The numerical solution of the many-body problem, which involves interacting electrons and ions, is a key challenge in condensed matter physics, chemistry, and materials science. Traditional methods to solve the multicomponent quantum Hamiltonian are usually specialized for one kind of particles—electrons or ions—and can suffer from a methodological gap when applied to the other ones. This work extends the self-consistent harmonic approximation, a proven successful technique for simulating quantum ions at finite temperatures in anharmonic crystals, to electrons. The approach minimizes the total free energy by optimizing an ansatz density matrix, solving a fermionic self-consistent harmonic Hamiltonian on a curved manifold parameterized through a neural network. This approach preserves an analytical expression for entropy, enabling the direct computation of free energies and phase diagrams of materials. By benchmarking this technique across several prototypical cases—a double-well potential, the hydrogen atom, and the H 2 dissociation—we demonstrate that it can address both the ground- and excited-state properties of electronic systems, capture quantum tunneling and static electronic correlations, and thereby provide a unified quantum framework of electrons and atomic nuclei.

This report bridges fundamental ideas from introductory calculus to advanced concepts in quantum mechanics and nonlinear dynamics. Beginning with the behavior of second derivatives in oscillatory and exponential functions, it introduces the Airy equation and the WKB approximation as mathematical tools for describing wave propagation and quantum tunneling near turning points—locations where transitions between oscillatory and exponential components occur. The analysis then extends to the non-dissipative Lorenz model, whose double-well potential and solitary-wave (sech-type) solutions reveal a deep mathematical connection with the nonlinear Schrödinger equation. Together, these examples highlight the universality of second-order differential equations in describing turning-point dynamics, encompassing physical phenomena ranging from quantum tunneling to coherent solitary-wave structures in fluid and atmospheric systems.

The microscopic structure of several amorphous substances often reveals complex patterns such as medium- or long-range order, spatial heterogeneity, and even local polycrystallinity. To capture all these features, models usually incorporate a refined description of the particle interaction that includes an ad hoc design of the inside of the system constituents and use temperature as a control parameter. We show that all these features can emerge from a minimal athermal two-dimensional model where particles interact isotropically by a double-well potential, which includes an excluded volume and a maximum coordination number. The rich variety of structural patterns shown by this simple geometrical model apply to a wide range of real systems including water, silicon, and different amorphous materials.

Abstract This paper investigates the non-relativistic energy eigenvalues of a quantum mechanical system described by a hyperbolic potential function which forms a modified double-well potential. In the literature, the eigenfunctions (wavefunctions) and the eigenvalues of such a system have been partly revealed analytically for a particular regime of the potential parameters by use of Confluent Heun equation. Furthermore, because of the limited asymptotic knowledge of Confluent Heun Functions, a Wronskian method has been employed for numerically achieving the eigenvalues of higher energy levels. As for this study, Asymptotic Iteration Method is utilized for obtaining the energy spectrum of this system in a closed form. By favour of usage of the method in the quasi-exactly solvable systems, it is allowed to obtain an analytical expression for the energy eigenvalues, for a parameter regime in which the potential parameters are not equal to zero. As a result, it will be possible to examine how each potential parameter independently affects the energy spectrum.

We propose a framework for topological soliton dynamics in trapped spinor superfluids, decomposing the force acting on the soliton by the surrounding fluid into the buoyancy force and spin corrections arising from the density depletion at soliton core and the coupling between the orbital motion and the spin mixing, respectively. Our formulation applies to large-amplitude soliton motion in general superfluids with spin degrees of freedom under arbitrary external potentials. For ferrodark solitons (FDSs) in spin-1 Bose-Einstein condensates, the spin correction could diverge, change the direction of the total force, and enable mapping the FDS motion in a harmonic trap to the atomic-mass particle dynamics in an emergent quartic potential. Initially placing a type-I FDS near the trap center, a single-sided oscillation happens, which maps to the particle moving around a local minimum of the emergent double-well potential. As the initial distance of a type-II FDS from the trap center increases, the motion exhibits three regimes: trap-centered harmonic and anharmonic oscillations followed by single-sided oscillations. Correspondingly the emergent quartic potential undergoes a transition from a single minimum to a double-well shape, where the particle motion shifts from oscillating around the single minimum to crossing between two minima via the local maximum, then the symmetry-breaking motion around one of the two minima. In a hard-wall trap with linear potential, the FDS motion maps to a harmonic oscillator.

Intrinsic, oriented electric fields inside protein active sites, quantified by vibrational Stark effect (VSE) measurements and electrostatic calculations have been implicated in catalytic preorganization, yet their systematic use to program drug-target lifetimes has remained underexplored. We advance a field-first paradigm in which the component of the protein field projected along a dominant reaction coordinate, denoted as (the component of the protein's electric field along the reaction coordinate), serves as a tunable design variable for off-rate and residence time (τ). Focusing on two mechanistic archetypes proton sharing (Ketosteroid Isomerase, KSI-like) and bond polarization/dissociation (human aldose reductase, hALR2-like) we show that realistic changes in can tilt barriers, alter curvatures, and modulate tunneling, yielding exponential leverage on kinetics. This reframes pharmacodynamic lifetime as an electrostatic, geometry-addressable property, complementary to affinity optimization and accessible through protein mutations or ligand substituents that re-orient local dipoles. We developed a predictive, quantum-mechanical framework grounded in analytical solutions to the one-dimensional time-independent Schrödinger equation. Two experimentally validated systems were modeled viz the proton-transfer dynamics in ketosteroid isomerase (KSI) using an asymmetric double-well potential, and carbonyl polarization/dissociation in human aldose reductase (hALR2) using a modified Morse potential. The intrinsic Stark field was incorporated via its projection onto the reaction coordinate ( ), coupling through molecular dipole and polarizability terms to tilt and reshape the potential energy landscape. The resulting eigenvalues and wavefunctions provided the parameters for a Grote-Hynes-corrected transition-state theory model with explicit quantum tunneling corrections. This approach quantitatively connects field strength to activation barriers, vibrational frequencies, and ultimately the off-rate ( ), enabling the prediction of how perturbations via mutagenesis or ligand design alter residence time.

Proton tunneling via hydrogen bonds is a widespread quantum phenomenon in chemistry and biochemistry. Modeling the dynamics of proton transfer is a challenging task due to the multidimensional nature of the problem. The picture becomes even more complex in tautomerisms where multiple protons are transferred simultaneously, as occurs in the base pairs of biological molecules. In this study, we investigate the dynamics of double proton tunneling by solving the Schrödinger equation using the time-dependent Fourier grid Hamiltonian method. This approach enables straightforward calculations of the tunneling probability and the dynamics of the tunneling protons in the classically forbidden region. Notably, in a semiclassical framework, the model allows the computation of the average and instantaneous tunneling velocity, the rate constant of the double transfer, the temporal variation of the Lagrangian at each barrier penetration step, and the transition state energy. The model is formulated for both symmetric and asymmetric four-well potentials. To evaluate the predictive capability of the model, a detailed investigation of double proton tunneling in the isolated formic acid dimer is performed. The time-dependent Schrödinger equation with a two-interacting double-well potential is solved using the Fourier grid Hamiltonian method. This approach leads to an algebraic equation that can be easily solved with standard mathematical software, such as Mathematica in the Wolfram language. The solutions are expressed as discretized time-dependent wave functions, which allow for the calculation of the tunneling probability as a function of the reaction coordinates. Using semiclassical approximation we can derive the action, as well as the mean and instantaneous velocities, and determine the rate constant of double tunneling.

Abstract Nanomechanical resonators driven parametrically enable binary information encoding based on the control of their two possible vibrational phases. We present a protocol to flip the parametric phase in a graphene nanomechanical resonator via annealing, offering a novel approach to nanomechanical logic. The core of our methodology involves driving the resonator with a parametric excitation near twice of its resonant frequency and applying an external drive to break the symmetry of the dynamical double-well potential of the bistable states. By introducing white force noise to anneal the resonator, its vibrational phase settles into the state with the lower potential. The phase can be deterministically prepared in one of two states, differing by approximately π radians, by controlling the phase of direct drive and annealing. The demonstrated protocol offers a promising approach for nanomechanical logic with potential advantages in effciency, error resilience and scalability.

Hydrogen atom dynamics in double-well hydrogen bond potentials: a case study of O–H...O bonds of potassium dihydrogen phosphate, tris-potassium hydrogen bisulphate and urea–phosphoric acid crystals

Abstract Based on the confluence properties of a second-order differential equation applied to the general Heun’s equation, an appropriate form of the confluent Heun differential equation is obtained. Then, under a general transformation derived from an auxiliary function, the confluent Heun equation is transformed into a Schrödinger-like equation. Such auxiliary function allows finding the appropriate change of variable to arrive at a Schrödinger equation with a multiparameter hyperbolic-type potential. Under an appropriate selection of the involved parameters, families of symmetric double-well potential are obtained. It is shown that a particular form of the parameters gives place to the Downing potential, which is a double-well potential given in terms of hyperbolic functions. Likewise, the proposal can be extended to the case of asymmetric double-well potentials and triple-well ones that could be useful in physics and quantum chemistry.

Nuclear quantum effects are associated with the finite spreading of the vibrational wavefunction, which may extend over different basins of attractions or minima in the underlying energy landscape. These effects add up to the thermal broadening that naturally occurs at finite temperature. In this paper we introduce different measures to quantify the extent of delocalisation in both spatial and energy domains. These measures rely on the statistics of inherent structures or local minima that are conveniently explored in numerical simulations based on path-integral approaches. The ideas are illustrated on a simple double-well one-dimensional potential, for which the thermal nuclear wavefunction can be determined exactly from the solution of the Schrödinger equation. Application to condensed para-hydrogen modelled by the Silvera-Goldman potential at various temperatures and across the melting transition shows how these indicators are robust against Trotter discretisation number. For this highly quantum system, landscape delocalisation is significant already in the solid state.

Context: Quantum mechanical proton tunneling is a recognized but often phenomenologically treated factor in enzymatic catalysis and transport. Its explicit role as a governing principle in drug-protein binding and pH-triggered release kinetics remains poorly understood, lacking a predictive theoretical framework that connects nanoscale cavity geometry to functional pharmacological outcomes. Method: We developed an analytical model for proton confinement in drug-protein cavities by solving the one-dimensional time-independent Schrödinger equation, reduced from a Nuclear Time Dependent Schrödinger equation for two fundamental potential classes viz an asymmetric double-well potentials representing localized enzymatic transfer and periodic Kronig-Penney potentials for delocalized conduction along proton wires. Analytical eigenproblems yield low-lying states and tunneling splittings, cross-checked by instanton/WKB estimates; KP/TB bands and a WKB bridge provide an effective hopping . Within the parameter ranges explored, our computations indicate that sub-ångström changes in donor–acceptor distance can modify tunneling probabilities and KIEs by factors on the order of ∼5−20, and produce apparent pKₐ shifts on the order of ∼0.3−0.5 units, leading to roughly one–to–two-order changes in and associated half-lives. In the extended sector, diagonal energetic disorder reduces localization length in a manner consistent with Anderson-type localization, and a modeled pH step on a proton wire can bias a confined cavity on ps–ns timescales, contingent on the adopted couplings. These results are model-based and meant to provide design implications not absolute predictions highlighting how cavity geometry and hydrogen-bond network order may be tuned to modulate tunneling-assisted kinetics.

With the rise of quantum computing, interest has grown in using two-state quantum systems (qubits) at the secondary level to foster students’ conceptual understanding. Quantum measurement, in particular, is central to quantum theory and its accurate conceptualization by students is crucial for grasping fundamental quantum principles. However, instructional methods typically make use of different contexts (i.e., different two-state systems), significantly affecting students’ conceptual development in quantum physics. In this paper, we report findings from a cluster-randomized field trial involving 181 students taught through three inquiry-based, two-state approaches: the which-path-encoded single-photon, the polarization, and the double-well potential approach. All three approaches supported students’ conceptual development, yet students taught using photon polarization and the double-well potential significantly outperformed those participating in a course following the which-path-encoded single-photon approach. Our findings indicate that students participating in the which-path-encoded single-photon approach often retain mixed-thinking frameworks, whereas those taught with photon polarization or the double-well potential approaches were more likely to develop toward quantum thinking. Thus, our findings underpin how influential the choice of (experimental) context is on students’ conceptual development (also) in quantum physics.

Abstract In this work, we introduce a PT -symmetric infinite-dimensional representation of the U z ( s l ( 2 , R ) ) Hopf algebra, and we analyse a multiparametric family of Hamiltonians constructed from such representation of the generators of this non-standard quantum algebra. It is shown that all these Hamiltonians can be mapped to equivalent systems endowed with a position-dependent mass. From the latter presentation, it is shown how appropriate point canonical transformations can be further defined in order to transform them into Hamiltonians with constant-mass over suitable domains. By following this approach, the bound-state spectrum and the corresponding eigenfunctions of the initial PT -symmetric Hamiltonians can be determined. It is worth stressing that a relevant feature of some of the new U z ( s l ( 2 , R ) ) systems here presented is found to be their connection with double-well and Pöschl-Teller potentials. In fact, as an application we present a particular Hamiltonian that can be expressed as an effective double-well trigonometric potential, which is commonly used to model several relevant systems in molecular physics.

Escape dynamics of active particles plays a critical role in understanding biochemical reactions. Previous experiments have revealed multimodal protein diffusion in live Escherichia coli, with a distribution of diffusion modes peaked in the subdiffusive regime. However, the influence of such multimodal diffusion on active transport dynamics remains unclear. To address this, we analyze the escape rates of active Brownian particles in a double-well potential by comparing particles with multiple diffusion modes to those with a single mode. For particles exhibiting a single diffusion mode, escape rates increase significantly in subdiffusive regimes as the active noise strength grows, while superdiffusive regimes show slight variations. This occurs because subdiffusive particles are more likely to be trapped in the potential well, which thereby reduces the probability of barrier crossing. Building on these results, we find that multiple diffusion modes can modulate average escape rates of active Brownian particles. For distributions centered on normal or superdiffusive modes, the average escape rate increases with the probability weight of the central mode. In contrast, when the central tendency of the diffusion-mode distribution lies in the subdiffusive regime, the average escape rate exhibits bidirectional behavior, either increasing or decreasing, which depends on the relative weights of diffusion modes. Furthermore, a comparative analysis of Gaussian diffusion-mode distributions against Laplace, Cauchy, and experimentally derived distributions reveals that heavy-tailed characteristics in the superdiffusive regime can enhance average escape rates beyond those predicted by the Gaussian distribution.

The introduction of an asymmetric term into the quantum Rabi model generally lifts energy-level degeneracies. However, when the asymmetry parameter takes specific multiples of the bosonic mode frequency, level degeneracies reappear-a phenomenon referred to as the hidden symmetry in the asymmetric quantum Rabi model. Identifying the origin of this hidden symmetry and its explicit operator form constitutes two central tasks in studying this system. Here, we investigate the origin of this hidden symmetry using the method of two successive diagonalizations, with a focus on physics in the regime where the ratio between the two-level splitting Δ and the mode frequency ω satisfies Δ/ω ≫ 1. We find that the hidden symmetry stems from energy-level matching within the asymmetric double-well potential, a picture strongly supported by the wavefunctions of both the ground and excited states. Moreover, the emergence of an excited-state quantum phase transition is identified and qualitatively discussed, which arises from the breaking and restoration of this hidden symmetry across different coupling regimes. Our results provide deeper insights into the physics of the asymmetric quantum Rabi model, particularly in the previously less-explored strong-coupling regime, where Δ/ω ≫ 1.

Hydrogen bonds in molecular crystals are often modeledas double-wellpotentials, yet direct evidence linking this potential form to vibrationalspectroscopic features remains elusive. In this study, we investigateα-glycine, a hydrogen-bonded crystal that exhibits pronouncedRaman anomalies without undergoing a structural phase transition.Through temperature- and polarization-dependent Raman spectroscopy,supported by isotope substitution and first-principles calculations,we identify two peaks whose behavior violates conventional Raman selectionrules. These peaks merge and narrow anomalously with temperature,an effect that cannot be explained by harmonic models or thermal broadening.Simulated spectra based on a weakly evolving asymmetric double-wellpotential reproduce this merging, indicating that both peaks originatefrom one double-well potential. Our results establish α-glycineas a model system directly linking microscopic hydrogen-bond potentialsto vibrational spectroscopic features.

Coarse-grained (CG) models provide an effective route to reduce the complexity of molecular simulations, but conventional approaches depend heavily on long, all-atom molecular dynamics trajectories to adequately sample the configurational space. This data dependence limits accuracy and generalizability, as unvisited configurations remain excluded from the resulting CG models. We introduce a fully data-free, generative framework for coarse-graining that directly targets the all-atom Boltzmann distribution. The model defines a structured latent space comprising slow collective variables, associated with multimodal marginal densities capturing metastable states, and fast variables, represented through simple, unimodal conditional distributions. A learnable, bijective map from latent space to atomistic coordinates enables the automatic and accurate reconstruction of molecular structures. Training relies solely on the interatomic potential and minimizes the reverse Kullback-Leibler (KL) divergence via an energy-based objective. To stabilize optimization and ensure mode coverage, we employ an adaptive tempering scheme that promotes the exploration of diverse configurations. Once trained, the model can generate independent, one-shot equilibrium samples at full atomic resolution. Validation on two synthetic systems, a double-well potential and a Gaussian mixture model, as well as on the benchmark alanine dipeptide, demonstrates that the method captures all relevant modes of the Boltzmann distribution, reconstructs atomic configurations with high fidelity, and automatically learns physically meaningful CG representations. These results suggest that the proposed framework provides a promising, data-free alternative to traditional CG techniques, offering both a principled approach to addressing the long-standing "chicken-and-egg" challenge in coarse-graining and an effective solution to the back-mapping problem by enabling the accurate reconstruction of all-atom configurations.

Unraveling the switching dynamics in a quantum double-well potential

Abstract We report the discovery of bistable polar states with switchable polarization in the Janus monolayer 1T-MoSSe, induced by symmetry breaking in its chalcogenide atomic layers. Our results demonstrate that Janus 1T-MoSSe exhibits two out-of-plane bistable polar states with switchable polarization, rather than a polarization emerging from a non-polar phase, which is an unconventional form of ferroelectric-like behavior. First-principles calculations and phenomenological models reveal that the inequivalent stacking of sulfur and selenium (S/Se) atoms breaks the central inversion symmetry, activating non-degenerate phonon modes at the K-point (K$_2$/K$_3$) that drive the structural transformation of metastable d1T$_\mathrm{S}$ and d1T$_\mathrm{Se}$ phases. This coupling leads to bipolar control of out-of-plane polarization through atomic displacements and charge redistribution, resulting in a polarization change of $\Delta P \approx \pm 0.3\ \mathsf{μ}\mathrm{C/cm}^2$. The Landau free energy model indicates that anharmonic terms and inter-mode coupling give rise to an asymmetric double-well potential, which is crucial for bistable polar states. Molecular dynamics simulations show that the d1T$_\mathrm{S}$-phase remains stable at high temperatures, whereas the d1T$_\mathrm{Se}$-phase undergoes an irreversible phase transition near 300 K, accompanied by a Peierls-like distortion of the Mo atomic chain. This phase transition is driven by differences in electronegativity, atomic radius, and d–p orbital hybridization between S and Se. Our findings provide a theoretical framework for designing nonlinear responses in 2D ferroelectrics and suggest that low-energy polarization reversal at room temperature can be achieved through strain or electric field control, with significant implications for non-volatile memory and piezoelectric sensors.