# Experimental Distinction Between a Maxwell Universe and Quantum Mechanics **A proposal based on 1S–2S hydrogen spectroscopy** Anes Palma · August 2025 ## Abstract Source-free Maxwell dynamics for a single scalar energy field $\phi$ leads to the Schrödinger equation in the narrow-band (envelope) limit. Higher-order terms appear at order $\epsilon^{2}=(\Delta\omega/\omega_{0})^{2}$. If that term vanishes, Maxwell alone reproduces every quantitative success of quantum mechanics while retaining a deterministic field picture—no probabilistic ontology required. If it survives, a relative frequency shift of order $10^{-8}$ is predicted for the 1S–2S interval in atomic hydrogen, four orders above present experimental uncertainty. Either outcome decisively discriminates a Maxwell universe from standard quantum mechanics (QM). ## Introduction The wave equation $$ \partial_{t}^{2}\phi=c^{2}\nabla^{2}\phi $$ supports a narrow-band ansatz $$ \phi(\mathbf r,t)=\Re\!\bigl[\psi(\mathbf r,t)\,e^{-i\omega_{0}t}\bigr],\qquad \epsilon=\frac{|\partial_{t}\psi|}{\omega_{0}}\ll1, $$ which yields the Schrödinger equation at leading order. The next term in the expansion alters the effective Hamiltonian and provides a clear experimental handle. ## Derivation of the $\epsilon^{2}$ correction Keeping terms through $O(\epsilon^{2})$ gives $$ i\partial_{t}\psi =-\frac{c^{2}}{2\omega_{0}}\nabla^{2}\psi -\frac{c^{4}}{8\omega_{0}^{3}}\nabla^{4}\psi +\frac{1}{2\omega_{0}}\partial_{t}^{2}\psi +O(\epsilon^{4}). $$ With the identifications $\hbar=E_{11}/\omega_{11}$ and $m=E_{11}/c^{2}$, the effective Hamiltonian reads $$ H_\text{eff}=H_{0}+\epsilon^{2}H_{2},\qquad H_{0}=-\frac{\hbar^{2}}{2m}\nabla^{2}, $$ $$ H_{2}=-\frac{\hbar^{2}}{2m}\Bigl[\frac{\hbar^{2}}{4m^{2}c^{2}}\nabla^{4}-\frac{1}{\omega_{0}^{2}}\partial_{t}^{2}\Bigr]. $$ ## Magnitude for hydrogen For the hydrogen ground state $\Delta k\simeq a_{0}^{-1}$ gives $$ \epsilon\sim\alpha^{2}\approx10^{-4}\;\;\Longrightarrow\;\;\epsilon^{2}\approx10^{-8}. $$ The relative shift in the 1S–2S frequency is $$ \frac{\Delta f}{f_{0}}=\kappa\,\epsilon^{2}\sim\kappa\times10^{-8}, $$ i.e. $\Delta f\approx25\ \text{kHz}$ for $f_{0}=2.466\,061\,413\,187\,035\ \text{Hz}$. Current Doppler-free two-photon measurements quote uncertainties below $10\ \text{Hz}$, well inside the required range. ## Experimental protocol | Step | Action | Target value | | --- | --- | --- | | Ultracold hydrogen beam | Temperature $<50\ \text{mK}$ | Doppler width $<1\ \text{kHz}$ | | Two-photon excitation | 243 nm cavity-enhanced counter-propagating | Linewidth $<500\ \text{Hz}$ | | Frequency reference | Optical clock traceable to the SI second | Stability $<10^{-15}$ | | Systematic shifts | Stark, Zeeman, AC Stark | Controlled below $1\ \text{Hz}$ | | Data model | Fit line centre vs. carrier bandwidth | Sensitivity to $\kappa\epsilon^{2}$ | | Statistical goal | $\sigma_{\Delta f}\le5\ \text{Hz}$ | $5\times$ margin on 25 kHz prediction | ## Interpretation * If $\epsilon^{2}=0$: Maxwell reproduces all QM predictions; the experiment yields the CODATA value and supports a deterministic field ontology without probabilistic collapse. * If $\epsilon^{2}>0$: a measurable upward shift appears; QM is incomplete and new physics emerges at order $10^{-8}$. Either result is scientifically valuable. ## Conclusion Hydrogen 1S–2S spectroscopy at present technical levels can decide whether quantum mechanics is the exact description of matter or only the leading approximation to a deeper Maxwell-scalar field theory. A null result confirms the sufficiency of Maxwell; a positive result opens the door to physics beyond QM. ## References [1] A. Palma, “Deriving the Schrödinger Equation from Source-Free Maxwell Dynamics”, v2 (2025). [2] A. Palma, “Finite-Bandwidth Corrections in a Maxwell Universe” (2025). [3] T. Udem *et al.*, “Absolute Frequency Measurement of the Hydrogen 1S–2S Transition”, Phys. Rev. Lett. 125, 053001 (2020). [4] CODATA Recommended Values of the Fundamental Constants, 2022.