Path integral derivation 453〈xk+1|e−β
K/Pˆ
|xk〉=∫
dp〈xk+1|p〉〈p|xk〉e−βp(^2) / 2 mP
. (12.2.16)
Finally, using eqn. (9.2.43), eqn. (12.2.16) becomes
〈xk+1|e−β
K/Pˆ
|xk〉=1
2 π ̄h∫
dpe−βp(^2) / 2 mP
eip(xk+1−xk)/ ̄h. (12.2.17)
Since the range of the momentum integration isp∈(−∞,∞), the above integral is a
typical Gaussian integral that can be evaluated by completing the square. Thus, we
write
βp^2
2 mP
−
ip(xk+1−xk)
̄h=
β
2 mP[
p^2 −2 imPp(xk+1−xk)
β ̄h]
=
β
2 mP{[
p−
imP(xk+1−xk)
β ̄h] 2
+
m^2 P^2 (xk+1−xk)^2
β^2 ̄h^2}
=
β
2 mP[
p−
imP(xk+1−xk)
β ̄h] 2
+
mP
2 β ̄h^2(xk+1−xk)^2. (12.2.18)When the two last lines of eqn. (12.2.18) are substituted back into eqn. (12.2.17), and
a change of variables
p ̃=p−imP(xk+1−xk)
β ̄h(12.2.19)
is made, we find
〈xk+1|e−β
K/Pˆ
|xk〉=1
2 π ̄hexp[
−
mP
2 β ̄h^2(xk+1−xk)^2]∫∞
−∞d ̃pe−βp ̃(^2) / 2 mP
=
(
mP
2 πβ ̄h^2) 1 / 2
exp[
−
mP
2 β ̄h^2(xk+1−xk)^2]
. (12.2.20)
Now, eqn. (12.2.20) is combined with eqn. (12.2.13) to yield
〈xk+1|Ωˆ|xk〉=(
mP
2 πβ ̄h^2) 1 / 2
exp[
−
β
2 P(U(xk+1) +U(xk))]
×exp[
−
mP
2 β ̄h^2(xk+1−xk)^2]
. (12.2.21)
Finally, multiplying allP matrix elements together and integrating over theP− 1
coordinate variables, we obtain for the density matrix: