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226 5. Some worked examples arising from physical problems

We see, therefore, that the small drag in this model leaves the pericentre unaffected
the eccentricity decreases towards zero (from its already assumed small
value) and the semi-major axis also approaches zero as
Thus the orbit gradually spirals in and, as it does so, it becomes more circular.

We have seen how we might tackle the problem of orbits that graze the atmosphere
of a planet, although here it was expedient to assume that the orbit was initially nearly
circular. If this is not the case, then will not be small and we face a more
exacting calculation, although the essential principles are unaltered.

5.3 PHYSICS OF PARTICLES AND OF LIGHT
In this section, we will examine some problems that arise from fairly elementary
physics; these will touch on quantum mechanics, light propagation and the move-
ment of particles. In particular, we discuss: E5.11 Perturbation of the bound states of
Schrödinger’s equation; E5.12 Light propagating through a slowly varying medium;
E5.13 Raman scattering: a damped Morse oscillator; E5.14 Quantum jumps: the ion
trap; E5.15 Low-pressure gas flow through a long tube.

E5.11 Perturbation of the bound states of Schrödinger’s equation
This is a classical problem in elementary quantum mechanics; it involves the time-
independent, one-dimensional Schrödinger equation

where is the given potential and E is the energy (i.e. the eigenvalues of the
differential equation). We seek solutions for which as and
is finite (and conventionally, we choose to provide a normalisation of
the eigenfunction, In this example, we choose

and this is to be a uniformly valid approximation as for The
problem then becomes

with as and
We seek a solution by assuming a straightforward expansion, and we will comment
on the conditions that ensure a uniform expansion valid for all x; see §2.3. Thus we
write