A Hermitian Operator is one which satisfies

(1) 
As shown in SturmLiouville Theory, if is SelfAdjoint 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 secondorder 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 ,
so
is Hermitian.
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 ,
so

(31) 
By further iterations, this can be generalized to

(32) 
See also Adjoint Operator, Hermitian Matrix, SelfAdjoint Operator,
SturmLiouville Theory
References
Arfken, G. ``Hermitian (SelfAdjoint) Operators.'' §9.2 in Mathematical Methods for Physicists, 3rd ed.
Orlando, FL: Academic Press, pp. 504506 and 510516, 1985.
© 19969 Eric W. Weisstein
19990525