Quantum Mechanical Equations Research Articles

Complex structure-preserving method for Schrödinger equations in quaternionic quantum mechanics

Complex structure-preserving method for Schrödinger equations in quaternionic quantum mechanics

Generalized Equations in Quantum Mechanics and Brownian Theory

We discuss formal analogies between a nonlinear Schrödinger equation derived by the author from the theory of scale relativity and the equations of Brownian theory. By using the Madelung transformation, the nonlinear Schrödinger equation takes the form of hydrodynamic equations involving a friction force, an effective thermal pressure, a pressure due to the self-interaction, and a quantum potential. These hydrodynamic equations have a form similar to the damped Euler equations obtained for self-interacting Brownian particles in the theory of simple liquids. In that case, the temperature is due to thermal motion and the pressure arises from spatial correlations between the particles. More generally, the correlations can be accounted for by using the dynamical density functional theory. We determine the excess free energy of Brownian particles that reproduces the standard quantum potential. We then consider a more general form of excess free energy functionals and propose a new class of generalized Schrödinger equations. For a certain form of excess free energy, we recover the generalized Schrödinger equation associated with the Tsallis entropy considered in a previous paper.

The continuity equation in the phase space quantum mechanics

A quantum phase space version of the continuity equation for systems with internal degrees of freedom is derived. The 1-D Dirac equation is introduced and its phase space counterpart is found. The phase space representation of free motion and of scattering in a nonrelativistic and relativistic case for setups with internal degrees of freedom is discussed and illustrated. Properties of Wigner functions of unbound states are analysed.

Impact of White Noise on the Exact Solutions of the Stochastic Riemann Wave Equation in Quantum Mechanics

In this article, the stochastic Riemann wave equation (SRWE) forced by white noise in the Itô sense is considered. The extended tanh function and mapping methods are applied to obtain new elliptic, rational, hyperbolic, and trigonometric stochastic solutions. Furthermore, we generalize some previous studies. The obtained solutions are important in explaining some exciting physical phenomena, since the SRWE is required for describing wave propagation. We plot numerous 3D and 2D graphical representations to explain how the multiplicative white noise influences the exact solutions of the SRWE. We can infer that the introduction of multiplicative white noise disrupts the symmetry of the solutions and serves to stabilize the solutions of the SRWE.

A decoherence explanation of the gallium neutrino anomaly

Gallium radioactive source experiments have reported a neutrino-induced event rate about 20% lower than expected with a high statistical significance. We present an explanation of this observation assuming quantum decoherence of the neutrinos in the gallium detectors at a scale of 2 m. This explanation is consistent with global data on neutrino oscillations, including solar neutrinos, if decoherence effects decrease quickly with energy, for instance with a power law E_\nu^{-r}Eν−r with r\simeq≃ 12. Our proposal does not require the presence of sterile neutrinos but implies a modification of the standard quantum mechanical evolution equations for active neutrinos.

Quantum Elliptic Curve

The present paper summarizes the author’s work on Elliptic Curve and its Rational Points. We derive new Quantum Mechanical Equation from the Quantum Mechanics first principles. We then describe the Quantum Entanglement Mechanism, based on the link between the two Quantum Elliptic Curves author first derived in the original version of the present paper in October 2020. The two Quantum Elliptic Curves represent states of the two entangled particles forming an EPR-pair (aka Bell state). Also we present two methods of manipulation of state of an EPR-pair. Together they provide algorithm for the Quantum Teleportation.

Global solutions and blow-up for Klein–Gordon equation with damping and logarithmic terms

In this paper, the initial boundary value problem for Klein–Gordon equation with weak and strong damping terms and nonlinear logarithmic term is investigated, which is known as one of the nonlinear wave equations in relativistic quantum mechanics and quantum field theory. Firstly, we prove the local existence and uniqueness of weak solution by using the Galerkin method and Contraction mapping principle. The global existence, energy decay and finite time blow-up of the solution with subcritical initial energy are established. Then these conclusions are extended to the critical initial energy. Besides, the finite time blow-up result with supercritical initial energy is shown.

Numerical Solution of Schrödinger Equation by using Crank-Nicolson Method

The Schrödinger equation is a fundamental equation in quantum mechanics that describes how wave functions evolve over time. The study explored specific focus on the Crank-Nicolson scheme, which is widely used and efficient method. By applying these methods to the one-dimensional Schrödinger equation, the work provided insights into the behavior of these systems. To confirm the accuracy and reliability of this method, a test problem is solved. The result obtained from numerical method is compared with analytical solution. The graph of compared result is shown with the help of computational software. The test demonstrates that the method is effective for solving the Schrödinger equation, even when an analytical solution is not possible or too difficult to obtain. Overall, the Crank-Nicolson difference scheme is a valuable tool for understanding the behavior of quantum systems and solving problems.

The propagation of Hermite–Gauss wave packets in optics and quantum mechanics

AbstractThe two‐dimensional paraxial equation of optics and the two‐dimensional time‐dependent Schrödinger equation, derived as approximations of the three‐dimensional Helmholtz equation and the three‐dimensional time‐independent Schrödinger equation, respectively, are identical. Here the free propagation in space and time of Hermite–Gauss wave packets (optics) or harmonic oscillator eigenfunctions (quantum mechanics) is examined in detail. The Gouy phase is shown to be a dynamic phase, appearing as the integral of the adiabatic eigenfrequency or eigenenergy. The wave packets propagate adiabatically in that at each space or time point they are solutions of the instantaneous harmonic problem. In both cases, it is shown that the form of the wave function is unchanged along the loci of the normals to wave fronts. This invariance along such trajectories is connected to the propagation of the invariant amplitude of the corresponding free wave number (optics) or momentum (quantum mechanics) wave packets. It is shown that the van Vleck classical density of trajectories function appears in the wave function amplitude over the complete trajectory. A transformation to the co‐moving frame along a trajectory gives a constant wave function multiplied by a simple energy or frequency phase factor. The Gouy phase becomes the proper time in this frame.Key PointsThis paper builds a bridge between quantum mechanics (QM) and classical optics in that the identity of the paraxial equation of optics and the Schrödinger equation of QM is shown, the Bohmian trajectories of QM are defined in optics, the Gouy phase of optics is defined in QM and given a new interpretation and The space and momentum wave functions are equivalent along a trajectory.

A review: Rise of PT-symmetry for laser applications

In quantum mechanics, with the groundbreaking discovery of parity-time symmetry (PT-symmetry), Bender and Boettcher unfold the nature of the non-Hermitian Hamiltonian to exhibit real eigenvalues. This opened a new doorway to efficiently explore all the previously ignored quantum non-Hermitian systems. The analogy between the Schrödinger equation in quantum mechanics and the paraxial wave equation in optics spurred researchers to apply the concept of PT-symmetry to optical systems. On exploring the PT-symmetric optics experimentally, by engineering loss and gain in the system, a lot of applications have been reported. It includes single-mode lasers, coherent perfect absorber lasers (CPA lasers), exceptional point sensors (EPS), spectral and spatial filters, etc. In addition to linear theoretical models on PT-symmetry, nonlinear theoretical models and the corresponding experimental results have also been reported in the last few decades. This review article begins with the fundamental idea of PT-symmetry for beginners. Subsequently, we elucidate its advancements in the field of optics. Besides, we present the recent developments of various laser systems using PT-symmetry. Thus, this review intends to focus on novel theoretical and experimental works on PT-symmetry.

The Lagrangian structure, the Euler equation, and second Newton’s law of ultrafast nonlinear optics

With a notable exception of space–time variable swap, pulse evolution equations of ultrafast optics are mathematically similar to the equations of quantum mechanics and beam dynamics. The Lagrangian structure of these equations is, however, much less intuitive. Here, we show that such a structure can be identified via a path-integral analysis of the pulse evolution equation, offering useful insights into the Lagrangian underpinning of ultrafast nonlinear optics. The Euler–Lagrange equation applied to the optical analog of the Lagrange function leads to a second-order differential equation that parallels in its structure second Newton’s law. Stationary-phase space–time paths found by solving such an equation reveal the hidden inner workings behind a vast class of field-waveform transformations in ultrafast optics, including the buildup of nonlinear phase, modulation instabilities, and soliton breather dynamics.

A theory of dark energy that matches dark matter

In this paper, a theory of dark energy is proposed that matches dark matter. The relativistic quantum mechanics equations reveal that free particles can have negative energies. We think that the negative energy is the dark energy, which behaves as dark photons with negative energies. In this work, the photon number states are extended to the cases where the photon number can be negative integers, called negative integer photon states, the physical meaning of which are that the photons in such a state are of negative energy, i.e., dark photons. The dark photons constitute dark radiation, also called negative radiation. The formalism of the statistical mechanics and thermodynamics of the dark radiation is presented. This version of dark energy is of negative temperature and negative pressure, the latter regarded as responsible for the accelerate expansion of the universe. It is believed that there is a symmetry of energy-dark energy in the universe. In our previous work, the theory of the motion of the matters with negative kinetic energy was presented. In our opinion, the negative kinetic energy matter is dark matter. In the present work, we demonstrate that the dark substances absorb and release dark energy. In this view, the dark matter and dark energy match. Therefore, there is a symmetry of matter-energy match and dark matter-dark energy match in the universe. We present the reasons why the negative kinetic energy systems and negative radiation are dark to us.

The Convergence of Symmetric Discretization Models for Nonlinear Schrödinger Equation in Dark Solitons’ Motion

The Schrödinger equation is one of the most basic equations in quantum mechanics. In this paper, we study the convergence of symmetric discretization models for the nonlinear Schrödinger equation in dark solitons’ motion and verify the theoretical results through numerical experiments. Via the second-order symmetric difference, we can obtain two popular space-symmetric discretization models of the nonlinear Schrödinger equation in dark solitons’ motion: the direct-discrete model and the Ablowitz–Ladik model. Furthermore, applying the midpoint scheme with symmetry to the space discretization models, we obtain two time–space discretization models: the Crank–Nicolson method and the new difference method. Secondly, we demonstrate that the solutions of the two space-symmetric discretization models converge to the solution of the nonlinear Schrödinger equation. Additionally, we prove that the convergence order of the two time–space discretization models is O(h2+τ2) in discrete L2-norm error estimates. Finally, we present some numerical experiments to verify the theoretical results and show that our numerical experiments agree well with the proven theoretical results.

Quantum Computing for Molecular Biology.

Molecular biology and biochemistry interpret microscopic processes in the living world in terms of molecular structures and their interactions, which are quantum mechanical by their very nature. Whereas the theoretical foundations of these interactions are well established, the computational solution of the relevant quantum mechanical equations is very hard. However, much of molecular function in biology can be understood in terms of classical mechanics, where the interactions of electrons and nuclei have been mapped onto effective classical surrogate potentials that model the interaction of atoms or even larger entities. The simple mathematical structure of these potentials offers huge computational advantages; however, this comes at the cost that all quantum correlations and the rigorous many-particle nature of the interactions are omitted. In this work, we discuss how quantum computation may advance the practical usefulness of the quantum foundations of molecular biology by offering computational advantages for simulations of biomolecules. We not only discuss typical quantum mechanical problems of the electronic structure of biomolecules in this context, but also consider the dominating classical problems (such as protein folding and drug design) as well as data-driven approaches of bioinformatics and the degree to which they might become amenable to quantum simulation and quantum computation.

Unidirectional beam splitting in acoustic metamaterial governed by double fractional stimulated Raman adiabatic passage

In this work, we take fractional stimulated Raman adiabatic passage (f-STIRAP) for the design of functional acoustic waveguide (WG) coupler into account. Assisted by the agreement in the form between Schrödinger equation in quantum mechanics and the coupled-mode equation of classical waves, the quantum three-level system is mapped onto the acoustic three-WG system with space-varying coupling actions between composing WGs. The output port of the coupler can be selected by adopting different superposition forms of incident waves, which is utilized to build a one-way acoustic mode converter based on double f-STIRAP. By further constructing a functional acoustic metamaterial arrayed by mode converters, a desired beam splitting behavior can be generated unidirectionally in a broadband. Our work bridges f-STIRAP and the design of acoustic metamaterial, which may have profound impacts on exploring quantum technologies for promoting advanced acoustic functional devices with simple configuration and excellent performance.

Solution of Klein-Gordon Equation in F(R) Theory of Gravity

The theory, as a modification of the general relativity theory, is frequently employed as an alternative theory of gravity and offers a promising avenue for addressing the challenges of formulating a quantum gravity theory. In this study, by applying the separation method of time, radial and angular variables, we derived the general solution of the Klein-Gordon equation in a curved space-time using modified Schwarzschild metric. We modified Ricci scalar form in Einstein’s action principle as a general function of Ricci scalar and formulated the general Schwarzschild metric. The solution of the time function was analytically obtained in exponential form, and the solution of the angular function in terms of Legendre polynomial depends on azimuthal and magnetic quantum numbers. The radial function in terms of a non-linear second-order differential equation was solved by a numerical method using Python. The solutions described the gravitational effect for a light particle on the area gravitationally has a strong interaction, represented by a spherically symmetric metric. For small (in Schwarzschild radius), the results analytically show that the gravitational effect in this region is massive. It follows that even light would be drawn into a black hole and unable to escape. For further research, it is expected to extend the Klein-Gordon equation in relativistic quantum mechanics to modified general relativity theory. This theory offers a different way of looking at the effects of gravity in quantum field theory.

Spectrum of the wave equation with Dirac damping on a non-compact star graph

We consider the wave equation on non-compact star graphs, subject to a distributional damping defined through a Robin-type vertex condition with complex coupling. It is shown that the non-self-adjoint generator of the evolution problem admits an abrupt change in its spectral properties for a special coupling related to the number of graph edges. As an application, we show that the evolution problem is highly unstable for the critical couplings. The relationship with the Dirac equation in non-relativistic quantum mechanics is also mentioned.

Anharmonicity and quantum nuclear effects in theoretical vibrational spectroscopy: a molecular tale of two cities

Anharmonic effects due to the shape of the molecular potential energy surface far from the equilibrium geometry are major responsible for the deviations of the actual frequencies of vibration from the harmonic estimates. However, anharmonic effects are not the solely responsible for this. Quantum nuclear effects also play a prominent role in theoretical vibrational spectroscopy as they contribute to drive away the molecular vibrational frequencies from their harmonic counterpart. The consequence of this is that anharmonicity and quantum effects may be difficult to separate spectroscopically and get often confused. In this work we show that anharmonicity can be detected by means of classical simulations, while quantum nuclear effects need to be identified by means of an approach originating from either the time independent or the time dependent Schroedinger equation of quantum mechanics. We show that classical methods are sensitive to the temperature or energy conditions under which they are undertaken. This leads to wrong frequency estimates, when dealing with few-Kelvin experiments, if one performs simulations simply matching the experimental temperature. Conversely, quantum approaches are not affected by this issue and they provide more and better information.

Modeling spin relaxation in complex radical systems using MolSpin.

Spin relaxation is an important aspect of the spin dynamics of free radicals and can have a significant impact on the outcome of their spin-selective reactions. Examples range from the use of radicals as spin qubits in quantum information processing to the radical pair reactions in proteins that may allow migratory birds to sense the direction of the Earth's magnetic field. Accurate modeling of spin relaxation, however, is non-trivial. Bloch-Redfield-Wangsness theory derives a quantum mechanical master equation from system-bath interactions in the Markovian limit that provides a comprehensive framework for describing spin relaxation. Unfortunately, the construction of the master equation is system-specific and often resource-heavy. To address this challenge, we introduce a generalized and efficient implementation of BRW theory as a new feature of the spin dynamics toolkit MolSpin which offers an easy-to-use approach for studying systems of reacting radicals of varying complexity.

The Helmholtz problem in slowly varying waveguides at locally resonant frequencies

This article aims to present a general study of the Helmholtz problem in slowly varying waveguides. This work is of particular interest at locally resonant frequencies, where a phenomenon close to the tunnel effect for Schrödinger equation in quantum mechanics can be observed. In this situation, locally resonant modes propagate in the waveguide under the form of Airy functions. Using previous mathematical results on the Schrödinger equation, we prove the existence of a unique solution to the Helmholtz source problem with outgoing conditions in such waveguides. We provide an explicit modal approximation of this solution, as well as a control of the approximation error in Hloc1. The main theorem is proved in the case of a waveguide with a monotonously varying profile and then generalized using a matching strategy. We finally validate the modal approximation by comparing it to numerical solutions based on the finite element method.