130_notes.dvi

Molecules canrotatelike classical rigid bodies subject to the constraint that angular momentum is
quantized in units of ̄h.

Erot=

1

2

L^2

I

=

ℓ(ℓ+ 1) ̄h^2

2 I

≈

̄h^2

2 Ma^20

=

m

M

α^2 mc^2

2

≈

m

M

E≈

1

1000

eV

1.35 Time Dependent Perturbation Theory

We have used time independent perturbation theory to find the energy shifts of states and to find
the change in energy eigenstates in the presence of a small perturbation. We now consider the case
of aperturbationV that is time dependent. Such a perturbationcan cause transitions
between energy eigenstates. We will calculate the rate of those transitions (See section 28).

We derive an equation for therate of change of the amplitude to be in thenth energy
eigenstate.

i ̄h

∂cn(t)

∂t

=

∑

k

Vnk(t)ck(t)ei(En−Ek)t/h ̄

Assuming that att= 0 the quantum system starts out in someinitial stateφi, we derive the
amplitude to be in a final stateφn.

cn(t) =

1

i ̄h

∫t

0

eiωnit

′

Vni(t′)dt′

An important case of a time dependent potential is a puresinusoidal oscillating (harmonic)
perturbation. We canmake up any time dependence from a linear combinationof sine
and cosine waves. With some calculation, we derive the transition rate in a harmonic potential of
frequencyω.

Γi→n≡

dPn

dt

=

2 πVni^2

̄h

δ(En−Ei+ ̄hω)

This contains a delta function of energy conservation. The delta function may seem strange. The
transition rate would be zero if energy is not conserved and infinite ifenergy is exactly conserved.
We can make sense of this if there is a distribution function ofP(ω) of the perturbing potential or if
there is a continuum of final states that we need to integrate over. In either case, the delta function
helps us do the integral simply.

1.36 Radiation in Atoms

The interaction of atoms with electromagnetic waves (See section 29) can be computed using time
dependent perturbation theory. The atomic problem is solved in theabsence of EM waves, then the
vector potential terms in the Hamiltonian can be treated as a perturbation.

H=

1

2 m

(

~p+

e

c

A~

) 2

+V(r).