Equivalence principle: tunnelling, quantized spectra and trajectories from the quantum HJ equation

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 ψ D / ψ of two real linearly independent solutions of the Schrödinger equation, and therefore of the trivializing map, is a local homeomorphism of the extended real line into itself, a consequence of the Möbius symmetry of the Schwarzian derivative. In this respect we prove a basic theorem relating the request of continuity at spatial infinity of ψ D / ψ, a consequence of the q↔ q −1 duality of the Schwarzian derivative, to the existence of L 2( R) solutions of the corresponding Schrödinger equation. As a result, while in the conventional approach one needs the Schrödinger 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.