New results by low momentum approximation from relativistic quantum mechanics equations and suggestion of experiments

TL;DR

This paper derives a negative kinetic energy Schrödinger equation from relativistic quantum mechanics equations using low momentum approximation, addressing longstanding issues and proposing experiments to detect negative energy electrons.

Contribution

It introduces a low momentum approximation approach to derive NKE Schrödinger equations from RQMEs, revealing new insights into negative energy states and potential stable electron systems.

Findings

Derived NKE Schrödinger equation from RQMEs.

Proposed that NKE electrons can form stable systems.

Suggested experiments to detect NKE electrons.

Abstract

A fundamental belief is that the formulism of relativistic quantum mechanics equations (RQMEs) should remain in low momentum motion. However, it is found that some formulas from RQMEs were lost in Schr\"odinger equation. For example, a free relativistic particle has positive and negative energy branches. The former includes positive kinetic energy (PKE) and the latter negative kinetic energy (NKE). The latter should be treated on an equal footing as the former. Nevertheless, from Schr\"odinger equation, a free particle can have only PKE. Starting from RQMEs and taking low momentum approximation, we derive NKE Schr\"odinger equation which is for the cases that free particles have NKE. Thus negative energy branch of RQMEs can be retained in low momentum motion. We point out a fact that whether Schr\"odinger equation is applicable in a region where a particle's energy E is less than potential V, E<V, has never been quantitatively verified. In such a region NKE Schr\"odinger equation should be employed. With the help of NKE Schr\"odinger equation, the lost formulas are recovered. The so-called difficulty of negative probability of Klein-Gordon equation for free particles is solved. A PKE (NKE) particle can have stationary motion only when it is subject to an attractive (repulsive) potential, which is determined by Virial theorem. Two NKE electrons in a potential can constitute a stablesystem, a new kind of possible mechanism for electron paring. The whole discussion stems from RQMEs with no any new postulation. Experiments are suggested, which may confirm that there are indeed NKE electrons.