Derivation of Fundamental Kinetic Equation in Quantum Mechanics

David Book

Huai‐Yu Wang

The Schrödinger equation, Klein‐Gordon equation (KGE), and Dirac equation are believed to be the fundamental equations of quantum mechanics. Schrödinger’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 Schrödinger equation, and the other is the NKE Schrödinger 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.

V. L. Kon'kov, V. N. Gorshenkov, Е. П. Богданов

Miriam Scharnke, Niclas Schlünzen, M. Bönitz

Time reversal symmetry is a fundamental property of many quantum mechanical systems. The relation between statistical physics and time reversal is subtle and not all statistical theories conserve this particular symmetry, most notably hydrodynamic equations and kinetic equations such as the Boltzmann equation. In this article, it is shown analytically that quantum kinetic generalizations of the Boltzmann equation that are derived using the nonequilibrium Green functions formalism as well as all approximations that stem from Φ-derivable self-energies are time reversal invariant.

H. W. Wyld, Burton D. Fried

Lu Huan, Wan Buyong, Liu Yao +1 more

The issues of the negative energy and the negative kinetic energy remain among the most profound and actively debated topics in physics. Based on the negative-kinetic-energy supersymmetric quantum mechanics (NKE-SUSYQM), this paper explores the supersymmetric algebra in depth. First, we begin with a concise overview of how the negative-kinetic-energy Schrödinger equation is derived and how NKE-SUSYQM is thereby established. Second, we analyze the fundamental quantum operators and elucidate the complete structure of the supersymmetric algebra within NKE-SUSYQM. Finally, we provide a detailed comparative study that contrasts the positive- and the negative-kinetic-energy SUSYQM, highlighting their key similarities and differences.