Why is that in quantum mechanics quite often the physical observable is represented by Hermitian operator?

Question

My knowledge in vector space and quantum mechanics is weak and I am trying to understand and make sense of the question that I asked.

It will be very helpful if someone could explain it to me in a conceptual and deeper sense.

Answer 1

Physical observables in quantum mechanics are always represented by Hermitian operators. The reason for this is that physical observables are real quantities, which means that the eigenvalues of the corresponding operators must be always real, which means that the operator should be Hermitian.