Computational Physics

Exercises 291

and the normalisation constraint

〈ψ|ψ〉=

∑

rs

CrCs〈χr|χs〉=

∑

rs

CrCsSrs

are given by

μC ̈r= 2

∑

s

(HrsCs− SrsCs).

We solve this equation of motion using the Verlet algorithm with friction. Note that
is determined by the normalisation condition. Therefore we first perform an
integration step with =0 and then calculate from(9.32). You are free to choose
μ, the frictional constant and the time steph, although they are not independent.
The energy should converge to 2.467 40 as found in Section 3.2.1.
9.2 [C] Extend the program of the previous problem to include more than one state. Each
stateψk(x)has its own set of coefficientsCrk:
ψk(x)=

∑

r

Crkχr(x).

We considerKstates. There are nowK(K+ 1 )/2 constraint equations:

〈ψk|ψl〉=

∑

rs

CrkSrsCsl=δkl

(interchangingkandlgives the same equation). The Euler–Lagrange equations now

become

μC ̈kr=

∑

l

∑

s

(Hrs− (^) kl)Csl.
Because we include friction in the problem, we can take (^) kldiagonal:
(^) kl=εkδkl,
and at each step we estimateεkas
εk=
∑
rs
CrkCskHrs
seeSection 9.4.1.
Implement these equations in a computer program and compare your results with
those presented inSection 3.2.1.
9.3 [C] Consider again the deep potential well of the previous two problems. We now use
the conjugate gradients method for finding the eigenvalues, by using it to minimise the
energy functional. It is assumed that you have a conjugate gradient routine available.
Let us first consider the ground state. This can be written as
ψG(x)=
∑
r
Crχr(x),
whereχris the basis consisting of polynomials vanishing on the boundaries of the
well. The energy functional
E[Cr]=
∑
rs
CrCsHrs