Unique Stationary Point

Suppose the functions f(q) and S{q) are infinitely differentiable on the interval [a, b] and for the function S(q) there exist only a unique stationary (critical or saddle) point q belonging to [a, b] and

then from 5.2 we have the relation: or

Here,

We show this and for definiteness we let So,

for q belonging to the interval [q,fr]. Hence,

for q belonging to the interval Q?,b]. Suppose -^[A] is examined on the intervals [я,T_x [A] and [A ]. For the integral Т_г [A] , we set

such that the fluctuation

and

Here,

So, from above,

Similarly, we do for .^[Я] and taking the sum of/j[A] and 5^е) [A], relation 5.50 is confirmed. So, the stationary phase approximation (or saddle point approximation) to the integral implies neglecting higher order terms encoded by о(Я^-1). We have shown that this approximation is true when the contribution due to о(Я'¹) for Я -> oo is indeed negligible. Иге exponential factor having the function S(f) is the dominant contribution to the function -^[a] from the classical path. The factor /"(Я) describes the contribution from the quantum corrections of the order о(Я"¹). Relation 5.50 in the stationary phase method, imitates the standard quantum mechanics and for large Я -*• oo , the quantity exp|;AS(q)| oscillates rapidly when the function S(q) changes. In this case, ДЯ) smoothly depends on ft. So, regions without critical points q contribute o^'^terms to the integral .F[A]. Relation 5.50 shows irrespective^of the neglected terms, there are some contributions due to small neighborhoods of the critical points 4 to the total integral of order о(Я^-1). So, in the limit Я -> oo , it is sufficient to evaluate at least the first quantum correction and hence, via the stationary phase approximation with reasonable accuracy the integral .F[A] can be evaluated.

We could also consider the case

where we have to evaluate the integral

Here,

and

The reader is expected to study this case.

We thus observe that for large Я, the region determining the integral is very small and so we may expand the function S(q) locally around the extremum q and replace/(q) by f(q). If S(q) has more than one extremum then we sum over the contributions of all extrema except one extremum is observed to be dominant.