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228 5. Some worked examples arising from physical problems

This is Hermite’s equation with solutions that guarantee at infinity only if
In particular,

and each of these solutions has been chosen to satisfy the normalisation condition on
(5.80a). A typical choice for is

but in order to ensure that is uniformly valid for all x, then (to avoid
a breakdown as and (to avoid a breakdown as for p > 2;
the case p = 2 and is equivalent to Let us calculate for m = 0
(so we have and and for a perturbation with
p = 2 and thus

and so the energy becomes

Observe that, at this order, we do not need to determine to find

Calculations of this type, for various and can be found in any good
text on quantum mechanics.

E5.12 Light propagating through a slowly varying medium
Fermat’s principle states that light travels between any two points on a path which
minimises the time of propagation. If the path, in two dimensions, is written as y =
y(x), and the speed of light at any point is c(x, y), then y(x) must satisfy

This equation can be obtained either from the eikonal equation (see Q4.27) for rays or
as the relevant Euler-Lagrange equation in the calculus of variations. [In the special case
where the medium varies only in x, so that c = c(x), we obtain