Physics
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More and more materials, with a growing variety of properties, are built into electronic devices. This is motivated both by increased device performance and by the studies of materials themselves. An important type of device is a Josephson junction based on the proximity effect between a quantum material and a superconductor, useful for fundamental research as well as for quantum and other technologies. When both junction contacts are placed on the same surface, such as a two-dimensional material, the junction is called ``planar". One outstanding challenge is that not all materials are amenable to the standard planar junction fabrication. The device quality, rather than the intrinsic characteristics, may be defining the results. Here, we introduce a technique in which nanowires are placed on the surface and act as a shadow mask for the superconductor. The advantages are that the smallest dimension is determined by the nanowire diameter and does not require lithography, and that the junction is not exposed to chemicals such as etchants. We demonstrate this method with an InAs quantum well, using two superconductors - Al and Sn, and two semiconductor nanowires - InAs and InSb. The junctions exhibit critical current levels consistent with transparent interfaces and uniform width. We show that the template nanowire can be operated as a self-aligned electrostatic gate. Beyond single junctions, we create SQUIDs with two gate-tunable junctions. We suggest that our method can be used for a large variety of quantum materials including van der Waals layers, topological insulators, Weyl semimetals and future materials for which proximity effect devices is a promising research avenue.
This study investigates the coupling effect of interface orientation and loading direction on the interface structure and evolution of Cu/Ag nanolayered composites by employing molecular dynamics (MD) simulations. Four distinct interface configurations Cu(001)/Ag(001), Cu( 11 ̅ 0 )/Ag( 11 ̅ 0 ), Cu(111)/Ag(111) and Cu(112 ̅ )/Ag(112 ̅ ) were subjected to tensile loading both perpendicular and parallel to the interface. Results show that the initial lattice mismatch leads to the formation of characteristic dislocation networks (square, triangular, and rectangular) at the interfaces, whose morphology is dictated by the specific orientation combination. The loading direction critically governs the subsequent defect nucleation and propagation pathways.Under perpendicular loading, dislocation nucleation preferentially initiates in the softerAg layer before transmitting into the Cu layer. In contrast, parallel loading promotes dislocation emission directly from the interface into both adjacent layers, with the Cu side often exhibiting more rapid plastic development. The mechanical response and the evolution of dislocation density, including the formation of sessile stair-rod dislocations, 2 / 30 are strongly dependent on both the interface type and the loading axis. Furthermore, the implications of varying dislocation densities on electrical resistivity are discussed. This work provides atomic-scale insights into the coupling effect of interface structure and loading condition, offering guidance for the interfacial design and processing of highstrength, high-conductivity layered composites.
Layered nickelates are believed to exhibit superconductivity similar to that found in the cuprates. However, the precise crystal structure of the superconducting phase of the layered nickelates has not been fully clarified. Here, I use first principles calculations to study the pressure dependence of the structural instabilities in the single-layer-trilayer La<sub>3</sub>Ni<sub>2</sub>O<sub>7</sub>, which is one member of the layered nickelates family that also shows signatures of superconductivity. I find a nearly dispersionless nondegenerate phonon branch in the parent P4/mmm phase that is unstable along the Brillouin zone edge M (½, ½, 0) → A (½, ½, ½) at all investigated pressures up to 20 GPa. Calculations show additional doubly-degenerate instabilities along the edge MA at lower pressures. I used group-theoretical analysis to identify the distinct low-symmetry distortions possible due to these instabilities and generated them using the eigenvectors of the unstable modes. Structural relaxations show that the lowest energy structures at 0 and 10 GPa involve condensation of both the nondegenerate and doubly-degenerate instabilities, which is in contrast to the experimental refinements that involve condensation of only the doubly-degenerate branch. I also find that structural distortions are energetically favorable at 20 GPa, contrary to the experiments that do not observe any distortions of the parent P4/mmm structure at high pressures.
Anti-site disorder (ASD), present in Heusler alloy thin films, is known to significantly affect their physical properties, but a complete understanding of the actual role of ASD is lacking. In this work, we systematically investigate the effect of ASD on electrical resistivity ρ(T ) and transverse magnetoresistance MR ⊥ in off-stoichiometric Co-Fe-Ti-Si (CFTS) thin films across the thermoelastic martensitic phase transformation (MPT). The CFTS films with A2 ASD exhibit a negative temperature coefficient of resistivity (n-TCR) and an upturn below ∼ 30 K. In sharp contrast, the partially L21-ordered films are metallic in nature, characterized by a resistivity minimum at low temperatures (Tmin ∼ = 30 K) and a positive TCR for T > Tmin. The change in the sign of TCR finds a straightforward explanation in terms of the competition between the quantum corrections (weak localization, electron-diffuson scattering) and the ballistic scattering mechanisms (electron-magnon and electron-phonon). We find that, stronger the atomic ASD, more prominent the quantum corrections and the weaker the scattering of e -m and e -p scattering. All the CFTS films exhibit a distinct thermal hysteresis and a significant drop in resistivity, symptomatic of a MPT, near the characteristic temperatures: martensite-end TMe ∼ = 300 K and austenite-begin TAb ∼ = 325 K. Regardless of the strength of ASD, in the martensite phase the anti-symmetric (ASMR) component of MR ⊥ (H) dominates over the symmetric (SMR) counterpart, whereas the reverse is true (SMR ≫ ASMR) for the austenite phase at temperatures TAb ∼ = 325 K ≤ T ≤ 375 K, where MR⊥ increases very sharply with temperature as the austenite phase grows rapidly at the expense of the martensite phase. The present results assert that the CFTS Heusler alloy thin films are promising candidates for shape-memory devices and for spintronic applications such as spin valves.
Topological insulators are generally believed to be highly immune to defects. However, its performance may become complicated depending on the specific application scenario. Taking monolayer mesoscale stanene ribbons as a model system, we investigated how the two realistic factors, the spatial distributions of vacancy defect and the device dimensions, would influence topological edge conductance. The electronic transport is calculated by an effective tight-binding model with parameters fitted through genetic algorithm. It is found that the edge states are more vulnerable than previously expected in the presence of random defects. For defect free ribbons, the calculated current is sharply localized at the edges with a typical ideal conductance of 2e2/h. It might be expected that the vacancy defect solely distributed in the edges will have larger effects on the edge conductance than those solely within the interior. On the contrary, the former have marginal effects as the current simply bypasses them, whereas the latter can lead to an appreciable decline in edge conductance via the hybridization of the interior defect states with the edge states. The most pronounced degradation of conductance occurs when defects are present at both edge and interior. The calculated local current reveals that the electrons traverse between edges through relay scattering among edge-interior defects, giving rise to back scattering and loop current. Increasing the ribbon width can help to reduce the inter-edge scattering and hence enhance the stability of the topological edge states. Whereas in a longer ribbon, the electrons will encounter more defects in the longer conduction path, increasing inter-edge scattering and weakening the topological edge conductance. Our study highlights the critical roles of spatial vacancy defect distribution and ribbon size in tuning the performance of topological devices.
Layered perovskites exhibit a variety of novel physical phenomena, many of which are closely related to the crystal-field splitting of the transition-metal d orbitals. However, several compounds (e.g., Sr2IrO4 and Sr2CrO4) show an unexpectedly different sign of the crystal field splitting from the one inferred by the conventional crystal-field theory. Whether this sign inversion is common for layered perovskites is not clear yet. In this work, we study the anomalous crystal-field splitting in the A2BL4 family by combining density functional theory calculations with low-energy tight-binding modeling based on Wannier functions. We show that in the compounds with such a sign inversion the layered structure together with the compression of A-cubes triumphs over the local elongation of BL6 octahedra in determining the crystal-field splitting. The results reveal that the anomalous crystal-field splitting is a common feature for layered perovskites. This work sets an important caution mark about the application of the conventional crystal-field theory to these materials.
One-dimensional stratified optical media provide a remarkably rich platform for exploring wave propagation, spectral theory, and topology within a mathematically transparent setting. In this Perspective, we present a unified framework for analyzing electromagnetic waves in layered structures based on the Hill operator, Floquet theory, and the monodromy (transfer) matrix formalism. We emphasize the central role of the monodromy matrix as a finite-dimensional representation of the Floquet operator. This representation establishes a link between dispersion relations, band formation, Bloch waves, and geometric phases. Building on this foundation, we develop the topological characterization of one-dimensional photonic bands through Zak phases and their symmetry protection, and establish a direct correspondence between geometric phases and the real-space structure of Wannier functions. This correspondence provides an intuitive explanation for topological interface states in dimerized bilayer photonic structures, placing stratified optical media on equal footing with paradigmatic models such as the Su--Schrieffer--Heeger chain. We further extend the formalism to time-modulated and space-time periodic media, where temporal Floquet harmonics generate synthetic dimensions and promote the Hill operator to a matrix-valued Floquet--Hill operator. Within this extended setting, phenomena such as frequency conversion, nonreciprocity, and topologically protected Floquet interface states arise naturally from the geometry of the underlying Floquet spectrum. By combining analytic operator methods, transfer-matrix techniques, and modern topological concepts, this Perspective highlights stratified photonic media as a versatile and conceptually unifying platform for studying wave physics across static, driven, and space-time periodic systems.
The potassium-doped ceramics Pr₀.₆Sr₀.₄₋ₓKₓMnO₃ (with x = 0.05 and x = 0.1) were investigated through comprehensive heat capacity measurements to evaluate their magnetocaloric properties under magnetic fields up to 2 T across a broad temperature range. Complementary magnetization analyses were also conducted to support the findings.
Heat capacity data revealed a paramagnetic-to-ferromagnetic phase transition occurring at 298 K for x = 0.05 and at 292 K for x = 0.1. Application of a magnetic field resulted in a shift of the transition temperature toward higher values in both compositions.
Indirect estimations of the magnetic entropy change (ΔSₘ) and adiabatic temperature change (ΔTad) at the maximum applied field yielded values of 2.29 J/kg·K and 1.22 K for x = 0.05, and 2.64 J/kg·K and 1.30 K for x = 0.1, respectively-highlighting the magnetocaloric potential of these materials. These entropy change values were further compared with those derived from magnetization measurements, showing good consistency.
Additionally, the relative cooling power (RCP) was estimated and benchmarked against representative manganite systems. The obtained RCP values were 50 J/kg for x = 0.05 and 66 J/kg for x = 0.1 under a 2 T field, aligning well with values reported for similar compounds.
The dependence of both ΔSₘ and ΔTad on the magnetic field followed a power-law behavior, in fair agreement with existing literature, reinforcing the reliability of the observed magnetocaloric effects.
Tantalum oxynitride (TaON) is a promising material for the photocatalytic water splitting reaction. However, the bandgap of its stable β-TaON form is too high for efficient visible-light absorption, while the anatase-type δ-TaON phase with a smaller gap of 2.1 eV is metastable in the bulk. In this study, anatase-type TaON was successfully synthesized via nitridation of Ta2O5 oxide particles with a nitridation–oxidation cycle treatment to achieve increased surface area and stabilize δ-TaON. Photocatalysis experiments demonstrated for the first time that δ-TaON, with an absorption edge at approximately 600 nm, shows clear photocatalytic activity for both hydrogen and oxygen evolution half-reactions with sacrificial reagents. Moreover, δ-TaON was an active Z-scheme oxygen evolution photocatalyst in the water splitting reaction. Our approach suggests a new method for surface area control and expands the potential range of applications of oxynitrides, including TaON.
Stimuli-responsive polymeric nanocarriers have emerged as a promising strategy for targeted and controlled drug delivery, addressing limitations such as premature drug release and low bioavailability. Developing systems that respond to multiple physiological triggers is crucial for improving therapeutic precision and reducing side effects. The present study introduces the development of a unique amphiphilic random copolymeric nanocarrier with pH- and redox-responsiveness for advanced drug delivery applications. The copolymer incorporates hydrophilic carboxylic acid groups, hydrophobic coumarin moieties, and disulfide linkages, enabling self-assembly into spherical aggregates with a coumarin-rich core and a hydrophilic surface. These aggregates exhibit a low critical micelle concentration (0.006 mg mL<sup>-1</sup>), ensuring stability in dilute environments. Spherical nanocarriers remain stable under acidic conditions (∼pH 2.0) and disassemble at neutral to basic pH (∼pH 7.4), mimicking gastrointestinal conditions. This property allows site-specific, stimulus-triggered drug release, with high drug-loading efficiency. The stimuli-responsiveness of the nanocarrier not only enhances oral drug delivery but also offers a versatile platform for encapsulating a wide range of therapeutics and diagnostics. This multifunctional nanocarrier system opens new avenues for personalized medicine and advanced material technologies, highlighting its potential for broader translational and industrial applications.
Conductive lubricants are crucial to resolving the conflict between friction wear and signal transmission in electrically powered components. Here, the excellent conductivity of ionic liquids was used to design and prepare an ionic supramolecular gel lubricant. An ionic liquid and urea-based functional groups were combined into an ionic gelator, which formed a 3D network in the base oil to form a gel that prevented migration of the base oil at the interface. The ionic supramolecular gel lubricant displayed exceptional mechanical and thermal responsiveness that facilitated reversible gel-to-sol transitions through modulation of the shear stress and temperature. Rheological tests demonstrated that the lubricant possessed high viscoelasticity, shear-thinning behavior, and creep recovery properties. The lubricant also demonstrated an exceptional tribological performance both with and without an applied electric field. Without the electric field, the lubricant reduced the friction coefficient by 42.1% and decreased the wear volume by 95.6%. Under the current-carrying conditions of 3 and 6 A, the lubricant maintained a stable friction coefficient demonstrating excellent adaptability. During carrier-driven friction tests, the ionic gelator accumulated extensively on the friction-pair surface, which resulted in enhanced adsorption properties and the formation of a denser lubricating film. The high conductivity and excellent tribological performance of the developed lubricant render it highly applicable to ensuring the stability of critical components in electronic devices and energy components.
Triboelectric nanogenerators (TENGs) have been widely explored for wearable motion monitoring, yet existing designs are mainly confined to single-site sensing, limiting their ability to capture coordinated multijoint movements. Here, a multinetwork conductive hydrogel with both robustness and conductivity was developed and further utilized to construct a TENG sensor, which demonstrated outstanding durability by maintaining stable output voltage over 3000 operating cycles with 6.8% signal decay after 15 days of storage (25 ± 2 °C), as well as rapid response and recovery times of 34 and 15 ms. To capture motion signals, TENG sensors were strategically deployed at the plantar, knee, shoulder, and wrist. Building upon this sensing platform, two integrated systems were established through the integration of a data processing hardware module and a personal computer (PC) software module: a real-time kinetic chain signal monitoring system (RKCSMS) and a wireless intelligent sports evaluation system (WISES). The two systems enabled accurate movement recognition, systematic assessment of kinetic chain patterns, and real-time feedback. This study demonstrated the strong potential of TENGs based on conductive hydrogels for applications in flexible smart wearables, rehabilitation assessment, sports training, and health management.
Na<sub>2</sub>FePO<sub>4</sub>F (NFPF) is a promising cathode material for sodium-ion batteries, yet it is still confronted with a cost-performance trade-off. In this study, a low-cost preparation method of high-performance NFPF cathode materials was proposed via an Al-doping strategy, using commercial FePO<sub>4</sub> without premodification. The effect of the Al content on the crystal structure, sodium-ion diffusion kinetics, and electrochemical performance of NFPF was investigated in detail. X-ray diffraction and Rietveld refinements confirmed the high-purity phase. Na<sub>1.98</sub>Fe<sub>0.98</sub>Al<sub>0.02</sub>PO<sub>4</sub>F/C (NFPF-Al0.02) showed an excellent rate capability (57.7 vs 44.5 mAh·g<sup>-1</sup> at 5.0 C) and better cycling stability (93.6% vs 66.7% after 100 cycles at 0.5 C) than the undoped NFPF sample. NFPF-Al0.02 also exhibited an excellent capacity retention of 84.6% after 1000 cycles at 5.0 C. GITT, CV, and EIS results revealed an order-of-magnitude increase in the sodium-ion diffusion coefficient of Al-doped NFPF-Al0.02 compared to that of pristine NFPF. A cost-performance comparison between this work and previous studies was conducted. In summary, this innovative method, which can produce high-performance NFPF cathode materials at low cost, is anticipated to significantly promote the commercialization of NFPF-based sodium-ion batteries.
Effective healing of infected wounds is hindered by the inability of single-function materials to dynamically address the distinct biological phases of infection control and tissue regeneration. Herein, we present a versatile drug-loaded nanocomposite hydrogel platform (ML-OCP) that employs a sequentially coordinated strategy for efficient infected wound repair. The ML-OCP hydrogel was fabricated by incorporating l-arginine-loaded mesoporous polydopamine nanoparticles (MPDA@l-Arg NPs) into a dual-network matrix, which was formed through the Schiff base linkage between oxidized hyaluronic acid (OHA) and carboxymethyl chitosan (CMCS), along with hydrogen bonding from polyvinylpyrrolidone (PVP). ML-OCP hydrogel exhibits excellent tissue adhesiveness, self-healing ability, and mechanical properties, enabling its adaptation to the dynamic wound environment and rapid hemostasis. Building upon this stable foundation, the hydrogel implements a sequentially coordinated therapeutic strategy: the MPDA@l-Arg NPs provide immediate antibacterial (via photothermal therapy, PTT) and antioxidant actions to control early infection, while the sustained release of l-arginine ensures continuous bioactive support for the subsequent proliferation and remodeling. In a rat model of infected full-thickness skin defect, ML-OCP hydrogel demonstrated significant antibacterial activity, promoted angiogenesis, and reduced inflammation. Collectively, this work provides a versatile hydrogel platform based on a sequentially coordinated strategy, offering a new therapeutic approach for infected wounds.
In order to implement a three-dimensional (3D) 2-transistor-0-capacitor (2T0C) DRAM cell using a thermally sensitive InGaZnO (IGZO) channel, the impact of electrical interactions between the write transistor (WTr) and the read transistor (RTr) in a novel structure was investigated, and a control methodology was established. First, the adoption of a discrete active island pattern led to the suppression of parasitic channels. As a result, the subthreshold hump was eliminated, and an excellent subthreshold swing characteristic of 154.4 mV dec<sup>-1</sup> was achieved. Second, memory characterization revealed that electrical coupling effects resulting from parasitic capacitance and electrostatic effects in the 3D stacked structure compromise the storage node voltage (<i>V</i><sub>SN</sub>) charging efficiency. These can be attributed to the extended BE, which serves to reduce the device footprint, and the voltage applied to activate the RTr to read the <i>V</i><sub>SN</sub> value, respectively. Finally, the strategic placement of the asymmetric S/D electrodes in the vertical-channel thin-film transistor ensured excellent operational stability with an SN variation below 0.03 V after 1000 s. Consequently, long-term linear multilevel operation of 3 bits was achieved under various write operation conditions.
Chitosan-based (CS-based) materials have attracted considerable attention owing to their excellent biocompatibility and intrinsic hemostatic activity, rendering them promising candidates for emergency hemorrhage control. Nevertheless, their clinical performance is often constrained by inadequate wettability and limited mechanical strength. In this study, we developed a superelastic hemostatic sponge (HMCT-NP) through a facile freeze-drying approach by incorporating hydrophobically modified CS, tannic acid (TA)-mediated cross-linking, and functional Fe-baicalin nanoparticles (Fe-Ba NPs). The grafted hydrophobic alkyl chains can insert into the membranes of red blood cells (RBCs) and platelets, thereby promoting their active adhesion and aggregation to accelerate rapid coagulation. TA enhances the mechanical properties of the sponge via hydrogen-bond-mediated cross-linking while also providing antibacterial and antioxidant functionalities. The incorporation of nanoparticles enhanced the antibacterial and antioxidant properties of the sponge and, notably, led to a significant improvement in its mechanical robustness. Through this modular design and synergistic functional enhancement, HMCT-NP effectively mitigates the intrinsic poor wettability of CS-based hemostatic sponges, demonstrating a water uptake capacity of approximately 95 g/g and a volumetric expansion greater than 200% upon hydration, thereby enabling rapid fluid imbibition and enhancing blood cell aggregation at the bleeding interface. Furthermore, its high compressibility and rapid fluid-triggered shape recovery enable effective deployment in narrow or deep wounds while maintaining biosafety and minimizing tissue irritation. In various bleeding models, HMCT-NP sponge demonstrated enhanced procoagulant activity and hemostatic performance. Meanwhile, the sponge effectively accelerated the healing of infected wounds. Collectively, these results underscore the potential of the HMCT-NP sponge as a versatile and promising strategy for clinical hemorrhage management.
Inflammatory bowel disease (IBD) is persistent or recurrent intestinal inflammation. The severity of IBD is highly associated with an imbalance between M1 and M2 macrophages. In this study, a novel strategy is designed to modulate macrophage polarization by reducing intracellular reactive oxygen species (ROS) levels and regulating mitochondrial function. A scavenging ROS nanozyme, titanium carbide (Ti<sub>3</sub>C<sub>2</sub>), is synthesized to self-assemble with mannose-modified trimethyl chitosan to form nanoparticles (TTCM). The synthesized TTCM exhibits an excellent biocompatibility. Additionally, it remains stable in simulated gastric fluid (SGF) and simulated intestinal fluid (SIF). In vitro experiments show that TTCM significantly reduced ROS levels, restored mitochondrial membrane potential, increased superoxide dismutase (SOD) activity, and suppressed the TLR4/NF-κB pathway, hence promoting the repolarization of pro-inflammatory M1 macrophages toward the anti-inflammatory M2 phenotype. In addition, we found that TTCM amplified antioxidant efficacy when compared with Ti<sub>3</sub>C<sub>2</sub>. These effects enhanced the intestinal barrier. In vivo experiments utilizing a dextran sulfate sodium (DSS)-induced acute colitis mouse model revealed the anti-inflammatory and intestinal-barrier-protective effects of TTCM, effectively mitigating IBD progression. Consequently, the findings suggest that the oral delivery of ROS-scavenging nanocarrier systems holds significant promise as a potential and effective therapeutic strategy for IBD treatment.
Transition-metal trichalcogenides, distinguished by their quasi-one-dimensional (quasi-1D) structures, hold substantial potential for advanced technologies, yet their supercapacitive behavior remains largely unexplored. Here, solution-exfoliated zirconium trisulfide (ZrS<sub>3</sub>) nanoribbons are introduced as a high-performance electrode material for supercapacitors. Density functional theory analysis reveals that quasi-1D ZrS<sub>3</sub> possesses a significant density of states near the band edges, giving rise to intrinsically high quantum capacitance as an electronic property. Solution-based exfoliation enables efficient production of ZrS<sub>3</sub> nanoribbons from bulk microparticles, yielding a quasi-1D structure with considerably increased surface area. In electrochemical measurements, the combination of the quasi-1D structure and large quantum capacitance allows the exfoliated ZrS<sub>3</sub> nanoribbons to achieve a high specific capacitance of 284 F g<sup>-1</sup>, exceeding that of the bulk microparticles by more than 40-fold. These results demonstrate that solution-based dimensional engineering enhances interfacial charge storage, while the intrinsically high quantum capacitance supports efficient electronic charge accommodation, highlighting ZrS<sub>3</sub> nanoribbons as promising supercapacitor electrode materials.
The development of efficient and stable near-infrared (NIR) phosphor-converted LEDs has significantly advanced NIR photonics. However, widely reported Cr<sup>3+</sup>-activated NIR luminescent materials remain limited in performance and fail to satisfy the increasing demands of diverse applications. Herein, a Cr-free alternative utilizing a rigid garnet host is proposed. Through the design and synthesis of a novel Fe<sup>3+</sup>-activated phosphor, Ca<sub>3</sub>Sn<sub>2</sub>Ga<sub>2</sub>SiO<sub>12</sub>:Fe<sup>3+</sup> (CSGS: Fe<sup>3+</sup>), efficient broadband NIR emission centered at 770 nm and spanning 600-1100 nm is successfully realized. This material achieves an internal quantum efficiency (IQE) of 62.38% and an external quantum efficiency (EQE) of 46.56%, while maintaining 62% of its emission intensity at 423 K. Its overall balanced performance exceeds that of most reported Fe<sup>3+</sup>-based systems. Supported by first-principles calculations, this study systematically clarifies the electronic structure, mechanical properties, site preference, and valence stability of Fe<sup>3+</sup>, elucidates the excited-state transition behavior, and proposes a crystal-field-induced site-selective luminescence mechanism. This work not only presents a high-performance Fe<sup>3+</sup>-activated NIR phosphor, but also provides theoretical insights and practical guidance for material design and optimization in solid-state lighting, nondestructive testing, and spectral analysis.
Abstract In a recent study (IJQF 12 (1), 379-399, 2025) \cite{1}, we developed a mathematical formulation of force as an essential quantum observable, connecting its quantum wave equation to the violation of translational symmetries that regulates the behavior of open quantum systems in a particularly coherent manner. In this letter, we demonstrate that, following the same methodological principles that underpin the well-established quantum wave equations, our proposed force wave equation constitutes a theoretically sound framework that emerges naturally when energy–momentum exchange, instigated by force, is viewed as the fundamental organizing principle for non-relativistic quantum dynamics. This validates the conceptual and mathematical foundations of the framework.
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