Page 94 - Electrical Properties of Materials

76 Bonds

instance because I am sure I would then exclude many actual or potential ap-
plications. Not being certain of the limitations of the approach, I would rather
give you a vague description, just a general idea of the concepts involved.
The coupled mode approach is concerned with the properties of coupled
oscillating systems like mechanical oscillators (e.g. pendulums), electric cir-
cuits, acoustic systems, molecular vibrations, and a number of other things
you might not immediately recognize as oscillating systems. The approach was
quite probably familiar to the better physicists of the nineteenth century but has
become fashionable only recently. Its essence is to divide the system up into its
components, investigate the properties of the individual components in isol-
ation, and then reach conclusions about the whole system by assuming that
the components are weakly coupled to each other. Mathematicians would call
it a perturbation solution because the system is perturbed by introducing the
coupling between the elements.
First of all we should derive the equations. These, not unexpectedly, turn out
to be coupled linear differential equations. Let us start again with Schrödinger’s
equation [eqn (3.4)], but put it in the operator form, that is
The operator in parentheses is usu- 2 ∂
2
ally called the Hamiltonian oper- – ∇ + V =i . (5.16)
ator and denoted by H. 2m ∂t
We may also write Schrödinger’s equation in the simple and elegant form
∂
H =i . (5.17)
∂t
We have attempted [eqn (3.7)] the solution of this partial differential
equation before by separating the variables,

= w(t)ψ(r). (5.18)

Let us try to do the same thing again but in the more general form,

= w j (t)ψ j (r), (5.19)
j
where a number of solutions (not necessarily finite) are superimposed.
Up to now we have given all our attention to the spatial variation of the
wave function. We have said that if an electron is in a certain state, it turns up
in various places with certain probabilities. Now we are going to change the
emphasis. We shall not enquire into the spatial variation of the probability at
all. We shall be satisfied with asking the much more limited question: what is
the probability that the electron (or more generally a set of particles) is in state
j at time t? We do not care what happens to the electron in state j as long as it
is in state j. We are interested only in the temporal variation, that is, we shall
confine our attention to the function w(t).
We shall get rid of the spatial variation in the following way. Let us
substitute eqn (5.19) into eqn (5.17)

dw j

w j Hψ j =i ψ j , (5.20)
dt
j j