Table of Contents

The previous chapter serves as a useful introduction to many of the basic concepts of quantum mechanics. In this chapter, we shall examine these concepts in a more systematic fashion. For the sake of simplicity, we shall concentrate on one-dimensional systems.

In this chapter, we shall extend the single particle, one-dimensional formulation of non-relativistic quantum mechanics, introduced in the previous chapters, in order to investigate one-dimensional systems containing multiple particles.

In this chapter, we shall investigate the interaction of a non-relativistic particle of mass m and energy E with various so-called central potentials, V(r) , where r is the radial distance from the origin. It is, of course, most convenient to work in spherical coordinates— r , θ , ϕ —during such an investigation.. Thus, we shall be searching for stationary wavefunctions, ψ(r,θ,ϕ) , that satisfy the time-independent Schrödinger equation.

Broadly speaking, a classical extended object (e.g., the Earth) can possess two different types of angular momentum. The first type is due to the rotation of the object’s center of mass about some fixed external point (e.g., the Sun)—this is generally known as orbital angular momentum. The second type is due to the object’s internal motion—this is generally known as spin angular momentum (because, for a rigid object, the internal motion consists of spinning about an axis passing through the cent

Historically, data regarding quantum phenomena has been obtained from two main sources. Firstly, from the study of spectroscopic lines, and, secondly, from scattering experiments. We have already developed theories that account for some aspects of the spectra of hydrogen, and hydrogen-like, atoms. Let us now examine the quantum theory of scattering.