QuantumPhysics.dvi

Putting all together, and usingδ(pb−pa) to carry out the integration overpb, we have

〈qb|U(tb−ta)|qa〉 =

∫

dpa

2 π ̄h

e−i(tb−ta)p

(^2) a/(2m ̄h)
eipa(qb−qa)/ ̄h

=

( m

2 πi(tb−ta) ̄h

)^12

exp

{

im(qb−qa)^2

2 ̄h(tb−ta)

}

(14.17)

The full significance of the exponential factor will become clear later. Here, we notice simply that
the quantity

q ̇=

qb−qa

tb−ta

(14.18)

is the velocity needed for a free particle in uniform motion departing from positionqaat timetato
reach position theqbat timetb. Its energyEis just its kinetic energy, and the associated actionS
are given by

E(qb,tb;qa,ta) =

m(qb−qa)^2

2(tb−ta)^2

S(qb,tb;qa,ta) =

m(qb−qa)^2

2(tb−ta)

(14.19)

For the free particle, the matrix elements of the evolution operator are given by

〈qb|U(tb−ta)|qa〉∼exp{iS(qb,tb;qa,ta)/ ̄h} (14.20)

Notice that for fixedqa,qbandtb−ta→0,Sis becomes very large, unlessqa∼qb. Astb−ta→0,
the evolution operator becomes local inqand tends towardsδ(qb−qa).

14.4 Derivation of the path integral

We now seek a formula for〈qb|U(tb−ta)|qa〉which will be valid for a general time-independent
HamiltonianH(Q,P). It is convenient to begin by dividing the time intervaltb−ta>0 intoN
consecutive segments of equal lengthε= (tb−ta)/N. Using the multiplicative property of the
exponential, we have

U(tb−ta) =U(ε)N (14.21)

Inserting N−1 times the identity operator, expressed as a completeness relation on position
eigenstates, we obtain,

〈qb|U(tb−ta)|qa〉=

(N− 1

∏

n=1

∫

R

dqn

) N

∏

n=1

〈qn|U(ε)|qn− 1 〉 (14.22)

where we have setq 0 ≡qaandqN≡qb. The remainingqnforn= 1,···,N−1 are integrated over
in this formula, as may be seen schematically on the Figure below.
WhenN→∞andε→0, we may expand the evolution operator in a power series inε,

U(ε) =I−i

ε

̄h

H+O(ε^2 ) (14.23)