The modified fundamental equations of quantum mechanics

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.

30 pages, 2 figures, 4 tables

References in corpus (2)

• Quantum simulation of the Klein paradox with trapped ions
• Fundamental formalism of statistical mechanics and thermodynamics of negative kinetic energy systems

Cited by in corpus (4)

• There is no vacuum zero-point energy in our universe for massive particles within the scope of relativistic quantum mechanics
• Evaluation of cross section of elastic scattering for non-relativistic and relativistic particles by means of fundamental scattering formulas
• Many-body theories for negative kinetic energy systems
• A physical mechanism of the generation of stable positive kinetic energy systems and a qualitative explanation of the proportions of the four ingredients in the universe