A Hermitian Operator
is one which satisfies
![\begin{displaymath}
\int_a^b v^*\tilde Lu\,dx = \int_a^b u\tilde Lv^*\, dx.
\end{displaymath}](h_1482.gif) |
(1) |
As shown in Sturm-Liouville Theory, if
is Self-Adjoint and satisfies
the boundary conditions
![\begin{displaymath}[v^*pu']_{x=a} = [v^*pu']_{x=b},
\end{displaymath}](h_1483.gif) |
(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
![\begin{displaymath}
\tilde Lu_i+\lambda_iwu_i=0.
\end{displaymath}](h_1484.gif) |
(3) |
Assume there is a second Eigenvalue
such that
![\begin{displaymath}
\tilde Lu_j+\lambda_jwu_j=0
\end{displaymath}](h_1486.gif) |
(4) |
![\begin{displaymath}
\tilde L{u_j}^*+{\lambda_j}^*w{u_j}^*=0.
\end{displaymath}](h_1487.gif) |
(5) |
Now multiply (3) by
and (5) by
![\begin{displaymath}
{u_j}^*\tilde Lu_i+{u_j}^*\lambda_i wu_i=0
\end{displaymath}](h_1490.gif) |
(6) |
![\begin{displaymath}
{u_i}\tilde L{u_j}^*+u_i{\lambda_j}^*w{u_j}^*=0
\end{displaymath}](h_1491.gif) |
(7) |
![\begin{displaymath}
{u_j}^*\tilde Lu_i-u_i\tilde L{u_j}^*=({\lambda_j}^*-\lambda_i)wu_i{u_j}^*.
\end{displaymath}](h_1492.gif) |
(8) |
Now integrate
![\begin{displaymath}
\int_a^b {u_j}^*\tilde Lu_i-\int_a^b u_i\tilde L{u_j}^*=({\lambda_j}^*-\lambda_i)\int_a^b wu_i{u_j}^*.
\end{displaymath}](h_1493.gif) |
(9) |
But because
is Hermitian, the left side vanishes.
![\begin{displaymath}
({\lambda_j}^*-\lambda_i)\int_a^b wu_i{u_j}^*=0.
\end{displaymath}](h_1494.gif) |
(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
.
![\begin{displaymath}
({\lambda_i}^*-\lambda_i)\int_a^b wu_i{u_i}^*=0.
\end{displaymath}](h_1498.gif) |
(11) |
The integral cannot vanish unless
, so we have
and the Eigenvalues
are real.
For a Hermitian operator
,
![\begin{displaymath}
\left\langle{\phi\vert\tilde O\psi}\right\rangle{} = \left\l...
...ngle{}^* = \left\langle{\tilde O\phi\vert\psi}\right\rangle{}.
\end{displaymath}](h_1502.gif) |
(12) |
In integral notation,
![\begin{displaymath}
\int(\tilde A\phi)^*\psi\,dx =\int\phi^*\tilde A\psi\,dx.
\end{displaymath}](h_1503.gif) |
(13) |
Given Hermitian operators
and
,
![\begin{displaymath}
\left\langle{\phi\vert\tilde A\tilde B\psi}\right\rangle{} =...
... \left\langle{\phi\vert\tilde B\tilde A\psi}\right\rangle{}^*.
\end{displaymath}](h_1506.gif) |
(14) |
Because, for a Hermitian operator
with Eigenvalue
,
![\begin{displaymath}
\left\langle{\psi\vert\tilde A\psi}\right\rangle{} = \left\langle{\tilde A\psi\vert\psi}\right\rangle{}
\end{displaymath}](h_1507.gif) |
(15) |
![\begin{displaymath}
a\left\langle{\psi\vert\psi}\right\rangle{} = a^*\left\langle{\psi\vert\psi}\right\rangle{}.
\end{displaymath}](h_1508.gif) |
(16) |
Therefore, either
or
. But
Iff
, so
![\begin{displaymath}
\left\langle{\psi\vert\psi}\right\rangle{}\not= 0,
\end{displaymath}](h_1512.gif) |
(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,
![\begin{displaymath}
\left\langle{\psi_n\vert\tilde A \psi_m}\right\rangle{} = \left\langle{\tilde A\psi_n\vert\psi_m}\right\rangle{}
\end{displaymath}](h_1513.gif) |
(18) |
![\begin{displaymath}
a_m\left\langle{\psi_n\vert\psi_m}\right\rangle{} = {a_n}^*\...
...\rangle{} = a_n\left\langle{\psi_n\vert\psi_m}\right\rangle{},
\end{displaymath}](h_1514.gif) |
(19) |
since
. Then
![\begin{displaymath}
(a_m-a_n)\left\langle{\psi_n\vert\psi_m}\right\rangle{} = 0
\end{displaymath}](h_1516.gif) |
(20) |
For
(i.e.,
),
![\begin{displaymath}
\left\langle{\psi_n\vert\psi_m}\right\rangle{} = 0.
\end{displaymath}](h_1519.gif) |
(21) |
For
(i.e.,
),
![\begin{displaymath}
\left\langle{\psi_n\vert\psi_m}\right\rangle{} = \left\langle{\psi_n\vert\psi_n}\right\rangle{}\equiv 1.
\end{displaymath}](h_1522.gif) |
(22) |
Therefore,
![\begin{displaymath}
\left\langle{\psi_n\vert\psi_m}\right\rangle{} =\delta_{nm},
\end{displaymath}](h_1523.gif) |
(23) |
so the basis of Eigenfunctions corresponding to a Hermitian operator are Orthonormal. Given two Hermitian operators
and
,
![\begin{displaymath}
(\tilde A \tilde B )^\dagger =\tilde B^\dagger\tilde A^\dagger
= \tilde B\tilde A =\tilde A \tilde B +[\tilde B ,\tilde A ],
\end{displaymath}](h_1524.gif) |
(24) |
the operator
equals
, and is therefore
Hermitian, only if
![\begin{displaymath}[\tilde B, \tilde A]= 0.
\end{displaymath}](h_1527.gif) |
(25) |
Given an arbitrary operator
,
so
is Hermitian.
so
is Hermitian. Similarly,
![\begin{displaymath}
\left\langle{\psi_1\vert(\tilde A\tilde A^\dagger)\psi_2}\ri...
...e{(\tilde A\tilde A^\dagger)\psi_1\vert\psi_2}\right\rangle{},
\end{displaymath}](h_1536.gif) |
(28) |
so
is Hermitian.
Define the Hermitian conjugate operator
by
![\begin{displaymath}
\left\langle{\tilde A \psi \vert\psi}\right\rangle{}\equiv \left\langle{\psi \vert\tilde A^\dagger\psi}\right\rangle{}.
\end{displaymath}](h_1539.gif) |
(29) |
For a Hermitian operator,
. Furthermore, given two Hermitian operators
and
,
so
![\begin{displaymath}
(\tilde A \tilde B)^\dagger =\tilde B^\dagger\tilde A^\dagger.
\end{displaymath}](h_1544.gif) |
(31) |
By further iterations, this can be generalized to
![\begin{displaymath}
(\tilde A \tilde B \cdots\tilde Z)^\dagger =\tilde Z^\dagger\cdots\tilde B^\dagger\tilde A^\dagger.
\end{displaymath}](h_1545.gif) |
(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