Foundations of Non-linear Quantum Mechanics

TL;DR

This paper develops a foundational framework for non-linear quantum mechanics based on six postulates, introducing non-linear operators and equations, and predicts novel non-commuting properties of non-linear observables.

Contribution

It establishes the theoretical basis for non-linear quantum mechanics using new postulates and propositions, extending traditional linear quantum theory.

Findings

Non-linear solutions accurately describe free particles.

Position and non-linear momentum operators do not commute.

Implicit wave functions can be derived from non-linear PDEs.

Abstract

The foundations of non-linear quantum mechanics are based on six postulates and five propositions. On a first quantised level, these approaches are built on non-linear differential operators, non-linear eigenvalue equations, and the notion of non-linear observables and non-linear states. The present theory predicts that the non-linear function solution of a non-linear partial differential equation for a free particle is correct and that the commutator of the position operator and the non-linear momentum operator does not commute. Moreover, the implicit wave function of linear quantum mechanics can be determined by a non-linear wave function solution of non-linear partial differential equation, which also verifies the non-linear quantum mechanics.