0198506961.pdf

124 The interaction of atoms with radiation

functions ofH 0 ; the unperturbed eigenvalues and eigenfunctions ofH 0

are just the atomic energy levels and wavefunctions that we found in

previous chapters. We write the wavefunction for the level with energy

Enas

Ψn(r,t)=ψn(r)e−iEnt/
. (7.3)

For a system with only two levels, the spatial wavefunctions satisfy

H 0 ψ 1 (r)=E 1 ψ 1 (r),

H 0 ψ 2 (r)=E 2 ψ 2 (r).

(7.4)

These atomic wavefunctions are not stationary states of the full Hamil-

tonian,H 0 +HI(t),but the wavefunction at any instant of time can be

expressed in terms of them as follows:

Ψ(r,t)=c 1 (t)ψ 1 (r)e−iE^1 t/
+c 2 (t)ψ 2 (r)e−iE^2 t/
, (7.5)

or, in concise Dirac ket notation (shorteningc 1 (t)toc 1 ,ω 1 =E 1 /
,

etc.),

Ψ(r,t)=c 1 | 1 〉e−iω^1 t+c 2 | 2 〉e−iω^2 t. (7.6)

Normalisation requires that the two time-dependent coefficients satisfy

|c 1 |^2 +|c 2 |^2 =1. (7.7)

7.1.1 Perturbation by an oscillating electric field

The oscillating electric fieldE=E 0 cos (ωt) of electromagnetic radiation

produces a perturbation described by the Hamiltonian

HI(t)=er·E 0 cos (ωt). (7.8)

This corresponds to the energy of an electric dipole−erin the electric

field, whereris the position of the electron with respect to the atom’s

(^2) Note thatE 0 cos (ωt) is not replaced centre of mass. (^2) Note that we have assumed that the electric dipole
by a complex quantityE 0 e−iωt,be-
cause a complex convention is built into
quantum mechanics and we must not
confuse one thing with another.
moment arises from a single electron but the treatment can easily be
generalised by summing over all of the atom’s electrons. The interaction
mixes the two states with energiesE 1 andE 2. Substitution of eqn 7.6
into the time-dependent Schr ̈odinger eqn 7.1 leads to
i

.

c 1 =Ωcos(ωt)e−iω^0 tc 2 , (7.9)

i

.

c 2 =Ω∗cos (ωt)eiω^0 tc 1 , (7.10)

whereω 0 =(E 2 −E 1 )/
and theRabi frequencyΩ is defined by

Ω=

〈 1 |er·E 0 | 2 〉

=

e

∫

ψ∗ 1 (r)r·E 0 ψ 2 (r)d^3 r. (7.11)

Theelectricfieldhasalmostuniformamplitudeovertheatomicwave-

function so we take the amplitude|E 0 |outside the integral.^3 Thus, for

(^3) Thisdipoleapproximation holds when
the radiation has a wavelength greater
than the size of the atom, i.e.λ a 0 ,
as discussed in Section 2.2.
radiation linearly polarized along thex-axis,E=|E 0 |̂excos (ωt), we
obtain^4
(^4) Section 2.2 on selection rules shows
how to treat other polarizations.