Exercises 419
in the path-integral formalism, we should include all paths satisfying these
boundary conditions. Using completeness, we can write, with 0<τ′<τ:〈r 1 ,0|r 2 ,τ〉=∫
d^3 r′〈r 1 ,0|r′,τ′〉〈r′,τ′|r 2 ,τ〉.Show that the integrand in this equation can be written as〈r 1 ,0|r′,τ′〉〈r′,τ′|r 2 ,τ〉=〈r 10 |r 2 ,τ〉
1
( 2 πστ′)^3 /^2
e−[r′−r(τ′)] (^2) /( 2 στ′)
,
with
στ′=
τ′(τ−τ′)
τ
and r(τ′)=r 1 +
τ′
τ
(r 2 −r 1 ).
12.3 In this problem we consider the cumulant expansion analysis for the Coulomb
potential [4, 22] using the result of the previous problem.
The cumulant expansion is a well-known expansion in statistical physics [31]. It
replaces the Gaussian average of an exponent by the exponent of a sum of averages:
〈eτV〉=eτ〈V〉+
(^12) (τ (^2) 〈V (^2) 〉−〈V〉 (^2) )+···
.
First we note that the matrix between two positionsr 1 andr 2 separated by an
imaginary timeτcan be written in the following way:
〈r 1 ,0|exp
(
−
∫τ
0
V(rτ′)dτ′
)
|r 2 ,τ〉.
where the time evolution leading from 0 toτis that of a free particle and the
expression is to be evaluated in a time-ordered fashion.
If we evaluate this in the cumulant expansion approximation retaining only the
first term, it is clear that we must calculate
∫τ
0
dτ′
∫
d^3 r′〈r 1 ,0|r′,τ′〉V(r′)〈r′,τ′|r 2 ,τ〉.
This is done in this problem.
(a) Show that the Fourier transform of the Coulomb potential isV(k)= 2 π/k^2.
(b) Show that the Fourier transform of the expression derived in Problem 12.2 is
given by
e−ik·r(τ
′)−στ′k (^2) / 2
,
withστ′andr(τ )as given in the previous problem.
(c) Show, by transforming back to ther-representation, that the cumulant potential
is given by
Vcumulant(r 1 ,r 2 ;τ)=
∫τ
0
erf[r(τ′)/
√
2 στ′]
r(τ′)
dτ′.
12.4 In the path-integral simulation for the hydrogen atom we use a table in which the
cumulant expression for the potential is stored and we want to linearly interpolate
this table.