Lecture 22 Highlights
We now consider the problem of how an atom makes a transition from one state
to another when it is stimulated (perturbed) by an electromagnetic field. Consider a
hydrogen atom in its 1s ground state. The light exerts a force on the electron dominated
by the electric field, )cos(
ˆ
0txEE x
ω
=
r
, which is arbitrarily assumed to be polarized along
the x-direction. We assume that the wavelength of the (visible) light (
λ
~ 500 nm) is
much greater than the size of the atom, which is the scale of the Bohr diameter ~ 0.1 nm.
Therefore the atom experiences a uniform-in-space electric field, as written above, to
good approximation.
The potential associated with the (conservative) electric force is:
, which yields
')(),(
0
dxEetxV x
x
∫
−−=
(
)
titi
xeexE
e
txV
ωω
−
+= 0
2
),( . We treat this potential
as the time-dependent perturbation. Assume that the hydrogen atom is left alone in the 1s
state for all times before t=0. At t=0 the light turns on and the perturbation begins. At
time t the light is turned off. Now the question is which state does the hydrogen atom
find itself in, and with what probability? This is a job for time-dependent perturbation
theory.
The transition probability can be calculated from the transition amplitude rate
from state n to state j:
xdxtxxe
i
anj
tEEi
nj nj 3*
/)( )(),(')(
00 rrr
h
&h
ψψ
Η
−
=∫
−
In this case we get:
[]
xdxxxee
ieE
anj
tEEitEEi
x
nj njnj 3*
/)(/)(
0)()(
2
0000 rr
h
&hhhh
ψψ
ωω
∫
−−+− +
−
=
The last piece is the “dipole matrix element” xdxxxx njjn 3* )()(
r
r
ψψ
∫
=, which will give
rise to “selection rules” for the transitions.
Integrating up the transition amplitude rate gives the transition amplitude:
jn
nj
tEEi
nj
tEEi
x
nj x
EE
e
EE
e
eE
ta njnj
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−−
−
+
+−
−
=
−−+−
ωω
ωω
hh
hhhh
00
/)(
00
/)(
0
0000 11
2
)(
This quantity has two terms that get very large when . The
system starts in state n and makes a transition to state j. Hence the second term
corresponds to absorption of energy
ω
h±=− 00 nj EE
ω
hby the atom in moving from state n to state j.
The first term corresponds to the atom starting in a higher energy state n and giving up
energy
ω
hto the electromagnetic field and going into lower energy state j. This process
is known as stimulated emission, and will be investigated in more detail later.
We focus on the case of absorption of energy by the atom from the
electromagnetic field , which arises from the second term. After taking
the absolute square and using the trigonometric identity
ω
h+=− 00 nj EE
2
sin21cos 2z
z−= , we get the
absorption probability:
