418 Quantum Monte Carlo methods

where the quantum statistical partition functionZis given by

Z=Tre−βH.

We take
≡1.

(a) Show by using the Lie–Trotter–Suzuki formula that

E=

N

2 β

+

∫

dx (^0) ∫dx 1 ...dxM− 1 [−T+U]exp(−βScl)
dx 0 dx 1 ...dxM− 1 exp(−βScl)
with
T=
M
2 β^2
M∑− 1
m= 0
(xm−xm+ 1 )^2 ;
x 0 ≡xM;
U=
1
M
M∑− 1
m= 0
V(xm)
and
Scl=T+V.
(b) Show that
∫
dx 0 dx 1 ...dxN− 1
∑N− 1
∫ i=^0 xi(∂Scl/∂xi)[exp(−βScl)]
dx 0 dx 1 ...dxN− 1 exp(−βScl)

N
β
.
Hint: use partial integration.
(c) Show that
M∑− 1
m= 0
xm
∂T
∂xm
= 2 T.
(d) Show that the energy can also be determined by
E=
〈
1
N
N∑− 1
m= 0
[
V(xm)+
1
2
xm
∂V
∂xm
]〉
.
(e) Show that the generalisation to a three-dimensional particle is
E=
〈
1
N
M∑− 1
m= 0
[
V(rm)+
1
2
rm·
∂V
∂rm
]〉
.
12.2 A particle moves in three dimensions. It experiences no potential:V(r)=0. At
imaginary timeτ=0 the particle is localised atr 1.
(a) What is the wave functionψ 0 (r,τ)of the particle forτ′>0?
(b) We assume that the particle moves fromr 1 at time 0 tor 2 at timeτ. When we
want to evaluate the matrix element
〈r 1 ,0|r 2 ,τ〉,