The modified fundamental equations of quantum mechanics

TL;DR

This paper proposes modified fundamental equations of quantum mechanics that decouple positive and negative kinetic energy solutions, addressing issues in Schrödinger, Klein-Gordon, and Dirac equations, and highlighting symmetry related to matter and dark matter.

Contribution

The paper introduces revised quantum equations that are symmetric, unitary, and decouple PKE and NKE, improving upon traditional formulations and resolving specific potential problems.

Findings

Modified equations are symmetric and unitary.

They resolve issues with step potential problems.

Highlight the symmetry between matter and dark matter.

Abstract

The Schrodinger equation, Klein-Gordon equation (KGE), and Dirac equation are believed to be the fundamental equations of quantum mechanics. Schrodinger's equation has a defect in that there are no negative kinetic energy (NKE) solutions. Dirac's equation has positive kinetic energy (PKE) and NKE branches. Both branches should have low-momentum, or nonrelativistic, approximations: One is the Schrodinger equation, and the other is the NKE Schrodinger equation. The KGE has two problems: It is an equation of the second time derivative so that the calculated density is not definitely positive, and it is not a Hamiltonian form. To overcome these problems, the equation should be revised as PKE- and NKE-decoupled KGEs. The fundamental equations of quantum mechanics after the modification have at least two merits. They are unitary in that all contain the first time derivative and are symmetric with respect to PKE and NKE. This reflects the symmetry of the PKE and NKE matters, as well as, in the author's opinion, the matter and dark matter of our universe. The problems of one-dimensional step potentials are resolved by utilizing the modified fundamental equations for a nonrelativistic particle.