![\begin{displaymath}
\vert\left\langle{\psi_1\vert\psi_2}\right\rangle{}\vert^2 \...
...\right\rangle{}\left\langle{\psi_2\vert\psi_2}\right\rangle{}.
\end{displaymath}](s1_360.gif) |
(1) |
Written out explicitly
![\begin{displaymath}
\left[{\int_a^b \psi_1(x)\psi_2(x)\,dx}\right]^2\leq \int_a^b [\psi_1(x)]^2\,dx \int_a^b [\psi_2(x)]^2\,dx,
\end{displaymath}](s1_361.gif) |
(2) |
with equality Iff
with
a constant. To derive, let
be a
Complex function and
a Complex constant such that
for some
and
. Since
,
![\begin{displaymath}
\int \psi^*\psi\,dx = \int f^*f\,dx+\lambda\int f^*g\,dx+\lambda^*\int g^*f\,dx+\lambda\lambda^* \int g^*g\,dx \geq 0,
\end{displaymath}](s1_370.gif) |
(3) |
with equality when
. Set
![\begin{displaymath}
\lambda=-\int g^*f\,dx
\end{displaymath}](s1_372.gif) |
(4) |
so that
![\begin{displaymath}
\lambda^*=-\int f^*g\,dx.
\end{displaymath}](s1_373.gif) |
(5) |
Plugging (5) and (4) into (3) then gives
|
|
|
(6) |
![\begin{displaymath}
\int g^* f\,dx\int f^*g\,dx\leq \int f^*f\,dx\int g^*g\,dx
\end{displaymath}](s1_376.gif) |
(7) |
![\begin{displaymath}
\left\vert{\int g^*f\,dx}\right\vert = \left\vert{\int f^*g\,dx}\right\vert \leq \int f^*f\,dx\int g^*g\,dx
\end{displaymath}](s1_377.gif) |
(8) |
so
![\begin{displaymath}
\vert\left\langle{f\vert g}\right\rangle{}\vert^2\leq \left\...
...f\vert f}\right\rangle{}\left\langle{g\vert g}\right\rangle{}.
\end{displaymath}](s1_378.gif) |
(9) |
Bessel's Inequality can be derived from this.
References
Abramowitz, M. and Stegun, C. A. (Eds.).
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, p. 11, 1972.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 527-529, 1985.
© 1996-9 Eric W. Weisstein
1999-05-26