How Does Hermitian Property Lead to \langle A^2 \rangle in Quantum Mechanics?

[spaghetti3451]

I know that <Aψ|Aψ> = <A²>. Can anyone tell please tell me how you get one line to the another?

 

[spaghetti3451]

Ok so, <Aψ|Aψ> = <AAψ|ψ> because A is self adjoint (hermitian). What next?

 

[Amok]

Ok so, <Aψ|Aψ> = <AAψ|ψ> because A is self adjoint (hermitian). What next?

$$\langle A\psi \mid A\psi \rangle = \langle \psi \mid A^{\dagger}A\mid \psi \rangle$$

If A is Hermintian:

$$\langle \psi \mid A^{\dagger}A\mid \psi \rangle = \langle \psi \mid A^{2}\mid \psi \rangle$$

And that's it :)