# Spectral Analysis of Proton Eigenfunctions in Crystalline Environments

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*Quantum Reports*in 2024–2025)

## Abstract

**:**

## 1. Introduction

## 2. Single-Particle Bound States in a Finite Quadratic Periodic Potential

## 3. Three-Dimensional Analysis

## 4. Quasi-Harmonic Case

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Periodic 1D potential with 40 sites with lattice spacing $d=\frac{a}{\sqrt{2}}=2.5\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{-8}\phantom{\rule{4.pt}{0ex}}\mathrm{cm}=1.27\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{-3}\phantom{\rule{4.pt}{0ex}}{\mathrm{eV}}^{-1}$ along the base vector ${\overrightarrow{a}}_{1}$.

**Figure 2.**One particle energy spectrum calculated using the parameters of Table 1. Note the harmonic oscillator-like energy spacing of the bound states and their degeneracy, equal to the number of lattice sites ${N}_{w}$. The free eigenvalues have a Brillouin-like structure. Different bans correspond to different colors. The finite number of wells causes intermediate eigenvalues to appear between adjacent bands (see sub-figure).

**Figure 4.**Linear fit of the photon energies associated with the dipolar transitions between adjacent bound energy levels (the last two points to the right are excluded from the fit).

**Figure 5.**Eigenvalues of the radial equation for all quantum numbers of the bound states. The confirmation of independence from angular momentum, as displayed by the eigenstates of the full harmonic oscillator, persists even for eigenvalues very close to the free spectrum.

Parameter | Symbol | Value |
---|---|---|

oscillation frequency | ${\omega}^{\prime}$ | 0.41 eV |

proton mass | m | 938 MeV |

lattice spacing | d | 2.5 Å |

number of points of a single cell | ${N}_{c}$ | 257 |

number of points of the potential wells | ${N}_{w}$ | 101 |

spatial discretization | $\epsilon $ | 0.0097 Å |

number of crystal sites | ${N}_{s}$ | 41 |

number of points inside the crystal | ${M}_{c}$ | 10,537 |

total number of points | M | 12,537 |

**Table 2.**Calculated and theoretical energies for bound states with ${N}_{\infty}=4096$, ${R}_{\infty}=7.5\phantom{\rule{4pt}{0ex}}\AA $, and ${R}_{w}=0.5125\phantom{\rule{4pt}{0ex}}\AA $.

$\mathit{n}$ | $\mathit{nr}$ | $\mathit{l}$ | Computed Energy (eV) | Theoretical Energy (eV) (Equation (12)) |
---|---|---|---|---|

0 | 0 | 0 | $-4.876$ | $-4.877$ |

1 | 0 | 1 | $-4.460$ | $-4.460$ |

2 | 1 | 0 | $-4.042$ | $-4.043$ |

2 | 0 | 2 | $-4.043$ | $-4.043$ |

3 | 1 | 1 | $-3.626$ | $-3.626$ |

3 | 0 | 3 | $-3.626$ | $-3.626$ |

4 | 2 | 0 | $-3.208$ | $-3.209$ |

4 | 1 | 2 | $-3.209$ | $-3.209$ |

4 | 0 | 4 | $-3.209$ | $-3.209$ |

5 | 2 | 1 | $-2.792$ | $-2.792$ |

5 | 1 | 3 | $-2.792$ | $-2.792$ |

5 | 0 | 5 | $-2.792$ | $-2.792$ |

6 | 3 | 0 | $-2.374$ | $-2.375$ |

6 | 2 | 2 | $-2.375$ | $-2.375$ |

6 | 1 | 4 | $-2.375$ | $-2.375$ |

6 | 0 | 6 | $-2.375$ | $-2.375$ |

7 | 3 | 1 | $-1.958$ | $-1.958$ |

7 | 2 | 3 | $-1.958$ | $-1.958$ |

7 | 1 | 5 | $-1.958$ | $-1.958$ |

7 | 0 | 7 | $-1.958$ | $-1.958$ |

8 | 4 | 0 | $-1.540$ | $-1.541$ |

8 | 3 | 2 | $-1.541$ | $-1.541$ |

8 | 2 | 4 | $-1.541$ | $-1.541$ |

8 | 1 | 6 | $-1.541$ | $-1.541$ |

8 | 0 | 8 | $-1.541$ | $-1.541$ |

9 | 4 | 1 | $-1.126$ | $-1.124$ |

9 | 3 | 3 | $-1.125$ | $-1.124$ |

9 | 2 | 5 | $-1.125$ | $-1.124$ |

9 | 1 | 7 | $-1.124$ | $-1.124$ |

9 | 0 | 9 | $-1.124$ | $-1.124$ |

10 | 5 | 0 | $-0.713$ | $-0.706$ |

10 | 4 | 2 | $-0.713$ | $-0.706$ |

10 | 3 | 4 | $-0.711$ | $-0.706$ |

10 | 2 | 6 | $-0.709$ | $-0.706$ |

10 | 1 | 8 | $-0.707$ | $-0.706$ |

10 | 0 | 10 | $-0.707$ | $-0.706$ |

11 | 5 | 1 | $-0.316$ | $-0.289$ |

11 | 4 | 3 | $-0.310$ | $-0.289$ |

11 | 3 | 5 | $-0.303$ | $-0.289$ |

11 | 2 | 7 | $-0.296$ | $-0.289$ |

11 | 1 | 9 | $-0.292$ | $-0.289$ |

11 | 0 | 11 | $-0.290$ | $-0.289$ |

12 | 6 | 0 | $-0.002$ | $0.128$ |

12 | 5 | 2 | $0.001$ | $0.128$ |

12 | 4 | 4 | $0.002$ | $0.128$ |

12 | 3 | 6 | $0.004$ | $0.128$ |

12 | 2 | 8 | $0.006$ | $0.128$ |

12 | 1 | 10 | $0.008$ | $0.128$ |

12 | 0 | 12 | $0.011$ | $0.128$ |

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**MDPI and ACS Style**

Gamberale, L.; Modanese, G.
Spectral Analysis of Proton Eigenfunctions in Crystalline Environments. *Quantum Rep.* **2024**, *6*, 172-183.
https://doi.org/10.3390/quantum6020014

**AMA Style**

Gamberale L, Modanese G.
Spectral Analysis of Proton Eigenfunctions in Crystalline Environments. *Quantum Reports*. 2024; 6(2):172-183.
https://doi.org/10.3390/quantum6020014

**Chicago/Turabian Style**

Gamberale, Luca, and Giovanni Modanese.
2024. "Spectral Analysis of Proton Eigenfunctions in Crystalline Environments" *Quantum Reports* 6, no. 2: 172-183.
https://doi.org/10.3390/quantum6020014