0198506961.pdf

A.2 Interaction of classical oscillators of similar frequencies 299

When= 0 this leads to a matrix eigenvalue equation similar to that
used in the treatment of helium (eqn 3.14), but the formalism given here
enables us to treat both degenerate and non-degenerate cases as different
limits of the same equations. This two-level system is described by the
matrix equation

(

J+K

KJ−

)(

a

b

)

=E

(

a

b

)

. (A.2)

The eigenenergiesEare found from the determinantal equation (as in
eqn 7.91 or eqn 3.17):

E=J±

√

^2 +K^2. (A.3)

This exact solution is valid for all values ofK, not just small perturba-
tions. However, it is instructive to look at the approximate values for
weak and strong interactions.

(a)Degenerate perturbation theory,K 2 
IfK 2 then the levels are effectively degenerate, i.e. their energy
separation is small on the scale of the perturbation. For this strong
perturbation the approximate eigenvalues are

E=J±K, (A.4)

as in helium (Section 3.2). The two eigenvalues have a splitting of

2 Kand a mean energy ofJ. The eigenfunctions are admixtures of

the original states with equal amplitudes.

(b)Perturbation theory,K 2 
When the perturbation is weak the approximate eigenvalues ob-
tained by expanding eqn A.3 are

E=J±

(

+

K^2

2 

)

. (A.5)

This is a second-order perturbation, proportional toK^2 ,asineqns

6.36 and 7.92.

A.2 Interaction of classical oscillators of similar frequencies

In this section we examine the behaviour of two classical oscillators of
similar frequencies that interact with each other, e.g. the system of two
masses joined by three springs shown in Fig. 3.3, or alternatively the
system of three masses joined by two springs shown in Fig. A.1. The
mathematics is very similar in both cases but we shall study the latter
because it corresponds to a real system, namely a molecule of carbon
dioxide (withM 1 representing oxygen atoms andM 2 being a carbon