Mathematical Methods for Physics and Engineering : A Comprehensive Guide

QUANTUM OPERATORS

is to produce a scalar multiple of that ket, i.e.

A|ψ〉=λ|ψ〉, (19.3)

then, just as for matrices and differential equations,|ψ〉is called aneigenketor,

more usually, aneigenstateofA, with corresponding eigenvalueλ;tomarkthis

special property the state will normally be denoted by|λ〉, rather than by the

more general|ψ〉. Taking the Hermitian conjugate of this ket vector eigenequation

gives a bra vector equation,

〈ψ|A†=λ∗〈ψ|. (19.4)

It should be noted that the complex conjugate of the eigenvalue appears in this

equation. Should the action ofAon|ψ〉produce an unphysical state (usually

one whose wavefunction is identically zero, and is therefore unacceptable as a

quantum-mechanical wavefunction because of the required probability interpre-

tation) we denote the result either by 0 or by the ket vector|∅〉according to

context. Formally,|∅〉can be considered as an eigenket of any operator, but one

for which the eigenvalue is always zero.

If an operatorAis Hermitian (A†=A) then its eigenvalues are real and

the eigenstates can be chosen to be orthogonal; this can be shown in the same

way as in chapter 17 (but using a different notation). As indicated there, the

reality of their eigenvalues is one reason why Hermitian operators form the

basis of measurement in quantum mechanics; in that formulation of physics, the

eigenvalues of an operator are theonlypossible values that can be obtained when

a measurement of the physical quantity corresponding to the operator is made.

Actual individual measurements must always result in real values, even if they

are combined in a complex form (x+iyorreiθ) for final presentation or analysis,

and using only Hermitian operators ensures this. The proof of the reality of the

eigenvalues using the Dirac notation is given below in a worked example.

In the same notation the Hermitian property of an operatorAis represented

by the double equality

〈Aφ|ψ〉=〈φ|A|ψ〉=〈φ|Aψ〉.

It should be remembered that the definition of an Hermitian operator involves

specifying boundary conditions that the wavefunctions considered must satisfy.

Typically, they are that the wavefunctions vanish for large values of the spatial

variables upon which they depend; this deals with most physical systems since

they are nearly all formally infinite in extent. Some model systems require the

wavefunction to be periodic or to vanish at finite values of a spatial variable.

Depending on the nature of the physical system, the eigenvalues of a particular

linear operator may be discrete, part of a continuum, or a mixture of both. For

example, the energy levels of the bound proton–electron system (the hydrogen

atom) are discrete, but if the atom is ionised and the electron is free, the energy