Proving Commuter Operators: X & P

  • Thread starter Thread starter CPL.Luke
  • Start date Start date
  • Tags Tags
    Operators
CPL.Luke
Messages
440
Reaction score
0
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.
 
Physics news on Phys.org
Isn't it the idea to take the Poissonbrackets in classical mechanics and then "quantize" these brackets?
 
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 schrodinger equation from that, and a couple of the commutation relations.
 
Not an expert in QM. AFAIK, Schrödinger's equation is quite different from the classical wave equation. The former is an equation for the dynamics of the state of a (quantum?) system, the latter is an equation for the dynamics of a (classical) degree of freedom. As a matter of fact, Schrödinger's equation is first order in time derivatives, while the classical wave equation is second order. But, AFAIK, Schrödinger's equation is a wave equation; only its interpretation makes it non-classical...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
Is it possible, and fruitful, to use certain conceptual and technical tools from effective field theory (coarse-graining/integrating-out, power-counting, matching, RG) to think about the relationship between the fundamental (quantum) and the emergent (classical), both to account for the quasi-autonomy of the classical level and to quantify residual quantum corrections? By “emergent,” I mean the following: after integrating out fast/irrelevant quantum degrees of freedom (high-energy modes...

Similar threads

Back
Top