Computational Physics - Department of Physics

538 17 Bose-Einstein condensation and Diffusion Monte Carlo

any relevant physical properties of our system since it is generally independent of the choice

of the zero point of the energy. The effect on the projection operation becomes

Ψ(x,t) =e−(Ĥ−ET)tΨ(x) =∑

i

cie−(εi−ET)tφi(x) (17.2)

Consider the ideal situation of lettingETequal exactlyε 0 , resulting in

Ψ(x,t) =e−(Ĥ−ET)tΨ(x) =c 0 φ 0 +∑

i> 0

ciφi(x)e−(εi−ε^0 )t

Sinceεi>ε 0 fori 6 = 0 , all the remaining exponents become negative. In the limitt→∞, the
contributions from excited states must obviously vanish, so that propagatingΨaccording to
Eq. (17.1) gives
tlim→∞e−(Ĥ−ε^0 )tΨ(x) =c^0 φ^0
thus projecting out the ground state.
Without even considering how to do the time propagation in practice, the formulas already
indicate to us that approaching the problem first with VMC canhelp produce an initialΨ
close toφ 0 and, more importantly, a trial energyETbeing an upper bound to the true ground
state energyε 0. HopefullyETis close enough toε 0 to be smaller than the first excited energy.
Inserting such aETinto Eq. (17.2) while lettingt→∞will make all the excited states col-
lapse while the ground state blows up and dominates because of the positive exponent in the
coefficient.
From this consideration we also see that the evolution operator is not unitary. The norm of
Ψis not necessarily conserved with time. Depending on the value ofETit may grow without
limit or go to zero. Only in the case ofET=ε 0 the state tends to a constant. This proposes
a method of determining the ground state energy by adjustingETdynamically during time
propagation so that the state stays constant.
Writing out the imaginary time Schrödinger equation, Eq. (17.1), including the energy
offsetET, we get
∂
∂t
Ψ(x,t) =−K̂Ψ(x,t)−(V̂(x)−ET)Ψ(x,t)

= ̄h

2

2 m

∇^2 Ψ(x,t)−(V̂(x)−ET)Ψ(x,t) (17.3)

whereK̂andV̂is the kinetic and potential energy operator, respectively. We see that this is
of the form of an extended diffusion equation. We can therefore consider the wave function
Ψas a probability distribution evolving according to this equation. This greatly contrasts the
usual quantum mechanical interpretation of|Ψ|^2 being the actual probability density function
(PDF). We will see that the approach poses some potentially serious problems when the wave
functionΨwe seek has nodes and is partially negative and possibly complex. But to illustrate
the main mechanisms of the method we will at the moment focus on the simple cases of
bosonic ground states which usually are strictly positive and real.
In simple terms, the job consists of representing the initial stateΨby a collection of walk-
ers, much the same way as in VMC, and letting them evolve in time as a controlled diffusion
process governed by Eq. (17.3). Interpreting the rhs. termsof Eq. (17.3), we see that the first
term is a standard diffusion term with a diffusion constant of

D≡

̄h^2

2 m

(17.4)