Undergrad Hermitian Operator Proof - Question

[Jd_duarte]

Hi,

I am questioning about this specific proof -https://quantummechanics.ucsd.edu/ph130a/130_notes/node134.html.
Why to do this proof is needed to compute the complex conjugate of the expectation value of a physical variable? Why can't we just start with < H\psi \mid \psi > ?

 

[PeroK]

Hi,

I am questioning about this specific proof -https://quantummechanics.ucsd.edu/ph130a/130_notes/node134.html.
Why to do this proof is needed to compute the complex conjugate of the expectation value of a physical variable? Why can't we just start with < H\psi \mid \psi > ?

If that's a proof of anything, it escapes me.

Normally, one proves that the eigenvalues of a Hermitian operator are real and then attention is restricted to Hermitian operators for observables; even though there may be non-Hermitian operators with real eigenvalues.

 

[bobob]

Hi,

I am questioning about this specific proof -https://quantummechanics.ucsd.edu/ph130a/130_notes/node134.html.
Why to do this proof is needed to compute the complex conjugate of the expectation value of a physical variable? Why can't we just start with < H\psi \mid \psi > ?

Because you will need that result to prove that the eigenfunctions of an hermitian operator corresponding to different eigenvalues are orthogonal.