The 0.5 Ratio Limit and Geometry-Induced Missing Energy: Universal 3D Quantum Constraints on Fragment Distributions from Attosecond to Subatomic Scales

TL;DR

The paper shows that the kinetic-energy distribution of fragments after a sudden Coulomb explosion is determined by the 3D radial geometry of the initial quantum state. Using a Bohmian local-energy formalism on Slater-type orbitals, it derives a universal 0.5 Rule: for ground states, the peak detected energy $E_{peak}$ is always below half the integrated mean energy $\langle E \rangle$, due to $r^2$ volume weighting. The authors systematically explore how $R_E$ depends on the orbital exponent $\zeta$, principal quantum number $n$, and repulsive charge $Q$, finding that inner-shell localization (large $\zeta$) strongly suppresses the detected energy relative to the total energy, while excited states ($n>1$) produce secondary high-energy peaks that explain structured KER spectra in systems like $H_2^+$. They argue that the observed energy shortfall in various scales could reflect geometric effects rather than missing particles, proposing $R_E$ as a baseline correction for interpreting energy budgets in atomic, molecular, and subatomic processes.

Abstract

The sudden approximation is a fundamental tool for describing ultrafast quantum transitions, yet the relationship between the initial 3D wavefunction and the final kinetic energy release (KER) remains poorly understood. In this work, we demonstrate that the observed fragment energy distribution $P(E)$ following a sudden Coulomb explosion is a direct manifestation of the initial state's 3D radial geometry. By applying a Bohmian-inspired local energy framework to Slater-type orbitals, we identify a universal \textbf{``0.5 Rule''}: for ground-state configurations, the ratio of the peak detected energy to the integrated average energy ($R_E = E_{peak}/\langle E \rangle$) is strictly bounded below 0.5. Our systematic parametric study reveals that this ratio is highly sensitive to the orbital exponent $ζ$ and the principal quantum number $n$. Specifically, $R_E$ decreases significantly as $ζ$ increases, suggesting that the kinetic energy of fragments from inner-shell orbitals is far more suppressed than that of valence electrons. We further show that for excited states ($n > 1$), the emergence of secondary high-energy peaks provides a theoretical explanation for the structured KER spectra observed in $H_2^+$ ionization experiments. Finally, we propose that this geometric energy shift offers a new perspective on the ``missing energy'' problem in particle physics. Our findings suggest that a detected energy peak lower than the statistical mean may be an intrinsic signature of 3D quantum-spatial distribution ($r^2|ψ|^2$) rather than proof of undetected particles. This work provides a rigorous framework for recalibrating experimental energy signals across atomic, molecular, and subatomic scales.

Figures (5)

• Figure 1: Comparison of normalized energy distributions P(E) between the present model (solid) and tRecX TDSE simulations (dashed). Results are shown for repulsion charges $Q=0,0.5$, and using an initial 1s hydrogenic orbital ($\zeta=1,l=0$). Peak positions show high consistency across all charge values.
• Figure 2: Parametric evolution of the energy distribution $P(E)$. (Upper) Variation with repulsive Coulomb charge $Q\in[0,2]$ for a fixed 1s orbital ($\zeta=1$,$l=0$). Both the peak position and spectral width increase with Q. (Lower) Variation with orbital exponent $\zeta\in[1,4]$ at a fixed charge $Q=1$. The distribution broadens more rapidly with increasing $\zeta$ compared to $Q$, highlighting the sensitivity of the KER to initial state localization.
• Figure 3: Analytical fitting and energy comparison of P(E). (Upper) Results for $\zeta=1,Q=1,l=0$; (Lower) Results for $\zeta=5.2,Q=1.6,l=0$. The r-fit (green) and Gamma fit (cyan) show the lowest error (<2%), identifying the radial nature of the distribution. $E_p$ and $E_k$​ denote integrated potential and kinetic energies, while Escatter​ represents the total energy in the repulsive regime. In all cases, $E_{peak}$​ remains substantially lower than Escatter​.
• Figure 4: Universal scaling of the peak-to-mean energy ratio $R_E​=E_{peak}​/\langle E \rangle$. (Upper) Variance with repulsive charge Q at $\zeta=1$, showing monotonic convergence toward 0.5 as the repulsive potential dominates. (Middle) Variance with orbital exponent $\zeta$ at Q=1, showing a precipitous drop in detection efficiency for localized states. (Lower) Contour map of $R_E​(\zeta,Q)$. The ratio is strictly bounded below 0.5, with the greatest discrepancy occurring for inner-shell analogs (high $\zeta$). High $\zeta$ values correspond to deeper initial potentials where the "quantum pressure" ($T_{local}​∼\zeta^2$) significantly skews the average energy relative to the experimental peak.​.
• Figure 5: Influence of the principal quantum number $n$ on the energy distribution $P(E)$ for $Q=1, \zeta=1$. While the primary peak position remains consistent with the $n=1$ state, a secondary high-energy peak emerges for $n=2$ and $n=3$. The differences in the high-energy cutoff for varying $n$ are attributed to limited grid spacing in the numerical treatment. The growth of this secondary feature explains the extra peaks observed in experimental KER spectra where higher shells are populated.