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 ,
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, Self-Adjoint Operator,
Sturm-Liouville Theory
References
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.
© 1996-9 Eric W. Weisstein
1999-05-25