19

Quantum operators

Although the previous chapter was principally concerned with the use of linear

operators and their eigenfunctions in connection with the solution of given

differential equations, it is of interest to study the properties of the operators

themselves and determine which of them follow purely from the nature of the

operators, without reference to specific forms of eigenfunctions.

19.1 Operator formalism

The results we will obtain in this chapter have most of their applications in the

field of quantum mechanics and our descriptions of the methods will reflect this.

In particular, when we discuss a functionψthat depends upon variables such as

space coordinates and time, and possibly also on some non-classical variables,ψ

will usually be a quantum-mechanical wavefunction that is being used to describe

the state of a physical system. For example, the value of|ψ|^2 for a particular

set of values of the variables is interpreted in quantum mechanics as being the

probability that the system’s variables have that set of values.

To this end, we will be no more specific about the functions involved than

attaching just enough labels to them that a particular function, or a particular

set of functions, is identified. A convenient notation for this kind of approach

is that already hinted at, but not specifically stated, in subsection 17.1, where

the definition of an inner product is given. This notation, often called the Dirac

notation, denotes a state whose wavefunction isψby|ψ〉;sinceψbelongs to a

vector space of functions,|ψ〉is known as aket vector. Ket vectors, or simply kets,

must not be thought of as completely analogous to physical vectors. Quantum

mechanics associates the same physical state withkeiθ|ψ〉as it does with|ψ〉

for all realkandθand so there is no loss of generality in takingkas 1 andθ

as 0. On the other hand, the combinationc 1 |ψ 1 〉+c 2 |ψ 2 〉,where|ψ 1 〉and|ψ 2 〉