Quantum Mechanical Solution Research Articles

Similarity among quantum-mechanical states: Analysis and applications for central potentials

Abstract The similarity of quantum-mechanical solutions for central potentials is analytically determined and numerically explored for arbitrary dimensionalities. The study here provided focuses on hydrogenic systems and the harmonic oscillator, in respective non-relativistic frameworks. A diversity of analytical expressions for the Quantum Similarity Measure (QSM) and Index (QSI) are provided. Relevant conclusions are derived from the analyses grounded on state quantum numbers, space dimensionality and on the role played by the main characteristic parameters of these systems, namely the nuclear charge in the hydrogenic case, and the angular frequency for the oscillator. For this purpose, a statistical analysis of the QSI values has been performed for a large number both of states and combinations of them in each system. Considering the factorization of QSI into a radial and an angular part, particular attention is paid to the individual contribution of each part in both systems.

Giant caloric effects in spin-chain materials

The giant electro- and elastocaloric effects in spin-chain materials are predicted. The theory is based on the exact quantum mechanical solution of the problem. It is shown that the giant jumps in the entropy and the temperature caused by the caloric effect are weakly affected by the initial temperature. The effect can be used for the cooling of new quantum devices (like systems of qubits in quantum computers). On the other hand, since large changes are predicted in the narrow neighborhood of the critical point, the predicted effect can be used in ultrasensitive electric and stress sensors for modern microelectronics. Published by the American Physical Society 2024

Impact of Local and Nonlocal Vibronic Coupling on the Absorption and Emission Spectra of J- and H-Dimers.

The impact of exciton-vibrational (EV) coupling involving low-energy ("slow") intermolecular vibrations and higher-energy ("fast") intramolecular vibrations on the absorption and emission spectra of H- and J-dimers is studied theoretically for a pair of chromophores with excitonic coupling dominated by transition dipole-dipole coupling, JC. Exact quantum-mechanical solutions based on a Frenkel-Holstein-Peierls Hamiltonian reveal a fascinating interplay between the two coupling sources in determining the spectral line widths, Stoke shifts and radiative decay rates. It is shown that the ratio rules derived from the vibronic progression of the fast mode in molecular dimers remain valid under the influence of slow-mode EV coupling under most conditions. However, a highly unusual aggregate behavior occurs when the product of local and nonlocal couplings, |gLgNL|, exceeds 2ℏωs|JC|, where ℏωs is the energy of the slow mode. In this regime and when gL and gNL are in-phase, an H-dimer (JC > 0) becomes strongly emissive and can even be super-radiant, while a J-dimer (JC < 0) with out-of-phase gL and gNL values becomes subradiant. Such behaviors are in marked contrast to the predictions of Kasha theory and demonstrate the richness of the photophysical behavior resulting from EV coupling involving inter- and intramolecular vibrations.

Computation of the Spatial Distribution of Charge-Carrier Density in Disordered Media.

The space- and temperature-dependent electron distribution n(r,T) determines optoelectronic properties of disordered semiconductors. It is a challenging task to get access to n(r,T) in random potentials, while avoiding the time-consuming numerical solution of the Schrödinger equation. We present several numerical techniques targeted to fulfill this task. For a degenerate system with Fermi statistics, a numerical approach based on a matrix inversion and one based on a system of linear equations are developed. For a non-degenerate system with Boltzmann statistics, a numerical technique based on a universal low-pass filter and one based on random wave functions are introduced. The high accuracy of the approximate calculations are checked by comparison with the exact quantum-mechanical solutions.

Theoretical insight into clocking in a molecular mixed-valence cell of quantum cellular automata through the vibronic approach.

In this article, we develop a vibronic theory of clocking in molecular quantum cellular automata (QCA). The clocking mechanism is considered for a trigonal trimeric mixed-valence (MV) system with one mobile electron, which is shown to act as the dimeric unit encoding binary information (Boolean states 0 or 1) coupled to a third redox center (Null state). The model includes the electron transfer between the three centers; vibronic coupling of the mobile charge with the "breathing" modes, forming a double degenerate Jahn-Teller vibration of the molecular triangle; and two electric fields, one collinear to the dimeric unit, which controls the binary states, and the other perpendicular to this unit, performing clocking. In the framework of the adiabatic approximation, the potential surface of the trimeric system has been studied and the condition determining switching and clocking has been analyzed in terms of the two controlling fields and the vibronic and transfer parameters. A thorough understanding of the site populations is achieved through the quantum-mechanical solution of the vibronic problem, maintaining the adiabatic condition for the controlling fields. It is shown that a MV trimer can act as a molecular clocked QCA cell, with favorable conditions being a positive electron transfer parameter and sufficiently strong vibronic coupling.

О корректности размера позитрония

It is noted that initially the minimum size of the hydrogen atom was determined by the semiclassical method, which is still widely used today. When calculating the minimum size of a positronium atom, the method tested on the hydrogen atom was not used. Instead, in the formula for the radius, the mass was replaced by the reduced mass of positronium. There would be nothing wrong with this if the above “hydrogen” method gave the same result. However, it is not. The difference between the formula for positronium and the “hydrogen” formula is the coefficient “2”, due to the fact that the positive charge is not in the center of mass. In the literature, the minimum size of a positronium atom (Bohr radius of positronium) was determined in accordance with the two-body problem method. In the particular case under consideration, the method of the two-body problem is redundant and can be successfully replaced by solving the problem of the motion of one of the bodies. This allows you to obtain more visual and obvious solutions without excessive formalism with the reduced mass and displacement vector. The doubly refined value of the positronium radius was determined using the semiclassical method. However, positronium is a quantum object. Therefore, a quantum mechanical consideration is also presented. The textbook solutions are taken as a basis. The solution to the problem of two particles is found in the form of an equivalent solution for one of the particles moving relative to a stationary center of mass, especially since there is a reliable quantum mechanical solution for the case of a spherically symmetric force field, which assumes a stationary center of force. The latter is especially important, since the mobility of the force center (electric charge) can lead to radiation and, consequently, a change in the energy balance. It has been established that the calculated minimum radius of positronium (the distance from its center of mass at which the probability of finding an electron/positron is maximum) is equal to four, and not two Bohr radii of hydrogen, as indicated in the literature. In this case, the semiclassical and quantum solutions do not contradict each other.

Fokker-Planck treatment of nonlinearities in the dispersive coupling of an ion and an optical cavity

We complement previous studies of an ion coupled with an optical cavity in the dispersive regime, for a model which exhibits bistability of different configurations in the semiclassical description. Our approach is based on a truncated evolution in phase space and is intended to explore an especially interesting parameter region where the fully quantum-mechanical solution becomes hard but the crudest semiclassical approach fails to capture essential phenomena. We compare the results of our techniques with the ones from numerical diagonalization of the quantum evolution and find that although the treatment leads to a smoothening and a slight shift of the transitions in the system, it still provides a clear improvement over localized semiclassical approximations.

Spin and angular momentum in quaternionic quantum mechanics

We present two novel solutions of real Hilbert state quaternionic quantum mechanics . Firstly, we observe that the angular momentum operator admits two different classes of physically non-equivalent free particles. As a second result, we study the Larmor precession to observe that it has a quaternionic solution where a novel phenomenological interpretation is possible, as well as a different spin is possible, and these results may encourage experimental and theoretical investigations of the quaternionic theory.

Hadronic structure on the light front. IV. Heavy and light baryons

This work is a continuation in our series of papers, that addresses quark models of hadronic structure on the light front. The chief focus of this paper is the quantum-mechanical solution of the three-quark model Hamiltonian, describing baryons. In Jacobi coordinates, we use a harmonic oscillator basis for the transverse directions. For the longitudinal momentum fractions $x_i$, the pertinent basis follows from quantum mechanics in a "triangular cup" potential, which we solve exactly. We calculate the masses and light front wave functions for the flavor symmetric $\frac 32^+$ baryons $bbb,ccc,sss,uuu$.

Quantum states in disordered media. II. Spatial charge carrier distribution

The space- and temperature-dependent electron distribution $n(\mathbf r,T)$ is essential for the theoretical description of the opto-electronic properties of disordered semiconductors. We present two powerful techniques to access $n(\mathbf r,T)$ without solving the Schr\"odinger equation. First, we derive the density for non-degenerate electrons by applying the Hamiltonian recursively to random wave functions (RWF). Second, we obtain a temperature-dependent effective potential from the application of a universal low-pass filter (ULF) to the random potential acting on the charge carriers in disordered media. Thereby, the full quantum-mechanical problem is reduced to the quasi-classical description of $n(\mathbf r,T)$ in an effective potential. We numerically verify both approaches by comparison with the exact quantum-mechanical solution. Both approaches prove superior to the widely used localization landscape theory (LLT) when we compare our approximate results for the charge carrier density and mobility at elevated temperatures obtained by RWF, ULF, and LLT with those from the exact solution of the Schr\"odinger equation.

Machine learning based implicit solvent model for aqueous-solution alanine dipeptide molecular dynamics simulations.

Inspired by the recent work from Noé and coworkers on the development of machine learning based implicit solvent model for the simulation of solvated peptides [Chen et al., J. Chem. Phys., 2021, 155, 084101], here we report another investigation of the possibility of using machine learning (ML) techniques to "derive" an implicit solvent model directly from explicit solvent molecular dynamics (MD) simulations. For alanine dipeptide, a machine learning potential (MLP) based on the DeepPot-SE representation of the molecule was trained to capture its interactions with its average solvent environment configuration (ASEC). The predicted forces on the solute deviated only by an RMSD of 0.4 kcal mol-1 Å-1 from the reference values, and the MLP-based free energy surface differed from that obtained from explicit solvent MD simulations by an RMSD of less than 0.9 kcal mol-1. Our MLP training protocol could also accurately reproduce combined quantum mechanical molecular mechanical (QM/MM) forces on the quantum mechanical (QM) solute in ASEC environment, thus enabling the development of accurate ML-based implicit solvent models for ab initio-QM MD simulations. Such ML-based implicit solvent models for QM calculations are cost-effective in both the training stage, where the use of ASEC reduces the number of data points to be labelled, and the inference stage, where the MLP can be evaluated at a relatively small additional cost on top of the QM calculation of the solute.

On quantum and classical treatments of radiative recombination

The quantum-mechanical solution to the problem of radiative recombination of an electron in a Coulomb field, obtained in the approximation of the smallness of the electron coupling with the radiation field, has been known for a long time. However, in astrophysics, the classical approach, which does not explicitly use this smallness, is sometimes used to describe similar processes in systems of magnetic monopoles or self-interacting dark matter particles. The importance of such problems is determined by the fact that recombination processes play a crucial role in the evolution of the large-scale structure of the Universe. Therefore, of particular interest is the fact that the classical and quantum expressions for the recombination cross section differ significantly in magnitude. It is shown that the applicability of quantum and classical approaches to radiative recombination is closely related to the radiated angular momentum and its quantization. For situations where the classical approach is not suitable, a semi-classical approach based on the angular momentum quantization is proposed, in some respects an alternative to the well-known semi-classical Kramers' approach.

Driving a pure spin current from nuclear-polarization gradients

A pure spin current is predicted to occur when an external magnetic field and a linearly inhomogeneous spin-only field are appropriately aligned. Under these conditions (such as originate from nuclear contact hyperfine fields that do not affect orbital motion) a linear, spin-dependent dispersion for free electrons emerges from the Landau Hamiltonian. The result is that spins of opposite orientation flow in opposite directions giving rise to a pure spin current. A classical model of the spin and charge dynamics reveals intuitive aspects of the full quantum mechanical solution. We propose optical orientation or electrical polarization experiments to demonstrate this outcome.

Functional Properties of Tetrameric Molecular Cells for Quantum Cellular Automata: A Quantum-Mechanical Treatment Extended to the Range of Arbitrary Coulomb Repulsion

We discuss the problem of electron transfer (ET) in mixed valence (MV) molecules that is at the core of molecular Quantum Cellular Automata (QCA) functioning. Theoretical modelling of tetrameric bi-electronic MV molecular square (prototype of basic QCA cell) is reported. The model involves interelectronic Coulomb repulsion, vibronic coupling and ET between the neighboring redox sites. Unlike the majority of previous studies in which molecular QCA have been analyzed only for particular case when the Coulomb repulsion energy significantly exceeds the ET energy, here we do not imply assumptions on the relative strength of these two interactions. Moreover, in the present work we go beyond the adiabatic semiclassical approximation often used in theoretical analysis of such systems in spite of the fact that this approximation ignores such an important phenomenon as quantum tunneling. By analyzing the electronic density distributions in the cells and the ell-cell response functions obtained from a quantum-mechanical solution of a complex multimode vibronic problem we have concluded that such key features of QCA cell as bistability and switchability can be achieved even under failure of the condition of strong Coulomb repulsion provided that the vibronic coupling is strong enough. We also show that the semiclassical description of the cell-cell response functions loses its accuracy in the region of strong non-linearity, while the quantum-mechanical approach provides correct results for this critically important region.

Exact quantum-mechanical solution for the one-dimensional harmonic oscillator model asymmetrically confined into the infinite well

We constructed a new model of the non-relativistic quantum harmonic oscillator within the canonical approach. It is asymmetrically confined into the infinitely high potential well. Asymmetrical confinement effect is achieved thanks to the introduction of the effective mass that changes with position. It is shown that the model has exact solutions. Analytical expression of its wavefunctions of the stationary states is expressed through the Jacobi polynomials , whereas, energy spectrum is non-equidistant. At the limit, when confinement parameters a and b , defining the position of two walls of the potential well go to infinity, then obtained expressions of the wavefunctions also recover the wavefunctions expressed through the Hermite polynomials. • An exactly solvable new model of the non-relativistic quantum harmonic oscillator asymmetrically confined into the infinitely high potential well. • Asymmetrical confinement effect is achieved thanks to the introduction of the effective mass that changes with position. • Energy spectrum is non-equidistant. • The model has correct limit to the so-called Hermite oscillator when position of both walls of the potential well go to infinity.

Completeness and orthonormality of the energy eigenfunctions of the Dirac delta derivative potential

The orthonormality and completeness of the energy eigenfunctions solution for the Schrödinger equation are usually hard to prove explicitly. There are few examples treated in the literature that give an explicit demonstration of these properties. In this work, we are going to orthonormalize the energy eigenfunctions of the solutions of the Schrödinger equation with contact potential given by the Dirac delta derivative potential λδ′(x), where λ is the coupling constant and prove that they are complete. The solutions for this problem include both continuum and discrete energy spectra. This makes it a perfect choice in the sense that it covers all possible cases of the solutions in quantum mechanics. The bound states solutions are normalized as usual, while the scattering states are normalized in continuum sense. The scattering solutions are shown to be orthogonal to each other and to the bound states. Using these orthonormal eigenfunctions an explicit demonstration of the completeness is given. It is found that both the bound and scattering solutions should be considered to ensure completeness. The calculation shows that the scattering state solutions are complete by themselves except for a certain value of the energy eigenvalue. This energy eigenvalue was shown to be precisely consistent with the energy eigenvalue of the bound state.

VISUAL MACHINE LEARNING: INSIGHT THROUGH EIGENVECTORS, CHLADNI PATTERNS, AND COMMUNITY DETECTION IN 2D PARTICULATE STRUCTURES

Machine learning (ML) is quickly emerging as a powerful tool with diverse applications across an extremely broad spectrum of disciplines and commercial endeavors. Typically, ML is used as a black box that provides little illuminating rationalization of its output. In the current work, we aim to better understand the generic intuition underlying unsupervised ML with a focus on physical systems. The systems that are studied here as test cases comprise of six different 2-dimensional (2-D) particulate systems of different complexities. It is noted that the findings of this study are generic to any unsupervised ML problem and are not restricted to materials systems alone. Three rudimentary unsupervised ML techniques are employed on the adjacency (connectivity) matrix of the six studied systems: (i) using principal eigenvalue and eigenvectors of the adjacency matrix, (ii) spectral decomposition, and (iii) a Potts model based community detection technique in which a modularity function is maximized. We demonstrate that, while solving a completely classical problem, ML technique produces features that are distinctly connected to quantum mechanical solutions. Dissecting these features help us to understand the deep connection between the classical non-linear world and the quantum mechanical linear world through the kaleidoscope of ML technique, which might have far reaching consequences both in the arena of physical sciences and ML.

In a holistic perspective, time is absolute and relativity a direct consequence of the conservation of total energy

We are taught to think that the description of relativistic phenomena requires distorted time and distance. The message of this essay is that, in a holistic perspective, time and distance are universal coordinate quantities, and relativity is a direct consequence of the conservation of energy. Instead of the kinematics/metrics-based approach of the theory of relativity, the dynamic universe (DU) approach starts from the dynamics of space as a whole and expresses relativity in terms of locally available energy instead of locally distorted time and distance. In such an approach, e.g., the frequency of atomic clocks at different states of motion and gravitation is obtained from the quantum mechanical solution of the characteristic frequencies, and the unique status of the velocity of light becomes understood via its linkage to the rest of space. In the kinematic/metrics-based theory of relativity, we postulate the principle of relativity, Lorentz covariance, the equivalence principle, the constancy of the speed of light, and the rest energy of mass objects. The conservation of momentum and energy is honored in local frames of reference, and time and distance are parameters in frame-to-frame observations. In the dynamics-based DU, the whole space is studied as a closed energy system and the energy in local structures is derived conserving the overall energy balance. Any local state of motion and gravitation in space is related, through a system of nested energy frames, to the state of rest in hypothetical homogeneous space, which serves as the universal frame of reference. Relativity of observations appears as a direct consequence of the overall energy balance and the linkage of local to the whole—with time and distance as universal coordinate quantities. DU postulates spherically closed space and zero-energy balance of motion and gravitation. DU does not need the relativity principle or any other postulates of the theory of relativity. Primarily, the theory of relativity is an empirically driven mathematical description of observations, with postulates formulated to support the mathematics. DU relies on mathematics built on the conservation of an overall zero-energy balance as the primary law of nature, which makes DU more like a metaphysically driven theory. Both approaches produce precise predictions. The choice is philosophical—nature is not dependent on the way we describe it.

Quantum mechanical solution to spectral lineshape in strongly-coupled atom–nanocavity system

The strongly coupled system composed of atoms, molecules, molecule aggregates, and semiconductor quantum dots embedded within an optical microcavity/nanocavity with high quality factor and/or low modal volume has become an excellent platform to study cavity quantum electrodynamics (CQED), where a prominent quantum effect called Rabi splitting can occur due to strong interaction of cavity-mode single-photon with the two-level atomic states. In this paper, we build a new quantum model that can describe the optical response of the strongly-coupled system under the action of an external probing light and the spectral lineshape. We take the Hamiltonian for the strongly-coupled photon–atom system as the unperturbed Hamiltonian H 0 and the interaction Hamiltonian of the probe light upon the coupled-system quantum states as the perturbed Hamiltonian V . The theory yields a double Lorentzian lineshape for the permittivity function, which agrees well with experimental observation of Rabi splitting in terms of spectral splitting. This quantum theory will pave the way to construct a complete understanding for the microscopic strongly-coupled system that will become an important element for quantum information processing, nano-optical integrated circuits, and polariton chemistry.

Quaternionic Dirac free particle

In this paper, we solve the quaternionic Dirac equation [Formula: see text] in the real Hilbert space, and we ascertain that their free particle solutions set comprises eight elements in the case of a massive particle, and a four elements solutions set in the case of a massless particle, a richer situation when compared to the four elements solutions set of the usual complex Dirac equation [Formula: see text]. These free particle solutions were unknown in the previous solutions of anti-Hermitian quaternionic quantum mechanics, and constitute an essential element in order to build a quaternionic quantum field theory [Formula: see text].