HOMEWORK ASSIGNMENT 6
PHYS852 Quantum Mechanics II, Spring 2008
Topics covered:
Applications of timedependent perturbation theory, FermiGolden rule
Please feel free to keep only the term closest to resonance when computing amplitudes or probabilities.
1.
AC Stark Effect
: Consider a two level atom with bare Hamiltonian
H
0
=
E
e

e
)(
e

+
E
g

g
)(
g

. We
want to drive this atom with an oscillating electric field,
vector
E
(
vector
r, t
) =
E
0
cos(
ωt
)
vectore
z
.
The interaction between the atom and the field is the usual
V
=
−
vector
d
·
vector
E
(
t
). Here the dipole moment
operator is of course
vector
d
=
−
e
vector
R
.
a.) Find the expression for the perturbation operator as a twobytwo matrix in the basis
{
e
)
,

g
)}
.
Compute the matrix elements explicitly for the hydrogen atom transition 1S to 2S, and also for
the 1S to 2P transitions. What can you say about the ability to drive the 1S to 2S transition
via the electricdipole interaction?
b.) Consider the 1S and 2P levels as

g
)
and

e
)
, respectively. If the atom starts in state

g
)
, use
perturbation theory to compute is the probability as a function of time to make the transition
to state

e
)
. Your answer should be valid up to secondorder in the probabilities.
c.) As a function of the detuning, Δ =
ω
−
ω
a
, (where
ω
a
= (
E
e
−
E
g
)
/
planckover2pi1
), approximate the quantum
mechanical average energy as
E
avg
=
P
g
(
t
)
E
g
+
P
e
(
t
)
E
e
. Then compute the timeaverage of this
energy (specifically, averaged over an extremely long timescale). Show that the average energy
is always larger than
E
g
, and is proportional to the field intensity

E
0

2
. You can think of the
difference between
E
avg
and
E
g
as the AC analog to the stark shift to the ground state energy.
d.) Based on the previous results, would you expect an atom to be accelerated or decelerated when
moving into a region of increasing field intensity? Another way of looking at it is to see that if
the internal energy changes with position, then the atoms centerofmass motion sees a potential
V
(
vector
r
) =
E
internal
(
vector
r
). Then the question becomes is field intensity maximum a potential minimum
or a potential maximum.
e.) Consider that when the atom is in

e
)
, the field has one fewer photon than when the atom is in

g
)
, and that the photon carries an energy
E
=
planckover2pi1
ω
. Compute the total energy of the atomfield
system when the atom is in

e
)
. Subtract from this the total energy of the atomfield system
when the atom is in

g
)
. Multiply this by the probability to be in

e
)
at time
t
, and average the
result over time. This is the correct AC stark shift.
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 Spring '10
 MichaelMoore
 mechanics, Work, Fundamental physics concepts, Perturbation theory, Timedependent perturbation theory, E

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