Hermitian Operator -- from Wolfram MathWorld

A second-order linear Hermitian operator is an operator that satisfies

(1)

where denotes a complex conjugate. 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 biorthogonal system when is second-order and linear.

Note that the concept of Hermitian operator is somewhat extended in quantum mechanics to operators that need be neither second-order differential nor real. Simply assuming that the boundary conditions give sufficiently strongly vanishing near infinity or have periodic behavior allows an operator to be Hermitian in this extended sense if

(3)

which is identical to the previous definition except that quantities have been extended to be complex (Arfken 1985, p. 506).

In order to prove that eigenvalues must be real and eigenfunctions orthogonal, consider

(4)

Assume there is a second eigenvalue such that

(5)

(6)

Now multiply (4) by and (6) by

(7)

(8)

(9)

Now integrate

(10)

But because is Hermitian, the left side vanishes.

(11)

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 .

(12)

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

For a Hermitian operator ,

(13)

In integral notation,

(14)

Given Hermitian operators and ,

(15)

Because, for a Hermitian operator with eigenvalue ,

(16)

(17)

Therefore, either or . But iff , so

(18)

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,