10.1. (a) Show that (10.2.21) follows from (10.2.19) in the limiting case of a two-state

atom.

(b) Now consider, instead of the two-state model, a nonlinear electron oscillator

model in which (3.2.18b) is replaced by

d²x

dt²

+w√x + ax"

=

- Eo cos wt

m

in the case of an applied field of frequency w. Find, for n

= 2, the solution for

the induced dipole moment ex including terms up to second order in the field

amplitude Eo.

(c) For this nonlinear oscillator model with n = 3, find the solution for the

induced dipole moment including terms up to third order in Eo./n(50 points) Consider the nonlinear electron oscillator model

x+w²x + ax" = Eo cos wit

m

where Eo is the field amplitude, w₁ is the driving laser frequency, wo is the resonance,

and n is an integer. This problem is part (b) and (c) of question 10.1 from the textbook.

(a) Find, for n = 2, the solution for the induced dipole moment, ex, inluding terms up

to second order in the field amplitude Eo.

(b) For this nonlinear oscillator model with n = 3, find the solution for the induced

dipole moment including terms up to third order in Eo.

(c) Sketch (just sketch, don't need to plot) the potentials for n = 2 and n = 3.