Physical Chemistry Third Edition

1248 B Some Useful Mathematics

Therefore

aj

∫

fj∗fjdqaj∗

∫

fj∗fjdq (B-73)

and

aj∗aj (B-74)

A quantity equals its complex conjugate if and only if it is real, so thatajmust be real.

Property 5. Two eigenfunctions of a hermitian operator with different eigenvalues

are orthogonal to each other.

Two functionsfandgareorthogonalto each other if

∫

f∗gdq

∫

g∗fdq 0

(

definition of

orthogonality

)

(B-75)

wheref∗is the complex conjugate offandg∗is the complex conjugate ofg. The

two integrals in Eq. (B-75) are the complex conjugates of each other, so that if one

vanishes, so does the other.

We prove property 5 as follows: Multiply the eigenvalue equation, Eq. (B-69), by

fk∗, the complex conjugate of a different eigenfunction, and integrate, factoring the

constant eigenvalue out of the integral:

∫

fk∗̂Afjdqaj

∫

fk∗fjdq (B-76)

Now apply the hermitian property to the left-hand side of this equation:

∫

fk∗̂Afjdq

∫

(̂A∗fk∗)fjdqa∗k

∫

fk∗fjdqak

∫

fk∗fjdq (B-77)

where we have replaceda∗kbyakbecause we knowakto be real. The left-hand sides

of Eqs. (B-76) and (B-77) are equal, so the difference of the right-hand sides vanishes:

(aj−ak)

∫

fk∗fjdq 0 (B-78)

If the two eigenvalues are not equal to each other, the integral must vanish, and we

have proved the orthogonality offkandfj:

∫

fk∗fjdq 0 (B-79)

If two eigenfunctions have equal eigenvalues, they are not necessarily orthogonal to

each other, but linear combinations of the eigenfunctions can be constructed that are

orthogonal to each other.