Physics 6210/Spring 2007/Lecture 37
Lecture 37
Relevant sections in text:
§
5.7
Electric dipole transitions
Our transition probability (to ﬁrst order in perturbation theory) is
P
(
i
→
f
)
≈
4
π
2
α
m
2
¯
hω
2
fi
N
(
ω
fi
)
h
n
f
,l
f
,m
f

e

i

ω
fi

1
c
ˆ
n
·
~
X
ˆ
e
·
~
P

n
i
,l
i
,m
i
i
2
,
where
α
=
q
2
¯
hc
≈
1
137
is the
ﬁne structure constant
and
N
(
ω
) =
ω
2
c

A
(
ω
)

2
is the energy per unit area per unit frequency carried by the EM pulse characterized by
the vector potential in the radiation gauge with frequency components
A
(
ω
).
We now want to analyze the matrix element which appears. This factor reﬂects the
atomic structure and characterizes the response of the atom to the electromagnetic wave.
Let us begin by noting that the wavelength of the radiation absorbed/emitted is on
the order of 2
πc/ω
fi
∼
10

6
m
, while the atomic size is on the order of the Bohr radius
∼
10

8
m
. Thus one can try to expand the exponential in the matrix element:
h
n
f
,l
f
,m
f

e

i

ω
fi

1
c
ˆ
n
·
~
X
ˆ
e
·
~
P

n
i
,l
i
,m
i
i
=
h
n
f
,l
f
,m
f

(1

ip

ω
fi

1
c
ˆ
n
·
~
X
+
...
)ˆ
e
·
~
P

n
i
,l
i
,m
i
i
.
The ﬁrst term in this expansion, if nonzero, will be the dominant contribution to the
matrix element. Thus we can approximate
h
n
f
,l
f
,m
f

e

i

ω
fi

1
c
ˆ
n
·
~
X
ˆ
e
·
~
P

n
i
,l
i
,m
i
i ≈ h
n
f
,l
f
,m
f

ˆ
e
·
~
P

n
i
,l
i
,m
i
i
,
which is known as the
electric dipole approximation
. Transitions for which this matrix ele
ment is nonzero have the dominant probability; they are called
electric dipole transitions
.
We shall see why in a moment. Transitions for which the dipole matrix element vanishes
are often called “forbidden transitions”. This does not mean that they cannot occur, but
only that the probability is much smaller than that of transitions of the electric dipole
type, so they do not arise at the level of the approximation we are using.
If we restrict attention to the electric dipole approximation, the transition probability
is controlled by the matrix element
h
n
f
,l
f
,m
f

~
P

n
i
,l
i
,m
i
i
. To compute it, we use the fact
that
[
~
X,H
0
] =
i
¯
h
~
P
m
,
1
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and that
H
0

n,l,m
i
=
E
n

n,l,m
i
.
We get
h
n
f
,l
f
,m
f

~
P

n
i
,l
i
,m
i
i
=
m
i
¯
h
h
n
f
,l
f
,m
f

~
XH
0

H
0
~
X

n
i
,l
i
,m
i
i
=
imω
fi
h
n
f
,l
f
,m
f

~
X

n
i
,l
i
,m
i
i
.
Now perhaps you can see why this is called a dipole transition: the transition only occurs
according to whether or not the matrix elements of the (component along ˆ
e
of the) dipole
moment operator,
q
~
X
, are nonvanishing.
Selection rules for Electric Dipole Transitions
We have seen that the dominant transitions are of the electric dipole type. We now
consider some details of the dipole matrix elements
h
n
f
,l
f
,m
f

q
ˆ
e
·
~
X

n
i
,l
i
,m
i
i
.
In particular, we derive necessary conditions on
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 Spring '07
 M
 Physics, mechanics, Atom, Angular Momentum, Energy, Nf, electric dipole transitions

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