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Photoelectric Effect in Hydrogen
Michael Fowler 3/26/10
Introduction
In the photoelectric effect, incoming light causes an atom to eject an electron. We consider the
simplest possible scenario: that the atom is hydrogen in its ground state. The interesting question
is: for an ingoing light wave of definite frequency and amplitude, what is the probability of
ionization of a hydrogen atom in a given time?
In other words, assuming we can use time-
dependent perturbation theory, what is the ionization rate?
Formally, we know what to do.
We must find the interaction Hamiltonian
1
H
, then use Fermi’s
Golden Rule for the transition rate with a periodic perturbation:
(
)
2
1
2
i
f
f
i
R
f H
i
E
E
π
δ
ω
→
=
−
−
But it’s not that easy!
For one thing, the outgoing electron will be in some kind of plane wave
state, so whatever convention we adopt for normalizing such states appears in the rate.
But also
the
δ
function is tricky for excitation into the continuum: just how many of these plane wave
states satisfy
f
i
E
E
ω
=
+
?
We shall discover that with a consistent formalism, these two
difficulties cancel each other.
The Interaction Hamiltonian
Taking the incoming wave to be an electromagnetic field having vector potential
(
)
(
)
0
,
cos
A r t
A
k r
t
ω
=
⋅
−
The interaction Hamiltonian is given by replacing the electron kinetic energy term
2
/ 2
p
m
with
(
)
2
/
/ 2
p
qA c
m
−
.
The relevant new term is
(
)
(
)
(
)
(
)
1/ 2
/
/
m
q c
p A
A p
e mc A p
−
⋅
+
⋅
=
since
q
=
−
e
and
0
A
∇⋅
=
in our gauge.
Therefore
(
)
(
)
(
)
(
)
1
0
0
cos
.
2
i k r
t
i k r
t
e
H
k r
t
A
p
mc
e
e
e
A
p
mc
ω
ω
ω
⋅ −
−
⋅ −
=
⋅
−
⋅
=
+

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