Mathematical Methods for Physics and Engineering - Matematica.NET

17.3 PROPERTIES OF HERMITIAN OPERATORS17.3 Properties of Hermitian operatorsWe 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 remainderof this chapter, we will write out inner products in full. We note, however, thatthe inner product notation often provides a neat form in which to express results.)17.3.1 Reality of the eigenvaluesConsider an Hermitian operator for which (17.5) is satisfied by at least twoeigenfunctions y i (x) andy j (x), which have corresponding eigenvalues λ i and λ j ,so thatLy i = λ i ρ(x)y i , (17.18)Ly j = λ j ρ(x)y j , (17.19)where we have allowed for the presence of a weight function ρ(x). Multiplying(17.18) by yj ∗ and (17.19) by y∗ i and then integrating gives∫ b∫ byj ∗ Ly i dx = λ i yj ∗ y i ρdx, (17.20)a∫ baa∫ byi ∗ Ly j dx = λ j yi ∗ y j ρdx. (17.21)Remembering that we have required ρ(x) to be real, the complex conjugate of(17.20) becomes∫ bay j (Ly i ) ∗ dx = λ ∗ ia∫ bay ∗ i y j ρdx, (17.22)and using the definition of an Hermitian operator (17.16) it follows that the LHSof (17.22) is equal to the LHS of (17.21). Thus(λ ∗ i − λ j )∫ bay ∗ i y j ρdx=0. (17.23)If i = j then λ i = λ ∗ i (since ∫ ba y∗ i y iρdx ≠ 0), which is a statement that theeigenvalue λ i is real.17.3.2 Orthogonality and normalisation of the eigenfunctionsFrom (17.23), it is immediately apparent that two eigenfunctions y i and y j thatcorrespond to different eigenvalues, i.e. such that λ i ≠ λ j , satisfy∫ bay ∗ i y j ρdx=0, (17.24)561