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 andthe 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 representedby 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 particularlinear 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