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Hermitian Operators
As mentioned previously, the expectation value of an operator
is
given by

(55) 
and all physical observables are represented by such
expectation values. Obviously, the value of a physical observable such
as energy or density must be real, so we require <A> to be real.
This means that we must have
<A> = <A>^{*}, or

(56) 
Operators
which satisfy this condition are called Hermitian.
One can also show that for a Hermitian operator,

(57) 
for any two states
and .
An important property of Hermitian operators is that their eigenvalues
are real. We can see this as follows: if we have an eigenfunction of
with eigenvalue a, i.e.
,
then
for a Hermitian operator
Since
is never negative, we must have either a = a^{*} or
.
Since
is not an acceptable wavefunction,
a = a^{*}, so a is real.
Another important property of Hermitian operators is that their
eigenvectors are orthogonal (or can be chosen to be so). Suppose that
and
are eigenfunctions of
with eigenvalues
a and b, with .
If
is Hermitian then
since a = a^{*} as shown above. Because we assumed ,
we must
have
,
i.e.
and
are
orthogonal. Thus we have shown that eigenfunctions of a Hermitian operator
with different eigenvalues are orthogonal. In the case of degeneracy
(more than one eigenfunction with the same eigenvalue), we can choose
the eigenfunctions to be orthogonal. We can easily show this for the
case of two eigenfunctions of
with the same eigenvalue. Suppose
we have
We now want to take linear combinations of
and
to form two
new eigenfunctions
and ,
where
and
.
Now we want
and to be orthogonal, so
Thus we merely need to choose

(62) 
and we obtain orthogonal eigenfunctions. This Schmidtorthogonalization
procedure can be extended to the case of nfold degeneracy, so we have shown
that for a Hermitian operator, the eigenvectors can be made orthogonal.
Next: Unitary Operators
Up: Operators
Previous: Eigenfunctions and Eigenvalues