Schr\"odinger equation of general potential

TL;DR

This paper introduces a generalized Schrödinger equation that accounts for particles in arbitrary potentials depending on position, velocity, and time, expanding the scope of quantum mechanics beyond traditional potentials.

Contribution

It proposes a new quantum wave equation derived from a general Lagrangian, capable of describing particles in more complex potentials involving velocity dependence.

Findings

Derivation of a new quantum wave equation for general potentials

Potential applications across various fields of physics

Extension of quantum mechanics to non-traditional potential scenarios

Abstract

It is well known that the Schr\"odinger equation is only suitable for the particle in common potential $V(\vec{r},t)$. In this paper, a general Quantum Mechanics is proposed, where the Lagrangian is the general form. The new quantum wave equation can describe the particle which is in general potential $V(\vec{r}, \dot{\vec{r}}, t)$. We think these new quantum wave equations can be applied in many fields.