Quantum Mechanics

Inwe have introduced the basic concepts of quantum mechanics, and studied them for some simple, yet relevant, one-dimensional systems. In this chapter we take another step towards the description of real physical phenomena and generalise the concepts introduced so far to systems that evolve in more than one spatial dimension. The generalisation is straightforward and it will give us the opportunity to review some of the key ideas about physical states, observables and time evolution. In the process, we will encounter and highlight new features that were not present for one-dimensional systems. Once again let us emphasise that three-dimensional in this context refers to the dimension of the physical space in which the system is defined, and not to the dimensionality of the Hilbert space of states; the latter clearly will depend on the type of system that we consider. The three-dimensional formulation will allow us to discuss more realistic examples of physical systems. It will be clear as we progress through this chapter that everything we discuss can be generalised to an arbitrary number of dimensions. In some physical applications, where a quantum system is confined to a plane, a two-dimensional formulation will be useful. More generally, it is instructive to think about problems in arbitrary numbers of dimensions. In this respect, it is fundamental to be able to work with vectors, tensors, indices, and all that. Problems and examples in this chapter should help develop some confidence in using an index notation to deal with linear algebra.