This chapter discusses the problem of solving the Schrödinger equation for the potentials which are central: a potential V(r) is called central if it depends on the magnitude jrj of r and not on its orientation. We discuss, in particular, the problems of a free particle, potential well, hydrogen atom and of an isotropic harmonic oscillator.

In the case of central potentials, it is convenient and instructive to work in the spherical polar coordinate system discussed next.

Spherical Polar Coordinates

Let (x, y, z) be the coordinates of a point P with respect to the Cartesian system centred at O with ex, ey, ez as the unit vectors along the three axes.

The spherical polar system is described in terms of the coordinates (r, h, ɸ) where, as shown in Figure 14.1, r = jrj is the magnitude of the position vector of P, θ is the angle between OP and the z-axis, and _ is the angle between the x-axis and the projection of OP on the x − y plane. The spherical polar coordinates of P are related to its Cartesian coordinates by the relations

In spherical coordinate description, a potential is central if it is a function only of r. We list below some results useful for the discussion to follow.