The quantum mechanics based on a general kinetic energy

TL;DR

This paper generalizes the Schrödinger equation by introducing a broad class of kinetic energy operators, proving conservation laws, and exploring implications like probability teleportation and hidden non-Hermitian operators.

Contribution

It presents a unified framework for Schrödinger-type equations with general kinetic energy operators, including new insights into the Klein-Gordon and Dirac equations.

Findings

Conservation law and probability continuity are established for the generalized Schrödinger equation.

The Klein-Gordon equation contains a hidden non-Hermitian kinetic energy operator.

Probability teleportation is proposed as a new probability transportation mechanism.

Abstract

In this paper, we introduce the Schrodinger equation with a general kinetic energy operator. The conservation law is proved and the probability continuity equation is deducted in a general sense. Examples with a Hermitian kinetic energy operator include the standard Schrodinger equation, the relativistic Schrodinger equation, the fractional Schrodinger equation, the Dirac equation, and the deformed Schrodinger equation. We reveal that the Klein-Gordon equation has a hidden non-Hermitian kinetic energy operator. The probability continuity equation with sources indicates that there exists a different way of probability transportation, which is probability teleportation. An average formula is deducted from the relativistic Schrodinger equation, the Dirac equation, and the K-G equation.