Schroedinger Equation and the Associated Physics in Non-Inertial Frame

TL;DR

This paper develops a formalism to solve the Schrödinger equation in a non-inertial frame with acceleration, revealing effects analogous to Hawking radiation and Unruh effect, and demonstrating frame independence of two-body quantum systems.

Contribution

It introduces a Hamiltonian-based formalism for quantum mechanics in accelerated frames, extending previous work to include gravitational effects and non-inertial motion.

Findings

Solution relates gravity to electron emission phenomena.

Relative motion in quantum systems is frame-independent.

Analogies to Hawking radiation and Unruh effect are observed.

Abstract

In this article we have developed a formalism to obtain the solution of Schroedinger equation in a non-inertial frame. The frame is moving relative to an inertial frame with an acceleration. The formulation has been developed using Lagrangian formalism as discussed in Classical Mechanics book by Landau and Lifshitz [1]. Hence we have obtained the Hamiltonian of the nucleons. Then using the standard form of canonical quantization rule, we have setup the Schroedinger equation in non-inertial frame. In the present study we have considered only the accelerated rectilinear motion of the non-inertial frame. The rotation will be considered in some future work based on quantum field theory [2] (see also [3, 4]). We therefore drop the rotation part of the Hamiltonian in our calculation. The physically acceptable result on our work is basically the solution obtained by Fowler and Nordheim for the emission of electrons from cold metal surface induced by strong electric field [5] known as cold field emission of electrons. However in the present formulation, it is the gravity which acts on mass of the particle and causes emission. The centrifugal force acts like gravity. Hence we have got some flavor of Hawking radiation [6] and also Unruh effect [7] within the limited scope of our non-relativistic approach. We have also shown that relative motion of a two body quantum mechanical system does not depend on the nature of the frame of reference.