Equivalence Principle: Tunnelling, Quantized Spectra and Trajectories from the Quantum HJ Equation

TL;DR

This paper demonstrates that the equivalence principle, combined with the quantum Hamilton-Jacobi framework, inherently explains tunneling and quantized spectra without relying on the traditional wave function interpretation.

Contribution

It introduces a novel approach linking the equivalence principle to quantum mechanics, deriving tunneling and quantization directly from the quantum Hamilton-Jacobi equation without axiomatic wave function assumptions.

Findings

Derives the quantum Hamilton-Jacobi equation from the equivalence principle.

Shows the quantum potential as an intrinsic, non-vanishing energy.

Establishes a connection between Schwarzian symmetry and energy quantization.

Abstract

A basic aspect of the recently proposed approach to quantum mechanics is that no use of any axiomatic interpretation of the wave function is made. In particular, the quantum potential turns out to be an intrinsic potential energy of the particle, which, similarly to the relativistic rest energy, is never vanishing. This is related to the tunnel effect, a consequence of the fact that the conjugate momentum field is real even in the classically forbidden regions. The quantum stationary Hamilton-Jacobi equation is defined only if the ratio psi^D/psi of two real linearly independent solutions of the Schroedinger equation, and therefore of the trivializing map, is a local homeomorphism of the extended real line into itself, a consequence of the Moebius symmetry of the Schwarzian derivative. In this respect we prove a basic theorem relating the request of continuity at spatial infinity of psi^D/psi, a consequence of the q - 1/q duality of the Schwarzian derivative, to the existence of L^2(R) solutions of the corresponding Schroedinger equation. As a result, while in the conventional approach one needs the Schroedinger equation with the L^2(R) condition, consequence of the axiomatic interpretation of the wave function, the equivalence principle by itself implies a dynamical equation that does not need any assumption and reproduces both the tunnel effect and energy quantization.