The purpose of this chapter is to give you the mathematical language in which
quantum mechanics can be concisely stated. Just like it is difficult to understand the
locals in a foreign country if you can’t speak their language, this is an essential piece of
background if you want to understand quantum mechanics at more than a superficial
level. The best way to absorb this material is to practice using it by working the
problems and discussing it with your friends (thought not necessarily at dinner
parties…until you’ve got through a few more chapters).
Seriously, you should find that while the central topics of this chapter will at first
glance seem very unfamiliar (both in names and notation), and therefore possibly very
difficult, things will look much better at second (or third) glance. In particular, try
looking for two things to help in navigating through the material. First, watch for
analogies with mathematics that are already familiar to you, such as vectors, and linear
algebra, and matrices and matrix algebra. Second, pay attention to the links between the
qualitative concepts of quantum mechanics introduced in the previous chapter and the
mathematics introduced here: after all this is why
it is introduced!
We start in Section 2.2 with a discussion of the fact that valid wave functions
belong to a form of vector space, called state space (since the wave function describes the
state of a quantum mechanical system). This is essentially no different from the fact that
valid vectors (
) belong to Cartesian space. The basic properties of state vectors are
summarized and Dirac’s powerful “bra-ket” notation is introduced: it is essentially a
generalization of the notion for vectors.
The next section (2.3) deals with the very important idea of a basis expansion. A
basis expansion is to state space as the
axes are to real space. We discuss operators
in Section 2.4, followed by a discussion of Hermitian conjugation, which is a greatly
generalized form of complex conjugation. We then introduce the eigenvalues and
eigenvectors of operators in Section 2.5, with particular emphasis on Hermitian operators.
They have real eigenvalues and represent observables in quantum mechanics. The
eigenfunctions of Hermitian operators form a basis for state space.
2.2 Abstract vector spaces of bras and kets
There are 3 properties that together define a vector space: (1) closure is satisfied,
(2) a scalar product exists, and, (3) vectors can be represented in terms of a basis. As the
wavefunctions of quantum mechanics belong to a vector space (sometimes called Hilbert
space), there will be connections between the basic properties of a vector space, and some
of the important characteristics of wavefunctions that have physical implications.
Therefore we’ll examine these 3 basic properties in this and the following section.