Deformed quantum mechanics and q-Hermitian operators

TL;DR

This paper introduces a generalized q-deformed Schrödinger equation based on non-commutative q-differential calculus, linking quantum mechanics with q-deformed classical distributions and defining q-Hermitian operators.

Contribution

It develops a new q-deformed quantum framework with a generalized Schrödinger equation and natural q-Hermitian operator properties, extending traditional quantum mechanics.

Findings

Derivation of a q-deformed Schrödinger equation from q-differential calculus

Reproduction of q-deformed exponential stationary distribution at equilibrium

Natural emergence of q-Hermitian operators satisfying quantum principles

Abstract

Starting on the basis of the non-commutative q-differential calculus, we introduce a generalized q-deformed Schr\"odinger equation. It can be viewed as the quantum stochastic counterpart of a generalized classical kinetic equation, which reproduces at the equilibrium the well-known q-deformed exponential stationary distribution. In this framework, q-deformed adjoint of an operator and q-hermitian operator properties occur in a natural way in order to satisfy the basic quantum mechanics assumptions.