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## Homework Statement

If <h|Qh> = <Qh|h> for all functions h, show that <f|Qg> = <Qf|g> for all f and g.

f,g, and h are functions of x

Q is a hermitian operator

Hints: First let h=f+g, then let h=f+ig

## Homework Equations

<Q>=<Q>*

Q(f+g)= Qf+Qg

## The Attempt at a Solution

(All integral are with respect to x, and go from -∞ to +∞)

<(f+g)|Q(f+g)> = <Q(f+g)|(f+g)>

∫(f*+g*)(Qf+Qg) = ∫(Qf*+Qg*)(f+g)

∫f*Qf + ∫g*Qg + ∫f*Qg + ∫g*Qf = ∫(Qf*)f + ∫(Qg*)g + ∫(Qf*)g + ∫(Qg*)f

<f|Qf> + <g|Qg> + <f|Qg> + <g|Qf> = <Qf|f> + <Qg|g> + <Qf|g> + <Qg|f>

<f|Qg> + <g|Qf> = <Qf|g> + <Qg|f>

And then I'm stuck.

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