Exact Quantum Mechanical Solution Research Articles

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.

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.

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.

Quasi-relativistic description of a quantum particle moving through one-dimensional piecewise constant potentials

Using a novel quasi-relativistic wave equation, which can give precise results up to energies ~mc2, exact quantum mechanical solutions are found which corresponds to a particle with mass moving through one-dimensional piecewise constant potentials. As expected, at low particle’s speeds, the found solutions coincide with the solutions of the same problems calculated using the Schrödinger equation; however, as it should be, both solutions have a significative difference at quasi-relativistic speeds. Then, it is argued that the quasi-relativistic wave equation provides a simpler description than a fully relativistic theory or the perturbation approach for a quantum particle moving at quasi-relativistic energies through piecewise constant potentials.

Quasi-Relativistic Description of Hydrogen-Like Atoms

Using a novel wave equation, which is Galileo invariant but can give precise results up to energies as high as mc2, exact quasi-relativistic quantum mechanical solutions are found for the Hydrogen atom. It is shown that the exact solutions of the Grave de Peralta equation include the relativistic correction to the non-relativistic kinetic energies calculated using the Schrödinger equation.

The Finslerian quantum cosmology

We present a Friedmann–Robertson–Walker quantum cosmological model within the framework of Finslerian geometry. In this work, we consider a specific fluid. We obtain the corresponding Wheeler–DeWitt equation as the usual constraint equation as well as the Schrödinger equation following Dirac, although the approaches yield the same time-independent equation for the wave function of the universe. We provide exact classical and quantum mechanical solutions. We use eigenfunctions to study the time evolution of the expectation value of the scale factor. Finally, we discuss the physical meaning of the results.

Thermoelectric transport coefficients of a Dirac electron gas in high magnetic fields

We study the thermoelectric transport properties of a three-dimensional massive relativistic fermion gas with screened Coulomb impurities in high magnetic fields where only the lowest Landau levels contribute to the transport. Our results can be applied to experimental results of gapless and gapped Dirac materials. We focus on the effects of the mass term and we show the main differences that arise compared to the massless Dirac fermions. The different behavior is shown to be relevant at higher magnetic fields. The calculations are performed in the framework of the linear response theory using the exact quantum mechanical solution of the system in a constant magnetic field. We prove that the Mott formula and the Wiedemann-Franz law are valid at low temperatures and use them to calculate the thermoelectric transport coefficients. We show that the temperature range where the low temperature approximation is valid increases with increasing magnetic fields. The magnetic field dependence of measurable quantities (i.e. conductivity, Seebeck coefficient, Nernst coefficient and thermal conductivity) strongly depend on the magnetic field dependence of the scattering rate, thus the result relies on the proper treatment of the impurities. In this work they are included through the first Born approximation using screened charged impurities as impurity potential. We show that the electric conductivity does not change qualitatively in the case of finite mass term. On the other hand we find that the mass term causes significantly different behavior in the Seebeck and Nernst coefficients.

Analysis of two-color laser-induced electron emission from a biased metal surface using an exact quantum mechanical solution

Photoelectron emission from nanotips via strong-field lasers is a powerful technique for coherent control of ultrafast electron dynamics, and so is important in many areas, including time-resolved electron microscopy and free-electron lasers. The effects of dc bias are typically ignored, though. Exact solution of the time-dependent Schr\odinger equation reveals that, under large dc bias, strong modulation persists in the emission current, the magnitude of which increases significantly. In the authors' model, the dynamics of the multiphoton excited states depend strongly on the applied dc field. This work suggests a practical approach to strong control of high-current photoemission.

Benchmark First-Principles Calculations of Adsorbate Free Energies

Adsorbate free energies are fundamental quantities in the microkinetic modeling of catalytic reactions. In first-principles modeling, finite-temperature free energies are generally obtained by combining density functional theory energies with standard approximate models, such as the harmonic oscillator, the hindered translator, or the two-dimensional ideal gas. In this work, we calculate accurate free energies directly from first-principles potential energy surfaces combined with exact quantum mechanical solutions for the translational energy states to benchmark the reliability of common approximations. Through a series of case studies of monatomic adsorbates on metal surfaces, we show that no one free energy model performs satisfactorily in all cases. Moreover, even combinations of different approximations sometimes deviate significantly from the free energies calculated by our first-principles approach. Using observations from these case studies, we discuss how a full quantum mechanical approach can be extended to calculate accurate free energies for arbitrary adsorbate potential energy surfaces at computational cost similar to standard models.

Exact Classical and Quantum Mechanics of a Generalized Singular Equation of Quadratic Liénard Type

Authors introduce a generalized singular differential equation of quadratic Liénard type for study of exact classical and quantum mechanical solutions. The equation is shown to exhibit periodic solutions and to include the linear harmonic oscillator equation and the Painlevé-Gambier XVII equation as special cases. It is also shown that the equation may exhibit discrete eigenstates as quantum behavior under Nikiforov-Uvarov approach after several point transformations.

Three sources of errors in the Ehrenfest treatment of inelastic scattering and possible ways of resolving them.

In order to identify the origin of possible errors in the mixed quantum/classical approach to inelastic scattering [A. Semenov and D. Babikov, J. Chem. Phys. 140, 044306 (2014) and A. Semenov, M.-L. Dubernet, and D. Babikov, J. Chem. Phys. 141, 114304 (2014)], a simplified model is considered that consists of one intermolecular degree of freedom and two intramolecular states, coupled by a simple potential. For this system, analytic derivations are carried out to determine (i) the exact quantum mechanical solution of the inelastic scattering problem, (ii) a simplified version of it with all oscillatory terms neglected, and (iii) the Ehrenfest solution in which the translational motion is described by the mean-field trajectory while the internal molecular motion is treated by the time-dependent Schrodinger equation. It is shown that the appropriate choice of velocity for the mean-field trajectory permits to enforce microscopic reversibility and gives results in excellent agreement with full-quantum results. The average velocity method of Billing is rigorously derived as a limiting case (of this more general approach), when reversibility is enforced locally, at the initial moment of time only. It is demonstrated that errors of state-to-state transition probabilities in the Ehrenfest approach occur at lower values of total energy E if the magnitudes of excitation energy ΔE, potential energy difference between the two states ΔV, and coupling of two states V12 are large. Possible ways of applying this concept to rotational transitions in real molecules are explored, using examples from CO + CO inelastic scattering.

Hořava–Lifshitz quantum cosmology in the presence of Chaplygin gas: Exact solutions and the late-time acceleration

In this paper, we obtain new exact solutions and first-order late-time solutions for the generalized Chaplygin gas Hořava–Lifshitz quantum cosmology with the equation-of-state [Formula: see text]. Based on the Schutz formalism, we use the matter degree of freedom to recover the notion of time in quantum domain. This enables us to study time evolution of wave functions (wave packets) as well as their corresponding classical solutions. In particular, for [Formula: see text], we obtain new exact classical and quantum mechanical solutions for various gravitational cases which are valid for both early and late times. For [Formula: see text] and [Formula: see text] (the standard Chaplygin gas), and up to the first-order approximation, we solve the corresponding Schrödinger–Wheeler–DeWitt equation and find the late-time solutions in the classical and quantum domains. Also, we study the accelerating behavior of solutions at the late times and we compare the results with solutions raised from general relativity.

Excited-state quantum phase transition in the Rabi model

The Rabi model, a two-level atom coupled to a harmonic oscillator, can undergo a second-order quantum phase transition (QPT) [M. -J. Hwang et al, Phys. Rev. Lett. 115, 180404 (2015)]. Here we show that the Rabi QPT accompanies critical behavior in the higher energy excited states, i.e., the excited-state QPT (ESQPT). We derive analytic expressions for the semiclassical density of states, which shows a logarithmic divergence at a critical energy eigenvalue in the broken symmetry (superradiant) phase. Moreover, we find that the logarithmic singularities in the density of states leads to singularities in the relevant observables in the system such as photon number and atomic polarization. We corroborate our analytical semiclassical prediction of the ESQPT in the Rabi model with its numerically exact quantum mechanical solution.

Dynamics of interacting bosons using the Herman–Kluk semiclassical initial value representation

Recent experimental progress in monitoring the dynamics of ultracold gases in optical lattices necessitates a quantitative theoretical description for a significant number of bosons. In the present paper, we investigate if time-dependent semiclassical initial value methodology, with propagators expressed as integrals over phase space and using classical trajectories, is suitable to describe interacting bosons, concentrating on a single mode. Despite the nonlinear contribution from the self-interaction, the corresponding classical dynamics allows for a largely analytical treatment of the semiclassical propagator. We find that application of the Herman–Kluk (HK) propagator conserves unitarity in the semiclassical limit, but a decay of the norm is seen for low particle numbers n. The frozen Gaussian approximation (FGA) (HK with unit prefactor) is explicitly shown to violate unitarity in the present system for non-vanishing interaction strength, even in the semiclassical limit. Furthermore, we show by evaluating the phase space integral in steepest descent approximation, that the HK propagator reproduces the exact spectrum correctly in the semiclassical limit (). An error is, however, incurred in next-to-next-to-leading order (small parameter ), as seen upon numerical evaluation of the integral and confirmed analytically by considering finite n corrections to the steepest descent calculations. The FGA, in contrast, is only accurate to lowest order, and an erroneous next-to-leading order term in the energy spectrum was found analytically. Finally, as an example application, we study the dynamics of wave packets by computing the time evolution of the Wigner function. While the often-used truncated Wigner approximation cannot capture any interferences present in the exact quantum mechanical solution (known analytically), we find that the HK approach, despite also using classical information only, reproduces the salient features of the exact solution correctly.

Nonequilibrium Green's functions and atom-surface dynamics: Simple views from a simple model system

We employ Non-equilibrium Green's functions (NEGF) to describe the real-time dynamics of an adsorbate-surface model system exposed to ultrafast laser pulses. For a finite number of electronic orbitals, the system is solved exactly and within different levels of approximation. Specifically i) the full exact quantum mechanical solution for electron and nuclear degrees of freedom is used to benchmark ii) the Ehrenfest approximation (EA) for the nuclei, with the electron dynamics still treated exactly. Then, using the EA, electronic correlations are treated with NEGF within iii) 2nd Born and with iv) a recently introduced hybrid scheme, which mixes 2nd Born self-energies with non-perturbative, local exchange- correlation potentials of Density Functional Theory (DFT). Finally, the effect of a semi-infinite substrate is considered: we observe that a macroscopic number of de-excitation channels can hinder desorption. While very preliminary in character and based on a simple and rather specific model system, our results clearly illustrate the large potential of NEGF to investigate atomic desorption, and more generally, the non equilibrium dynamics of material surfaces subject to ultrafast laser fields.

Chaplygin gas Hořava-Lifshitz quantum cosmology

In this paper, we study the Chaplygin gas Ho\v{r}ava-Lifshitz quantum cosmology. Using Schutz formalism and Arnowitt-Deser-Misner decomposition, we obtain the corresponding Schr\"{o}dinger-Wheeler-DeWitt equation. We obtain exact classical and quantum mechanical solutions and construct wave packets to study the time evolution of the expectation value of the scale factor for two cases. We show that unlike classical solutions and upon choosing appropriate initial conditions, the expectation value of the scale factor never tends to the singular point which exhibits the singularity-free behavior of the solutions in the quantum domain.

Communication: Global flux surface hopping in Liouville space.

Recent years have witnessed substantial progress in the surface hopping (SH) formulation of non-adiabatic molecular dynamics. A generalization of the traditional fewest switches SH (FSSH), global flux SH (GFSH) utilizes the gross population flow between states to derive SH probabilities. The Liouville space formulation of FSSH puts state populations and coherences on equal footing, by shifting the hopping dynamics from Hilbert to Liouville space. Both ideas have shown superior results relative to the standard FSSH in Hilbert space, which has been the most popular approach over the past two and a half decades. By merging the two ideas, we develop GFSH in Liouville space. The new method is nearly as straightforward as the standard FSSH, and carries comparable computational expense. Tested with a representative super-exchange model, it gives the best performance among all existing techniques in the FSSH series. The obtained numerical results match almost perfectly the exact quantum mechanical solutions. Moreover, the results are nearly invariant under the choice of a basis state representation for SH, in contrast to the earlier techniques which exhibit notable basis set dependence. Unique to the developed approach, this property is particularly encouraging, because exact quantum dynamics is representation independent. GFSH in Liouville space significantly improves accuracy and applicability of SH for a broad range of chemical and physical processes.

Susceptibility of Fe atoms in Cu clusters

By putting a Kondo system of limited dimension inside an insulating matrix, the Kondo screening cloud is confined by the size of the system. This allows us to distinguish the bulk and nano-particle behaviors of such system. We have investigated the magnetic properties of 0.3 at. % Fe-doped Cu clusters with dimensions less than 20 nm embedded in a SiO2 matrix. The magnetic measurements are consistent with the Kondo interaction, and net antiferromagnetic Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions between several Fe atoms in one cluster are estimated to have a very small effect. To understand the low-temperature susceptibility reduction, we compare Brillouin functions with exact quantum-mechanical solutions for interacting spin-1 particles and with the Kondo predictions for confined nanoparticles.