Show that if H is hermitian, then <Ψ|H|Φ> = <HΨ|Φ>

Since H is hermitian, it is equal to its complex conjugate (shown below), a property which I used to do the proof:


[![it](https://i.stack.imgur.com/409uF.png)](https://i.stack.imgur.com/409uF.png)


I had to prove that <Ψ|H|Φ> = <HΨ|Φ>, which I did as below:


[![enter image description here](https://i.stack.imgur.com/mGOgM.png)](https://i.stack.imgur.com/mGOgM.png)


My question is whether my proof is valid - are the two things equal because the pink bits equal each other?




As the operator $H$ is linear, when taking it out of the bra symbol it must be replaced by its adjoint and placed on the right, i.e. $\langle H\psi|=\langle \psi|H^\dagger$ and so $\langle H^\dagger\psi|=\langle \psi|(H^\dagger)^\dagger=\langle\psi| H$. In you case the operator is hermitian so $H=H^\dagger$


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