Generalized Boundary Conditions for the qBounce Experiment

TL;DR

This work demonstrates that the self-adjointness of the Hamiltonian for ultracold neutrons in a linear gravitational potential on the half-line necessitates a one-parameter Robin boundary condition, parameterized by $\lambda$, which interpolates between Dirichlet and Neumann limits. It provides closed-form expressions for generalized energies $E_n(\lambda)=-\mathcal{E}_0\zeta_n(\lambda)$ and eigenfunctions $\psi_n(\xi,\lambda)=\mathcal{N}_n(\lambda)\mathrm{Ai}(\xi+\zeta_n(\lambda))$, along with recursion relations for matrix elements, sum rules, and a generalized uncertainty principle that explicitly depend on $\lambda$. The paper shows that boundary physics can bias local gravity measurements in qBounce and can mimic or mask short-range forces, with $\lambda$ functioning as an effective parameter that also captures other systematic effects. The results offer a broadly applicable framework for any system where boundary conditions govern observable spectra and dynamics, and they pave the way for boundary-engineering approaches to enhance or interpret precision quantum-gravitational experiments.

Abstract

Discrepancies between theory and recent qBounce data have prompted renewed scrutiny of how boundary conditions are implemented for ultracold neutrons bouncing above a mirror in Earth's gravity. We apply the theory of self-adjoint extensions to the linear gravitational potential on the half-line and derive the most general boundary condition that renders the Hamiltonian self-adjoint. This introduces a single real self-adjoint parameter $λ$ that continuously interpolates between the Dirichlet case and more general (Robin-type) reflecting surfaces. Building on this framework, we provide analytical expressions for the energy spectrum, eigenfunctions, relevant matrix elements, and a set of sum rules valid for arbitrary $λ$. We show how nontrivial boundary conditions can bias measurements of $g$ and can mimic or mask putative short-range ''fifth-force''. Our results emphasize that enforcing self-adjointness-and modeling the correct boundary physics-is essential for quantitative predictions in gravitational quantum states. Beyond neutron quantum bounces, the approach is broadly applicable to systems where boundaries and self-adjointness govern the observable spectra and dynamics.

Figures (6)

• Figure 1: Dirichlet $\rho_{n}^{D}(x) = \lvert \psi_{n}^{D}(x) \rvert^{2}$ and Neumann $\rho_{n}^{N}(x) = \lvert \psi_{n}^{N}(x) \rvert^{2}$ probability densities and general density $\rho_{n}(x,\lambda)=\lvert\psi_{n}(x,\lambda)\rvert^{2}$ with various $\lambda$ values for $n=1\text{–}4$. We set $\mathcal{E}_{0}=x_{0}=1$ for $n=1\text{–}4$ thus $\xi=x$.
• Figure 2: Generalized energies $E_{n}( \lambda)=-\mathcal{E}_{0} \zeta_{n}(\lambda)$ according to Eq. \ref{['generalized_energies']}. We see that the energy increases as $\lambda$ increases and vice versa. The dotted and dashed horizontal lines represent the Dirichlet $(\lambda=0)$ and Neumann $(\lambda = \infty)$ energies, respectively. We set $\mathcal{E}_{0}=x_{0}=1$ for $n=1\text{–}4$ thus $\xi=x$.
• Figure 3: Approximate generalized energies in the $0 <\lambda \ll 1$ and $\lambda \gg 1$ regimes corresponding to Eqs. \ref{['dim diri case zeros']} and \ref{['dim neu case zeros']}, respectively. The solid and dashed lines represent the numerically calculated generalized energies \ref{['generalized_energies']} and approximated energies (Eqs. \ref{['dim diri case zeros']} and \ref{['dim neu case zeros']}), respectively. We set $\mathcal{E}_{0}=x_{0}=1$ for $n=1\text{–}4$ thus $\xi=x$.
• Figure 4: Transition frequency $\nu_{n,n+1}(\lambda)=\nu_{n+1}(\lambda) - \nu_{n}(\lambda)$ as a function of the self-adjoint parameter $\lambda$. The vertical dashed lines are the transition frequencies with the Dirichlet condition. We use $g_{c}=9.804925 \text{ m}/\text{s}^{2}$ thus $\mathcal{E}_{0} = 0.6016 \, \text{peV}$.
• Figure 5: Transition frequencies $\nu_{1,6}(\lambda)$ and $\nu_{2,7}(\lambda)$ as a function of the self-adjoint parameter $\lambda$. The vertical dashed lines are the transition frequencies with the Dirichlet condition. We use $g_{c}=9.804925 \text{ m}/\text{s}^{2}$ thus $\mathcal{E}_{0} = 0.6016 \, \text{peV}$.