QuantumPhysics.dvi

where the combine measure is defined by

∫

Dq

∫

Dp= lim

N→∞

N∏− 1

n=1

∫

R

∫

R

dq(tn)dp(tn)

2 π ̄h

(14.34)

and the paths satisfy the following “boundary conditions”

q(ta) = qa

p(tb) = pb (14.35)

while the boundary conditions onp(ta) andq(tb) are now free. This form is particularly geometrical
because the measure is intrinsic and invariant under canonical transformations. The volume element

dq(tn)dp(tn)

2 π ̄h

(14.36)

gives a measure for the elementary volume of phase space in a quantum system. By the uncertainty
relation, there is in some sense one state per elementary phase space volume element ∆q∆p. The
above measure naturally puts a measure on the number of quantum states for the system.

14.5 Integrating out the canonical momentump

Whenever the Hamiltonian is simple enough so that its dependence on momentum and position
enter separately, momentum may be “integrated out”. Let us take the most customary form,

H=

p^2

2 m

+V(q) (14.37)

The combination under the integration is

qp ̇ −H(q,p) = ̇qp−

p^2

2 m

−V(q)

= −

1

2 m

(p−mq ̇)^2 +

1

2

mq ̇^2 −V(q) (14.38)

We now see that the integration overDpdoes not depend onq any more, after an innocuous
translation bymq ̇. The Gaussian integral involves an infinite number of integrations, so this gives
a constant, which we shall absorb into the definition of the integration measureDq,

∫

Dq= lim

N→∞

N∏− 1

n=1

(∫

R

dq(tn)

√

m

2 πi ̄hε

)

(14.39)

Our final result is thus,

〈qb|U(tb−ta)|qa〉=

∫

Dqexp

{

i

̄h

∫tb

ta

dtL(q,q ̇)

}

(14.40)