whereLis the standard Lagrangian, formulated in terms ofqand ̇q, and given by

L(q,q ̇) =

1

2

mq ̇^2 −V(q) (14.41)

The extra dynamics-dependent constant factor that appearsin the measure afterphas been inte-
grated out is a real nuisance and it is often “omitted”, in thesense that one gives up on computing
it. Instead, one writes that

〈qb|U(tb−ta)|qa〉=N

∫

Dqexp

{

i

̄h

∫tb

ta

dtL(q,q ̇)

}

(14.42)

whereNis a quantity which is independent ofqa,qb. Often,Ncan be determined by considering
limits ofqaorqbin which the integral simplifies, butNremains unchanged.

14.6 Dominant paths

The space of all paths that contribute to the path integral ishuge. Often, however, the physics of
the problem produces a natural distinction between the paths that contribute most and paths that
do not. We can see this by studying the free-particle weight factor in the path integral,

eiSm/ ̄h Sm=

m(qb−qa)^2

2(tb−ta)

(14.43)

Consider a particle traveling a distance ofqb−qa∼ 1 mmin a time oftb−ta∼ 1 sec. The phase
factors for a particle of massm= 1g, for an electron of massme∼ 10 −^27 g, and a proton of mass
mp∼ 2 × 10 −^24 gare,

Sm/ ̄h ∼ 0. 5 × 1025 ≫π

Sme/ ̄h ∼ 0. 005 ≪π

Smp/ ̄h ∼ 1 ∼π (14.44)

Clearly, for the particle of mass 1g, the interference of all the paths but the classical one willcancel
one another out. For the electron, paths other than the classical path interfere strongly with the
contribution of the classical path, and the case of the proton is intermediate.

14.7 Stationary phase approximation

When the classical path has an action which is much larger thanπ, it makes sense to treat the
functional integral by approximations. It is useful to consider an example of a rapidly oscillating
integral first.

∫+∞

−∞

dk ei(πak

(^2) − 2 πbk)

e−iπb
(^2) /a
√
−ia

(14.45)