Computational Physics

558 Appendix A

the field of numerical mathematics [2–6]and the interested reader is advised to
consult these.
This appendix serves as a refresher to those readers who have some knowledge
of the subject. Novice readers may catch an idea of the methods and can look up
the details in a specialised book. The choice of problems discussed in this appendix
is somewhat biased: although several methods described here are not used in the
rest of the book, the emphasis is on those that are.

A2 Iterative procedures for special functions

Physics abounds with special functions: functions which satisfy classes of differ-
ential equations or given by some other prescription, and which are usually more
complicated than simple sines, cosines or exponentials. Often we have an iterative
prescription for determining these functions. Such is the case for the solutions to
the radial Schrödinger equation for a free particle:
[
−

1

2

d^2

dr^2

+

l(l+ 1 )

2 r^2

]

[rRl(r)]=E[rRl(r)], (A.1)

where the units are chosen such that
^2 /m≡1 (it is always useful to choose such
natural units to avoid cumbersome exponents). The solutionsRlof (A.1) are known
as thespherical Bessel functions jl(kr)andnl(kr),k=

√

2 E. Thejlare regular for
r=0 and thenlare irregular (singular). Forl=0, 1, the spherical Bessel functions
are given by:

j 0 (x)=

sin(x)

x

; n 0 (x)=−

cos(x)

x

;

j 1 (x)=

sin(x)

x^2

−

cos(x)

x

; n 1 (x)=−

cos(x)

x^2

−

sin(x)

x

.

(A.2)

For higherl, we can find the functions by:

sl+ 1 (x)+sl− 1 (x)=

2 l+ 1

x

sl(x) (A.3)

whereslis eitherjlornl. Equation(A.3)gives us a procedure for determining these
functions numerically. Knowing for examplej 0 andj 1 ,Eq. (A.3)determinesj 2 and
so on. A three-point recursion relation has two independent solutions, and one of
these may grow strongly withl. If the solution we are after damps out and the other
one grows withl, the solution is unstable with respect to errors that will always
sneak into the solution owing to the fact that numbers are represented with finite
precision in the computer. It turns out thatjldamps out rapidly with increasingl,
so that it is sensitive to errors of this kind, especially whenlis significantly larger
thanx. One can avoid such inaccuracies by performing the recursion downwards