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

A basic aspect of the recently proposed approach to quantum mechanics is th at no use of any axiomatic interpretation of the wave function is made. In particular, the quantum potential turns out to be an intrinsic potential en ergy 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 for bidden regions. The quantum stationary Hamilton-Jacobi equation is defined only if the ratio psi(D)/psi of two real linearly independent solutions of the Schrodinger equation, and therefore of the trivializing map, is a local homeomorphism of the extended real line into itself, a consequence of the Mobius symmetry of the Schwarzian derivative. In this respect we prove a ba sic theorem relating the request of continuity at spatial infinity of psi(D )/psi, a consequence of the q <-> q(-1) duality of the Schwarzian derivativ e, to the existence of L-2(R) solutions of the corresponding Schrodinger eq uation. As a result, while in the conventional approach one needs the Schro dinger equation with the L-2(R) condition, consequence of the axiomatic int erpretation of the wave function, the equivalence principle by itself impli es a dynamical equation that does not need any assumption and reproduces bo th the tunnel effect and energy quantization. (C) 1999 Elsevier Science B.V . All rights reserved.