Of course, we knw how to do this integral exactly; it is just a Gaussian. But if we were just
interested in thea≫1 behavior of the integral, we can use stationary phase approximation. The
rule is to approximate the integral by the value it takes whenthe phase is stationary,
d
dk(πak^2 − 2 πbk) = 0 (14.46)which is alsoak−b= 0. Hence, in this approximation, we get
∫+∞−∞dk ei(πak(^2) − 2 πbk)
∼e−iπb
(^2) /a
(14.47)
Comparing with the exact result, we see that the exponentialis correct, though the prefactor is
missing. The prefactor is subdominant to the exponential though whena≪1, so this approxi-
mation is not too bad. The method is especially useful for integrals that we do not know how to
evaluate exactly, eg,
∫+∞
−∞
dk ei(πak
(^4) − 2 πbk)
∼e−^3 /^2 iπb(b/^2 a)
1 / 3
(14.48)
Here, we have assume that only the real stationary phase solution of 2ak^3 −b= 0 contributes.
In general, the issue of whether complex solutions contribute or not is a quite subtle one (see eg
Mathews and Walker).
We shall now be interested in applying the stationary phase method to the path integral
〈qb|U(tb−ta)|qa〉=
∫
Dqexp
{
i
̄h
∫tb
ta
dtL(q,q ̇)
}
(14.49)
We begin by assuming that there exists a classical trajectoryq 0 (t) such that
q 0 (ta) = qa
q 0 (tb) = qb (14.50)
and such thatq 0 (t) solves the Euler-Lagrange equations, and is thus a stationary path of the
action. For simplicity, we shall assume that the solutionq 0 (t) is unique. The stationary phase
approximation to the path integral is then,
〈qb|U(tb−ta)|qa〉∼exp
{
i
̄h
∫tb
ta
dtL(q 0 (t),q ̇ 0 (t))
}
(14.51)
In the next subsection, we shall improve upon this result.
14.8 Gaussian fluctuations
It is possible to systematically improve upon the stationary phase approximation. Here, we shall
discuss only the leading such correction, obtained by taking into account the Gaussian fluctuations