Exact prime density reproduced through resonant tunneling across a double barrier system

Abstract

A solid-state experiment based on quantum tunneling is proposed to reproduce the natural numbers and prime numbers as resonant tunneling energies in a double barrier system (DBS). For getting the prime numbers as eigenvalues the well potential is considered as the superposition of a smooth potential which is estimated using a semi-classical approach and a weak local fluctuating potential. We use the transfer matrix approach and finite element method by taking only the smooth part of the potential to obtain resonant energies which reproduces the local average prime density and the local average prime gap exactly. The methodology when applied to a quadratic potential of the well, produces whole numbers as eigenvalues except for a constant zero-point energy —the energy levels of a simple harmonic oscillator.