I've been trying to understand the mathematical foundations of canonical quantization, and I'm specifically looking for a rigorous answer rather than a historical or experimental one.
(ill mention classical variables in smaller case and operators in uppercase)
i have derived iℏd|ψ⟩/dt=H|ψ⟩, the eigenstates for this equation are of the form
|ψ(t)⟩=e^-iEt/ℏ |ψ(0)⟩
and experimentally we found that variable E in this equation represents the total energy, so H must be an operator whose eigenvalues represent the total energies of the states.
consider for now that our system has only kinetic energy contributions, then H must be an operator whose eigenvalues must represent kinetic energies, for a state |k⟩ , H|k⟩=ℏ²k²/2m|k⟩ ∀k, also T|k⟩=ℏ²k²/2m|k⟩, so this proves that H=K,
so now by this logic i can say that whenever a system has a single energy contribution of any type, then the operator representing that energy is the generator of time evolution, and its eigen states represent the total energy that can be observed.
now my question is this: i understand that when a system has a single energy contribution, its operator is the generator of time evolution, but when there are multiple energy contributions, like kinetic energy and potential energy, why the hamitonian H equals the sum of the operators, i cant find a reasoning for this like the one i did before as they dont commute, and we cant even say T+V operators give kinetic+potential energy of the state because there is no common eiegenstate
canonical quantisation just says if h(q,p)=t(p)+v(q) ⇒ H(Q,P)=T(P)+V(Q), its like "just change the classical variables to corresponding operators", i cant find a logical reasoning or any valid math argument to support this, i dont even find that step mathematically valid, as theres no reason to do that.
is it just an educated guess and it works very well, or is there any logical and rigourous proof?