Common sense suggests that a particle must have a definite origin if its full path information is available. In quantum mechanics, the knowledge of path information is captured through the well-established duality relation between path distinguishability and interference visibility. If visibility is zero, high path distinguishability can be achieved, which enables one to determine with high predictive power where the particle originates. We investigate the complementarity between path information and interference visibility through an experiment involving three sources emitting into identical modes. Our findings challenge the classical intuition that a particle can be traced back to its origin through its trajectory when full path information is available. By grouping the crystals in different ways, we demonstrate that it is impossible to ascribe a definite physical origin to the photon pair, even if the emission probability of one individual source is zero and full path information is available. Our results shed new light on the physical interpretation of probability assignment and path information beyond its mathematical meaning and show that the interpretation of path information in quantum mechanics is subjective.

Combining classical density functional theory (cDFT) with quantum mechanics (QM) methods offers a computationally efficient alternative to traditional QM/molecular mechanics (MM) approaches for modeling mixed quantum-classical systems at finite temperatures. However, both QM/MM and QM/cDFT rely on somewhat ambiguous approximations, the two major ones being: (i) the definition of the QM and MM regions as well as the description of their coupling, and (ii) the choice of the methods and levels of approximation made to describe each region. This paper addresses the second point and develops an exact theoretical framework that allows us to clarify the approximations involved in the QM/cDFT formulation. We, therefore, establish a comprehensive density functional theory (DFT) framework for mixed quantum-classical systems within the canonical ensemble. We start by recalling the expression of the adiabatic equilibrium density matrix for a mixed system made of Nqm quantum and Nmm classical particles, which can be related to a partial Wigner transform. Then, we propose a variational formulation of the Helmholtz free energy in terms of the full, non-equilibrium, QM/MM density matrix. Taking advantage of permutational symmetry and thanks to constrained-search methods, we reformulate the computation of the Helmholtz free energy using only the quantum and classical one-body densities. Therefore, this paper generalizes both cDFT and electronic DFT (eDFT) to QM/MM systems. We then reformulate the functional to make the standard eDFT and cDFT Levy-Lieb functionals explicitly appear, together with a new universal correlation functional for QM/MM systems. A mean-field approximation is finally introduced in the context of solvation problems, and we discuss its connection with several existing mixed cDFT-eDFT schemes. An extension to the semi-grand canonical ensemble, where the number of classical particles is allowed to fluctuate, is provided in the supplementary material.

In this journal, Moses Gomberg's 1900 revelation, "An Instance of Trivalent Carbon: Triphenylmethyl", lauded on a centennial National Historic Chemical Landmark for challenging the "prevailing belief that carbon can only have four bonds", shifts its place in our imaginations as the facts given here are accommodated. In 1898 Gomberg presumed that he had made a molecular complex of bromotriphenylmethane and two neutral I2 molecules. But he was mistaken. Instead, Gomberg produced a mixture of three persistent single crystals of the triphenylmethyl cation before he published his aforementioned, controversial paper. Trigonal carbon coordination was the crack in the valency rules that had organized chemistry prior to the invention of quantum mechanics. Gomberg did not recognize the wealth of trivalent carbon compounds he had in hand before the proposition of the radical and corresponding cation thereafter. This work alludes to a counterfactual history.

Quantum key distribution (QKD) provides a foundation for information-theoretic security based on quantum mechanics, yet its practical deployment is often constrained by intrinsically low secure key generation rates, particularly in high-bandwidth or low-latency settings. This work introduces a hybrid cryptographic technique that integrates conventional QKD with deterministic chaos, modeled using the Lorenz attractor, to provide a software-based enhancement of the effective key expansion rate. From a short 20-bit QKD seed, the system generates long bitstreams within milliseconds; although these streams exhibit high empirical randomness, their fundamental entropy remains bounded by the seed, consistent with standard cryptographic principles. The method employs the exponential divergence of chaotic trajectories, such that even minute uncertainties in an adversary's estimate of the initial state lead to rapid desynchronization and, as established in Appendix A, an exponential decay of Eve's mutual information with respect to the expanded key. Simulation results confirm this theoretical behavior and demonstrate an effective rate amplification exceeding two orders of magnitude over the baseline QKD seed rate. The proposed chaotic expansion operates entirely in software and requires no modifications to existing QKD hardware, offering a practical pathway to enhance throughput for applications ranging from secure video communication to low-latency IoT and edge-computing environments.

Abstract Quantum decoherence, conventionally modeled as a stochastic process, remains the primary obstacle to scalable quantum computation. In this work, a conceptual framework grounded in the hydrodynamic formulation of quantum mechanics (Madelung-Bohm formalism) is presented, which reinterprets decoherence as a process governed by deterministic flow dynamics rather than fundamental
randomness. A comprehensive numerical simulation of a qubit coupled to an environmental oscillator at strong coupling (c = 1.5) has been performed, from which an ensemble of Bohmian trajectories has been extracted and analyzed. The strong coupling regime ensures conclusive, unambiguous demonstration of complete decoherence within accessible simulation times, with system purity
exhibiting clear, monotonic decay from a pure to a maximally mixed state without residual oscillations.
The results provide direct visualization of decoherence as a continuous dynamical evolution while perfectly reproducing the statistical predictions of standard open quantum systems theory, including Born rule agreement and decoherence-induced pointer state formation. The key finding is that the environmental influence—the “decoherence force” Fdec = −∇Q derived from the quantum potential possesses a deterministic, non-random structure with identifiable spectral signatures.
This insight provides the theoretical foundation for a class of Noise-Informed Control Strategies.
It is argued that by experimentally characterizing this deterministic signature via quantum noise spectroscopy, bespoke control protocols can be designed to suppress or manipulate decoherence. This work shifts the perspective from combating fundamentally random noise to controlling a structured, deterministic interaction, aligning with emerging paradigms in quantum error correction and the information-selection principles of quantum Darwinism.

Abstract Quantum mechanics has revolutionised our understanding of reality. At the heart of this theory lies Planck’s constant h , which plays a fundamental role in describing the behaviour of matter on the smallest scales. In this paper we present a teaching approach that combines a concise historical overview of the origin of h with a simple experiment in which students determine its value from measurements on a photovoltaic panel. The article integrates methodologies from Science, Technology, Engineering, and Mathematics and problem-based learning to demonstrate how contemporary educational techniques can not only aid in understanding quantum concepts, but also enable the experimental determination of one of the most fundamental constants of nature. Using low-cost components available in school and introductory university laboratories, students measure the threshold voltage of a photovoltaic cell for different wavelengths of light and, by analysing the linear dependence of the electron energy e U on c / λ to obtain an experimental estimate of h . In contrast to other works where light-emitting diodes have been used to determine Planck’s constant (Chakarvarti and Sharma 1988 Phys. Educ. 23 416; Ward 2013 Phys. Educ. 48 20–1; Morton and Abraham 1986 Phys. Educ. 21 414; Korpal et al 2023 Phys. Educ. 58 045003), this article presents a method of experimentally determining h , by measuring the threshold voltage of a photovoltaic panel as a function of the frequency of incident light. The proposed experiment can be easily conducted in a school setting, for example as a group project under the guidance of a teacher. It illustrates that core quantum concepts can be explored experimentally even at the introductory level. Furthermore, this experiment can complement university-level lectures or laboratory courses.

Machine-learned potentials (MLPs) have emerged as transformative tools for modeling metal-organic frameworks (MOFs), bridging the accuracy of quantum mechanics with the efficiency required for large-scale molecular simulations. By learning the potential energy surface directly from quantum-mechanical reference data, MLPs enable a unified description of the complex nature of MOFs and their interactions with guest molecules across multiple length and time scales. Recent developments have demonstrated the capability of MLPs to model intrinsic MOF properties such as lattice dynamics, thermal expansion, and mechanical response, as well as to describe adsorption thermodynamics, diffusion, and cooperative host-guest behavior in flexible frameworks. Developing reliable and transferable MLPs for MOFs remains a significant challenge due to the vast chemical and structural diversity of MOFs and the complexity of sampling guest-framework configurations. The lack of openly shared, standardized, and user-friendly MLP implementations also limits their broader adoption. This review focuses on the current progress in MLP-based modeling of MOFs, highlighting methodological advances, data-generation strategies, and active-learning protocols, while outlining the key challenges and future directions for developing transferable, accessible, and universal MLPs for the predictive design and discovery of MOFs.

Solid-liquid interfaces are ubiquitous in nature and engineering, and their frictional behavior remains a key factor limiting performance gains in surface engineering. However, conventional tribology has largely focused on the effect of macroscopic variables such as surface topography, which do not account for the microscopic essence of ultra-low-friction phenomena at the nanoscale. Recently, the role of quantum-scale excitations, such as electrons and phonons, in micro-/nanoscale solid-liquid friction has been increasingly emphasized. By using in situ detection techniques such as terahertz time-domain spectroscopy and non-contact atomic force microscopy, the quantum-scale friction has been observed. Its essence stems from the energy and momentum transfer induced by fluctuations in liquid charge density or electron or phonon excitations within solids. However, limited capabilities in simultaneously probing multiple physical quantities at sub-nanometer and femtosecond resolutions hinder a comprehensive understanding of the quantum origins and applications of solid-liquid interfacial friction. This review synthesizes the cutting-edge theories and experimental advances in quantum-scale solid-liquid friction and proposes a potential breakthrough path based on deep integration of simulation and experiment to address core gaps, including incomplete theoretical frameworks and constrained detection capabilities. Despite multidimensional challenges, quantum-scale friction research demonstrates substantial potential for transformative technologies,such as low-power nanofluidic devices, high-efficiency energy storage, intelligent drug delivery, and super-lubrication materials,underscoring its significance for the convergence of interfacial science, quantum mechanics, and micro/nanofluidics.

In quantum mechanics, the wavefunction of a free electron intrinsically embodies wave-particle duality, exhibiting wave- and (or) particle-like characteristics upon measurement. In solids, electrons depart from this idealized description; their low-energy excitations are quasiparticles that, within a single-particle perspective, govern material behavior. Here, we present a comparative analysis of the normal states within the many-body localization (MBL) and Landau-Fermi liquid (LFL) frameworks. We propose a phenomenological correspondence between LFL quasiparticles and the local integral of motions (LIOMs) in an MBL phase, which we term the MBL/LFL duality, by analogy to the wave-particle duality of electron in free space. Within this heuristic framework, we argue that instabilities of an MBL 'normal state' can admit analogues of Fermi-liquid instabilities, such as LIOM pairing, and we outline both mean-field and renormalization-group perspectives that highlight possible routes to exotic nonequilibrium phases. These results are primarily suggestive: where small-scale numerical illustrations are presented, they serve to exemplify the proposed scenarios rather than constitute exhaustive numerical proof. Finally, our work offers a broader perspective in which MBL may represent not only a stable endpoint but also a setting for normal-state instabilities that parallel those of conventional Fermi liquids. We hope that the MBL/LFL duality can provide a new avenue for understanding exotic nonequilibrium phases that may emerge from the interplay between localization and interaction-driven instabilities.

This article presents a novel theoretical framework that reinterprets the fundamental nature of space and time. I propose that they are not independent, pre-existing continua but are emergent as orthogonal and anti-parallel projections of a single, conserved state vector in a higher-dimensional space. The model is built upon the core axiom of an invariant norm, ||Ψ||^2 = constant, and a strong geometric condition: the vectorial projections for space (S) and time (T) are equal in magnitude but opposite in direction, expressed as S = -κT, where κ is a fundamental constant. From this foundation, I demonstrate how key features of modern physics emerge naturally. The Lorentz transformations and phenomena of time dilation are derived from the compensatory exchange between the S and T components during state evolution. The mass-energy equivalence E=mc² is reformulated as a geometric conversion law, with the speed of light c acting as the exchange constant κ. Furthermore, the cosmological arrow of time is linked to a global drift of the state vector from an initial condition of high temporal potential toward increased spatial expression. The framework offers integrated explanations for causality, black hole structure, and provides pathways for unification with quantum mechanics, suggesting that spacetime itself is a quantum-informational construct.

We study the deformed supersymmetric quantum mechanics with a polynomial superpotential with ℏ correction. In the minimal chamber, where all turning points are real and distinct, it was shown that the exact Wentzel–Kramers–Brillouin periods obey the Z 4 -extended thermodynamic Bethe ansatz (TBA) equations of the undeformed potential. By changing the energy parameter above/below the critical points, the turning points become complex, and the moduli are outside of the minimal chamber. We study the wall crossing of the Z 4 -extended TBA equations by this change of moduli and show that the Z 4 structure is preserved after the wall crossing. In particular, the TBA equations for the cubic superpotential are studied in detail, where there are two chambers (minimal and maximal). At the maximally symmetric point in the maximal chamber, the TBA system becomes the two sets of the D 3 -type TBA equations, which are regarded as the Z 4 extension of the A 3 / Z 2 -type TBA equation.

The reflection of light from a dielectric interface follows well-known optical principles, such as the equality of the angles of incidence and reflection, as well as Snell's law of refraction. These laws hold precisely when considering the behavior of a single plane wave at an optical interface. However, real optical beams, which have finite spatial and angular extents, exhibit deviations from these idealized laws. Specifically, reflected beams can undergo spatial displacements and angular deflections, collectively referred to as beam shifts. The Goos–Hänchen shift displaces the beam in a direction perpendicular to the plane of incidence, while the Imbert–Fedorov shift introduces a transverse displacement for circularly polarized light. These effects arise from the angular dispersion of the reflection coefficients and from the spin–orbit interaction of circularly polarized light. These shifts can be described using classical electromagnetism; however, this paper presents the theory behind these shifts using a highly compact quantum mechanical formulation. The shifts are small, only a fraction of the wavelength of light. By applying the weak measurement technique known from quantum mechanics, these shifts can be significantly enhanced. We introduce an experimental setup that allows such amplified shifts to be easily measured, even in student laboratories.

Consciousness and the measurement problem.

Quantum mechanics modeling on the micro-mechanical property of refractory high-entropy alloy

Bridging quantum mechanics and thermodynamics: Irreversible correlations as the arrow of time

Lifshitz quantum mechanics and anisotropic Josephson junction

Chemistry has evolved from empirical pattern recognition to a unified, physics-informed science governed by universal principles. This perspective traces the conceptual progression of chemical thought from Mendeleev’s periodic classification to thermodynamics, quantum mechanics and the emerging systems view of self-organisation and complexity. By dividing this trajectory into four historical phases viz. (i) thermodynamic and kinetic universality, (ii) nonideal solution theory and ionic interactions, (iii) quantum-mechanical interpretation of matter and bonding, and (iv) self-organisation in far-from-equilibrium systems. Each phase contributed to a deeper understanding of matter-energy relationships and strengthened the theoretical foundations of chemistry. Emphasis is placed on the interplay between the macroscopic laws and microscopic models, with recurring themes of order, symmetry and energy flow serving as unifying principles across both equilibrium and non-equilibrium phenomena. This conceptual synthesis illustrates the natural convergence of thermodynamics, statistical mechanics, and quantum theory, giving rise to systems chemistry and the modern study of emergent behaviour. Beyond its historical narrative, the work asserts that an analysis of chemistry through its evolving paradigms reveals a coherent scientific continuum integrating atomic theory, information and complexity, thereby positioning chemistry as a central discipline for elucidating organisational principles in natural systems.

This work demonstrates that an approach which makes use of magnetic circular dichroism (MCD) together with electronic circular dichroism (ECD) brings one to a rapid, sensitive, nondestructive, and inexpensive determination of the electronic structure of diamagnetic, chiral, and flexible molecular systems. The subject of this study is cobalamins (Cbls), including vitamin B12, the unique and intricate structure of which determines their selective and strong protein binding. Their existence in two forms (base-on and base-off) not only causes significant structural changes but also influences the reactivity of B12 derivatives in biologically important organometallic reactions. Therefore, recognizing the Cbl forms and understanding how they switch between them is essential. Notably, this study is the first to show that combining MCD and ECD, supported by quantum mechanics calculations, allows differentiation between base-on and base-off Cbls in aqueous environment at pH 7.4 and in acidic conditions, respectively. Furthermore, these techniques are sensitive to Cbl modifications at the meso position of the corrin macrocycle or in the axial upper ligands.

Abstract In this paper, we study the skew curvature of ruled surface in Minkowski 3-space. The skew curvature is closely related to quantum mechanics in the study of the dynamics, and is derived from Schrödinger equation on a surface. First of all, we prove that there is no linear Weingarten type ruled surfaces with non-null ruling in terms of the Gaussian, mean and skew curvatures. Also, we show that the ruled surface with null ruling (shortly, null scroll) has zero skew curvature. Finally, we give an application to construct null scrolls with zero skew curvature.

Ramsauer–townsend effect in muon scattering by nuclei in relativistic and nonrelativistic modified quantum mechanics