## Hermitian Operator

A Hermitian Operator is one which satisfies

 (1)

As shown in Sturm-Liouville Theory, if is Self-Adjoint and satisfies the boundary conditions
 (2)

then it is automatically Hermitian. Hermitian operators have Real Eigenvalues, Orthogonal Eigenfunctions, and the corresponding Eigenfunctions form a Complete set when is second-order and linear. In order to prove that Eigenvalues must be Real and Eigenfunctions Orthogonal, consider
 (3)

Assume there is a second Eigenvalue such that
 (4)

 (5)

Now multiply (3) by and (5) by
 (6)

 (7)

 (8)

Now integrate
 (9)

But because is Hermitian, the left side vanishes.
 (10)

If Eigenvalues and are not degenerate, then , so the Eigenfunctions are Orthogonal. If the Eigenvalues are degenerate, the Eigenfunctions are not necessarily orthogonal. Now take .
 (11)

The integral cannot vanish unless , so we have and the Eigenvalues are real.

For a Hermitian operator ,

 (12)

In integral notation,
 (13)

Given Hermitian operators and ,
 (14)

Because, for a Hermitian operator with Eigenvalue ,
 (15)

 (16)

Therefore, either or . But Iff , so
 (17)

for a nontrivial Eigenfunction. This means that , namely that Hermitian operators produce Real expectation values. Every observable must therefore have a corresponding Hermitian operator. Furthermore,
 (18)

 (19)

since . Then
 (20)

For (i.e., ),
 (21)

For (i.e., ),
 (22)

Therefore,
 (23)

so the basis of Eigenfunctions corresponding to a Hermitian operator are Orthonormal. Given two Hermitian operators and ,
 (24)

the operator equals , and is therefore Hermitian, only if
 (25)

Given an arbitrary operator ,
 (26)

so is Hermitian.
 (27)

so is Hermitian. Similarly,
 (28)

so is Hermitian.

Define the Hermitian conjugate operator by

 (29)

For a Hermitian operator, . Furthermore, given two Hermitian operators and ,
 (30)

so
 (31)

By further iterations, this can be generalized to
 (32)

Arfken, G. Hermitian (Self-Adjoint) Operators.'' §9.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 504-506 and 510-516, 1985.