Many of the important differential equations of physics can be cast in the form of a linear, second-order equation:

0 + * ’ (* )„ = 5( * ) , (3.1)

where 5 is an inhomogeneous ( “driving” ) term and k2 is a real function. When k2 is positive, the solutions of the homogeneous equation (i.e., 5 = 0) are oscillatory with local wavenumber fc, while when k2 is negative, the solutions grow or decay exponentially at a local rate (—&2) 2. For example, consider trying to find the electrostatic potential, $ , generated by a localized charge distribution, p(r). Poisson’s equation is

V 2$ = -4?r/>, (3.2)

which, for a spherically symmetric p and $, simplifies to 1 d f 2 d$\ A

The standard substitution

$ (r) = 7*” 1<^ (r)

then results in d2 6 •frS = - 4*V>, (3.4)

which is of the form (3.1) with k2 = 0 and 5 = —Airrp. In a similar manner, the quantum mechanical wave function for a particle of mass m

and energy E moving in a central potential V(r) can be written as

$(r) = r - xR{r)YLM{r) , where Y l m is a spherical harmonic and the radial wave function R satisfies

+ k\r)R = 0 ; k2 (r) = 2E _ L(L + l)h _

This is also of the form (3.1), with S = 0. The equations discussed above appear unremarkable and readily

treated by the methods discussed in Chapter 2, except for two points. First, the boundary conditions imposed by the physics often appear as constraints on the dependent variable at two separate points of the inde­ pendent variable, so that solution as an initial value problem is not obvi­ ously possible. Moreover, the Schroedinger equation (3.5) is an eigenvalue equation in which we must find the energies that lead to physically ac­ ceptable solutions satisfying the appropriate boundary conditions. This chapter is concerned with methods for treating such problems. We begin by deriving an integration algorithm particularly well suited to equations of the form (3.1), and then discuss boundary value and eigenvalue prob­ lems in turn.