Mathematical Methods for Physics and Engineering : A Comprehensive Guide

17.3 PROPERTIES OF HERMITIAN OPERATORS

17.3 Properties of Hermitian operators

We now provide proofs of some of the useful properties of Hermitian operators.

Again much of the analysis is similar to that for Hermitian matrices in chapter 8,

although the present section stands alone. (Here, and throughout the remainder

of this chapter, we will write out inner products in full. We note, however, that

the inner product notation often provides a neat form in which to express results.)

17.3.1 Reality of the eigenvalues

Consider an Hermitian operator for which (17.5) is satisfied by at least two

eigenfunctionsyi(x)andyj(x), which have corresponding eigenvaluesλiandλj,

so that

Lyi=λiρ(x)yi, (17.18)

Lyj=λjρ(x)yj, (17.19)

where we have allowed for the presence of a weight functionρ(x). Multiplying

(17.18) byy∗jand (17.19) byy∗iand then integrating gives

∫b

a

yj∗Lyidx=λi

∫b

a

y∗jyiρdx, (17.20)

∫b

a

yi∗Lyjdx=λj

∫b

a

y∗iyjρdx. (17.21)

Remembering that we have requiredρ(x) to be real, the complex conjugate of

(17.20) becomes
∫b

a

yj(Lyi)∗dx=λ∗i

∫b

a

y∗iyjρdx, (17.22)

and using the definition of an Hermitian operator (17.16) it follows that the LHS

of (17.22) is equal to the LHS of (17.21). Thus

(λ∗i−λj)

∫b

a

y∗iyjρdx=0. (17.23)

Ifi=jthenλi=λ∗i (since

∫b

ay

∗
iyiρdx= 0), which is a statement that the
eigenvalueλiis real.

17.3.2 Orthogonality and normalisation of the eigenfunctions

From (17.23), it is immediately apparent that two eigenfunctionsyiandyjthat

correspond to different eigenvalues, i.e. such thatλi=λj, satisfy

∫b

a

y∗iyjρdx=0, (17.24)