I am a freshman trying to understand the very basics of quantum mechanics but I met barriers at the beginning. What really matters is the postulates of quantum mechanics and their relationship with self-adjointness.
Postulate 1) Every observable of a physical system is represented in the mathematical formalism of quantum mechanics by a linear adjoint operator which acts in the Hilbert space associated with the physical system considered.
As far as I know, self adjointness of a given form of operator is dependent on the format of Hilbert space. For example, momentum operator, represented by first-derivative, is not self adjoint in infinite well but in free space. Then for the given observable (here, momentum), the form of operator correspond to the observable should be changed to make the operator self adjoint when we use it in different kinds of Hilbert spaces??
Furthermore if someone wants quantum mechanics to be used with strict mathematical formalism, should he/she always check self-adjointness of the given operator everytime he/she uses it in different Hilbert spaces?
Postulate 2) If B is a Hermitian operator that represents physically observable property, then the eigenfunctions of B form a complete set for the Hilbert space considered.
If postulate 1 is true, shouldn't we change the word "Hermitian" used in postulate 2 with "self adjoint"??
And.... after we change the word, is it still correct that two commuting (self adjoint) operators share common eigenfunctions?
Postulate 3) The time dependence of the state of an undisturbed quantum mechanical system is given by Schrodinger equation..... and if we assume that the state is stationary, the form of the equation is Hf = Ef where f is the wave function, H is Hamiltonian, and E is energy of the system.
Because energy is a sort of observables, this postulate tells us that Hamiltonian operator is always self adjoint regardless of the choice of Hilbert spaces if postulate 1 is true??
And.. If we solve the (Schrodinger) equation and get a set of eigenfunctions (of Hamiltonian) for the system, do the eigenfunctions have all the information about the system? If the eigenfunctions are not the eigenfunctions of... say, p, which means that p don't commute with H, the eigenfunctions of p cannot be one of the possible states of the system?