QuantumPhysics.dvi

=

∫

dpn

2 π ̄h

〈qn|pn〉〈pn|qn− 1 〉exp

{

−i

ε

̄h

H(qn,pn)

}

=

∫ dp

n

2 π ̄h

exp

{

i(qn−qn− 1 )pn/ ̄h−i

ε

̄h

H(qn,pn)

}

(14.27)

The final step consists of a change of notation,

tn = t+nε

qn = q(tn)

pn = p(tn) (14.28)

The argument of the exponential may now be recast as follows,

i

̄h

(qn−qn− 1 )pn−i

ε

̄h

H(qn,pn) =

i

̄h

ε

(

q(tn)−q(tn−ε)

ε

p(tn)−H(qn,pn)

)

=

i

̄h

∫tn

tn− 1

dt

(

q ̇(t)p(t)−H(q(t),p(t))

)

(14.29)

Putting all together,

〈qb|U(tb−ta)|qa〉=

∫

Dq

∫

Dpexp

{

i

̄h

∫tb

ta

dt

(

q ̇(t)p(t)−H(q(t),p(t))

)}

(14.30)

where the measures are defined by

∫

Dq = lim

N→∞

N∏− 1

n=1

∫

R

dq(tn)

∫

Dp = lim

N→∞

∏N

n=1

∫

R

dp(tn)

2 π ̄h

(14.31)

and the paths satisfy the following “boundary conditions”

q(ta) = qa

q(tb) = qb (14.32)

Notice that there are no boundary conditions onp(t). The combination ̇qp−H(q,p) is nothing but
the Lagrangian written in canonical coordinatesqandp, and the integral overτthat enters the
path integral formula above is just the action in terms ofqandp.

An alternative formula is obtained by expressing〈qb|as the Fourier transform of a momentum
state,

〈pb|U(tb−ta)|qa〉=

∫

Dq

∫

Dpexp

{

i

̄h

∫tb

ta

dt

(

q ̇(t)p(t)−H(q(t),p(t))

)}

(14.33)