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232 5. Some worked examples arising from physical problems

we therefore obtain the periodicity condition

Finally, the evaluation of the integral (which is left as an exercise) yields

and so we have, after solving for

where We observe that as and that

with This describes the evolution (shift) of the frequency, as the
damping progressively affects the solution.

E5.14 Quantum jumps: the ion trap
In the study of the discontinuous emission or absorption of energy (quantum jumps),
a single ion is trapped (in an electromagnetic device called a Paul trap; see Cook, 1990)
and its motion is governed by an equation of the form

Here, is a parameter, v(x) is a given function (sufficient for the existence of
and we seek the complex-valued function for In this case, we see
that the oscillatory term on the right oscillates rapidly and so we use the method of
multiple scales in the form

which gives the equation

We seek a solution