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and so equation (5.77) gives

with

and

We assume that a solution exists for will give an example shortly—then
(5.78b) can be written

which is integrated over all x. After using integration by parts on the first term, and
invoking the decay conditions at infinity, we obtain

which reduces to

when we make use of (5.78a) and (5.80b). Thus the correction to the energy is known
(and we assume that and are such as to ensure that this correction is finite). The
same procedure applied to equation (5.78c) yields

A simple potential is that associated with the harmonic oscillator, namely
then equation (5.78a) becomes

which can be rewritten in terms of to give