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
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.