Spectral Analysis of Proton Eigenfunctions in Crystalline Environments

Author:

Gamberale Luca12,Modanese Giovanni3ORCID

Affiliation:

1. Quantumatter Inc., Dover, DE 19901, USA

2. LEDA srl, Università Milano Bicocca, I-20126 Milano, Italy

3. Faculty of Engineering, Free University of Bozen-Bolzano, I-39100 Bolzano, Italy

Abstract

The Schrödinger equation and Bloch theorem are applied to examine a system of protons confined within a periodic potential, accounting for deviations from ideal harmonic behavior due to real-world conditions like truncated and non-quadratic potentials, in both one-dimensional and three-dimensional scenarios. Numerical computation of the energy spectrum of bound eigenfunctions in both cases reveals intriguing structures, including bound states with degeneracy matching the site number Nw, reminiscent of a finite harmonic oscillator spectrum. In contrast to electronic energy bands, the proton system displays a greater number of possible bound states due to the significant mass of protons. Extending previous research, this study rigorously determines the constraints on the energy gap and oscillation amplitude of the previously identified coherent states. The deviations in energy level spacing identified in the computed spectrum, leading to the minor splitting of electromagnetic modes, are analyzed and found not to hinder the onset of coherence. Finally, a more precise value of the energy gap is determined for the proton coherent states, ensuring their stability against thermal decoherence up to the melting temperature of the hosting metal.

Funder

Free University of Bozen-Bolzano

Publisher

MDPI AG

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