Computational Physics - Department of Physics

546 17 Bose-Einstein condensation and Diffusion Monte Carlo

for the above trial wave function of Eq. (17.22). Compute also the analytic expression for the

drift force to be used in importance sampling

F=^2 ∇ΨT

ΨT

. (17.26)

The tricky part is to find an analytic expressions for the derivative of the trial wave function

1

ΨT(R)

N

∑

i

∇^2 iΨT(R),

for the above trial wave function of Eq. (17.22). We rewrite

ΨT(R) =ΨT(r 1 ,r 2 ,...rN,α,β) =∏

i

g(α,β,ri)∏

i<j

f(a,|ri−rj|),

as

ΨT(R) =∏

i

g(α,β,ri)e∑i<ju(ri j)

where we have definedri j=|ri−rj|and

f(ri j) =e∑i<ju(ri j),

and in our case

g(α,β,ri) =e−α(x

(^2) i+y (^2) i+z (^2) i)
=φ(ri).
The first derivative becomes
∇kΨT(R) =∇kφ(rk)

[

∏

i 6 =k

φ(ri)

]

e∑i<ju(ri j)+∏

i

φ(ri)e∑i<ju(ri j)∑

j 6 =k

∇ku(ri j)

We leave it as an exercise for the reader to find the expressionfor the sceond derivative. The

final expression is

1

ΨT(R)∇

2

kΨT(R) =

∇^2 kφ(rk)

φ(rk) +

∇kφ(rk)

φ(rk)

(

∑

j 6 =k

rk

rku

′(ri j)

)

+

∑

i j 6 =k

(rk−ri)(rk−rj)

rkirk j

u′(rki)u′(rk j)+∑

j 6 =k

(

u′′(rk j)+

2

rk j

u′(rk j)

)

You need to get the analytic expression for this expression using the harmonic oscillator wave
functions and the correlation term defined in the project.
b) Write a Variational Monte Carlo program which uses standard Metropolis sampling and com-
pute the ground state energy of a spherical harmonic oscillator (β= 1 ) with no interaction.
Use natural units and make an analysis of your calculations using both the analytic expres-
sion for the local energy and a numerical calculation of the kinetic energy using numerical
derivation. Compare the CPU time difference. You should also parallelize your code. The only
variational parameter isα. Perform these calculations forN= 10 , 100 and 500 atoms. Compare
your results with the exact answer.
c) We turn now to the elliptic trap with a hard core interaction. We fix, as in Refs. [143,147]
a/aho= 0. 0043. Introduce lengths in units ofaho,r→r/ahoand energy in units ofh ̄ωho. Show
then that the original Hamiltonian can be rewritten as

H=

N

∑

i= 1

1

2

(

−∇^2 i+x^2 i+y^2 i+γ^2 z^2 i

)

+∑

i<j

Vint(|ri−rj|). (17.27)