
= 1
}
=
e
mc
2
π
¯
hc
2
ωL
3
f

e
ik
·
r
p
·
ˆ

i
Squaring this and using Fermi’s golden rule yields
w
=
(2
π
)
2
e
2
m
2
ωL
3

f

e
ik
·
r
p
·
ˆ

i

2
=
(2
π
)
2
e
2
m
2
ωL
3
m

k
f

¯
h
2
L
2
π
3

f

e
ik
·
r
p
·
ˆ

i

2
ρ
f
(
E
f
)
=
αc

k
f

2
πm
¯
hω

f

e
ik
·
r
p
·
ˆ

i

2
d
Ω
where, for the final state, we have used the free electron density of states
ρ
(
E
f
) =
m

k
f

¯
h
2
L
2
π
3
d
Ω
1
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Finally, this may be converted to a differential cross section by dividing by the incident
(single photon) ﬂux
c/L
3
. This gives the desired result
dσ
d
Ω
=
α

k
f

2
πm
¯
hω
L
3

f

e
ik
·
r
p
·
ˆ

i

2
Note that this matrix element is only for the electron states. Moreover, the final ejected
electron wavefunction must be normalized as
ψ
∼
1
/L
3
/
2
. Taking this into account,
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 Winter '03
 Unknow
 Physics, Momentum, Work, Photon, Quantum Field Theory, Polarization, matrix element, equal time commutator, kleingordon field

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