- #1
latentcorpse
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If [itex]\hat{O}[/itex] is hermitian, show that [itex]\hat{O}^2[/itex] is hermitian.
we have [itex]<\psi|\hat{O}^2|\psi>^* = <\psi|\hat{O}\hat{O}|\phi>^*=<\phi|\hat{O}^{\dagger} \hat{O}^{\dagger}|\psi>=<\phi|\hat{O}\hat{O}|\psi>=<\phi|\hat{O}^2|\psi>[/itex]
which works (hopefully)!
to do this in integral notation is the following ok:
[itex]\left(\int \psi^* \hat{O} \hat{O} \phi dx \right)^* = \int \left(\hat{O}\hat{O} \phi \right)^* \psi dx = \int \phi^* \hat{O} \hat{O} \psi dx[/itex]
?
we have [itex]<\psi|\hat{O}^2|\psi>^* = <\psi|\hat{O}\hat{O}|\phi>^*=<\phi|\hat{O}^{\dagger} \hat{O}^{\dagger}|\psi>=<\phi|\hat{O}\hat{O}|\psi>=<\phi|\hat{O}^2|\psi>[/itex]
which works (hopefully)!
to do this in integral notation is the following ok:
[itex]\left(\int \psi^* \hat{O} \hat{O} \phi dx \right)^* = \int \left(\hat{O}\hat{O} \phi \right)^* \psi dx = \int \phi^* \hat{O} \hat{O} \psi dx[/itex]
?