The Radial Equation for Central Force Fields

Abstract

We saw in the previous chapter that for spherically symmetric force fields solutions to Schrödinger’s equation are

$$u\left( r \right) = R\left( r \right)Y\left( {\theta ,\phi } \right)$$

where Y(λ, ø) satisfies Eq. (5.9). The general solution can be written as an infinite sum of such products.¹ The radial dependence of u(r) is contained in R(r) which is described by the differential equation²

$$\frac{d}{{dr}}\left( {{r^2}\frac{d}{{dr}}R\left( r \right)} \right) + \left\{ {\frac{{2m{r^2}}}{{{h^2}}}\left[ {E - V\left( R \right)} \right] - l(l + 1)} \right\}R(r) = 0$$

(6.1)

The function V(r) represents the potential energy of the particle in the central field.