Computational Physics

582 Appendix A

matrix equation. We need to solve two tridiagonal matrix equations for|φ〉and|χ〉
to obtain the solution to the problem with periodic boundaries.
Finally, we describe a fourth method which can be applied to initial value
problems: thesplit operatormethod. The idea of this method is based on prin-
ciples which have been used extensively in Chapter 12. It consists of splitting the
Hamiltonian into the kinetic and potential operator, and using representations in
which these operators are diagonal. To be specific, the operatorV(x)is diagonal
in thex-representation, whereas the kinetic energy term,T=p^2 /( 2 m)is diagonal
in thep-representation. The two representations are connected through a Fourier
transform:

〈x|p〉=

1

√

2 π

eipx/
. (A.84)

The time-evolution operator is split into three terms:

e−it(V+T)/
=e−itV/(^2 
)e−itT/
e−itV/(^2 
)+O(t^3 ) (A.85)

Suppose we know the initial state in thex-representation. Then we act on it with the
front term (exponent of the potential) which is diagonal. Then we Fourier-transform
the result and multiply it by the second term (exponent of the kinetic energy) and
then, after Fourier-transforming back again, we multiply the result by the last factor
(potential). This method has a similar efficiency to the Crank–Nicholson method.

Boundary value problems

We consider the Poisson equation for the potential of a charge distribution as a
typical example of this category:

∇^2 ψ(r)=−ρ(r). (A.86)

∇^2 is the Laplace operator andψis the potential resulting from the charge distri-
butionρ. Furthermore, there are boundary conditions, which are generally of the
Von Neumann or of the Dirichlet type (according to whether the derivative or the
value of the potential is given at the boundary respectively). In this section we shall
discuss iterative methods for solving this type of equation. For more details the
readerisreferredtostandardbooksonthesubject[ 14 , 15 ].
We restrict ourselves to two dimensions, and we can discretise the Laplace
equation on a square lattice, analogous to the way in which this was done for
the Schrödinger equation in the previous subsection:

∇^2 ψ(r)=

(

∂^2

∂x^2

+

∂^2

∂y^2

)

ψ(r); (A.87a)

∇^2 Dψi,j=

1

x^2

(ψi+1,j+ψi−1,j+ψi,j+ 1 +ψi,j− 1 − 4 ψij)

=∇^2 ψ(r)+O(x^2 ). (A.87b)