Proving Commuter Operators: X & P

[CPL.Luke]

I forgot the exact terminology for these types of operators but here goes.

take for example the operators x, and p.

the commuter equals i(h bar), and the eigenvectors are Fourier transforms of each other.

my question is, how do you go about proving at least one of the properties listed above without referencing any functional form of x or p aka start with the most basic axioms possible.

 

[haushofer]

Isn't it the idea to take the Poissonbrackets in classical mechanics and then "quantize" these brackets?

 

[CPL.Luke]

I remember that being one ofthe methods that was used, I really am curious about the different methods. or what's necessary and sufficient condition effectively.

For instance one professor of mine effectivley made it an axiom that the inner product of a x state vector and a p vector was equal to EXP[ipx] and went on to derive the Schrödinger equation from that, and a couple of the commutation relations.