- #1
Sergio Rodriguez
- 14
- 4
Homework Statement
Show that if ##\left( \Omega f\right) ^* = -\Omega f ^* ##
then ##\left< \Omega \right> = 0 ## for any real function f. where ##\Omega## is an operator
Homework Equations
It's a self test of the completeness relation --Molecular quantum mechanics (Atkins)--
so the equation is
$$ \sum_s \left| s \right> \left< s \right| = 1 $$
The Attempt at a Solution
## \left< \Omega \right> = \left<m\left|\Omega\right|n\right>## for any functions ##f_m## and ##f_n##
if ## f_m = f_n = f ##, then ## \left<\Omega\right> = \left<f\left|\Omega\right|f\right>
##
and as f is a real function ## f^* = f##, so:
##\left( \Omega f\right) ^* = -\Omega f ^* = -\Omega f ##
##\left( \Omega f\right) ^* = \Omega^* f^* = \Omega^* f ##
so ##\Omega^* f = -\Omega f ##
##\Omega^* \left|f\right> = -\Omega \left|f\right>##
##\left<f \left| \Omega^* \right|f\right> = - \left< f\left|\Omega \right|f\right>##
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