300 Appendix A: Perturbation theory
Fig. A.1An illustration of degener-
ate perturbation theory in classical me-
chanics, closely related to that shown
in Fig. 3.3. (a) The ‘unperturbed’ sys-
tem corresponds to two harmonic oscil-
lators, each of which has a massM 1
attached to one end of a spring and
with the other end fixed. These inde-
pendent oscillators both have the same
resonant frequencyω 1 .(b)Asystem
of three masses joined by two springs
corresponds to two coupled harmonic
oscillators. WhenM 2 M 1 the cou-
pling is weak. The displacementsx 1 ,
x 2 andx 3 are measured from the rest
position of each mass (and are taken as
being positive to the right). One eigen-
mode of the system corresponds to a
symmetric stretch in which the central
mass does not move; therefore this mo-
tion has frequencyω 1 (independent of
the value ofM 2 ). (c) The asymmetric
stretching mode has a frequency higher
thanω 1. (This system gives a simple
model of molecules such as carbon diox-
ide; a low-frequency bending mode also
arises in such molecules.)
(a)(b)(c)atom). The equations of motion for this ‘ball-and-spring’ molecular
model areM 1..
x 1 =κ(x 2 −x 1 ), (A.6)
M 2..
x 2 =−κ(x 2 −x 1 )+κ(x 3 −x 2 ), (A.7)
M 1..
x 3 =−κ(x 3 −x 2 ), (A.8)whereκis the spring constant (of the bond between the carbon and
oxygen atoms). Thex-coordinates are the displacement of the masses
from their equilibrium positions. Adding all three equations gives zero
on the right-hand side since all the internal forces are equal and opposite
and there is no acceleration of the centre of mass of the system. This
constraint reduces the number of degrees of freedom to two. It is con-
venient to use the variablesu=x 2 −x 1 andv=x 3 −x 2 .Substituting
κ/M 1 =ω^21 andκ/M 2 =ω^22 leads to the matrix equation(..
u..
v)
=
(
−
(
ω 12 +ω^22)
ω^22
ω 22 −(
ω 12 +ω 22)
)(
u
v)
. (A.9)
A suitable trial solution is
(
u
v)
=
(
a
b)
e−iωt. (A.10)This leads to an equation with the same form as eqn A.2 with=0(cf.
eqn 3.14).