Momentum Operators Research Articles

Numerical simulation of a fractional Davey–Stewartson model via a conservative scheme: preservation of mass, energy and momenta

In the present manuscript, we depart from a two-dimensional form of the nonlinear Davey–Stewartson system from fluid dynamics, consisting of two partial differential equations with two unknown functions, one of them is complex-valued and the other real-valued. This model possesses four conserved quantities, namely, the mass, the energy and two momenta. In a first stage, the mathematical model is generalized to the fractional scenario considering Riesz fractional operators and three different orders. We prove mathematically that this system also possesses four quantities that are preserved in time, and which are fractional extensions of the classical mass, energy and momentum operators. Motivated by this fact, we introduce a numerical scheme for the mathematical model by making use of fractional-order centred differences. By using similar arguments as in the analysis of the continuous system, we propose some discretizations of the mass, the energy and the momenta, and we proved theoretically that these quantities are also conserved in the discrete case. In our theoretical analysis, we establish mathematically the properties of consistency for the mathematical model as well as for the discrete conserved quantities. Computationally, the finite-difference scheme is implemented using a fixed-point algorithm. Various computer simulations are performed to illustrate the conservation of the discrete quantities. Finally, we provide some interesting perspectives of research at the end of this work. This report is the first work in which a conservative finite-difference scheme for the Davey–Stewartson system is designed and rigorously analysed for the conservation properties, for both the integer-order and fractional cases.

Large Orbital Moment and Dynamical Jahn-Teller Effect of AlCl-Phthalocyanine on Cu(100).

Submonolayer amounts of chloroaluminum-phthalocyanine on Cu(100) were studied with scanning tunneling spectroscopy. The molecule can be prepared in a fourfold symmetric state whose conductance spectrum exhibits a zero-bias feature similar to a Kondo resonance. In magnetic fields, however, this resonance splits far more than expected from the spin of a single electron. Density functional theory calculations reveal a charge transfer of 1.3 electrons to the degenerate lowest unoccupied molecular orbitals. These orbitals are mixed by the orbital momentum operator L[over ^]_{z} with a large matrix element corresponding to m_{L}≈2.7. Dehydrogenation of a ligand lifts the degerenracy of the lowest unoccupied molecular orbital, reduces the splitting in magnetic fields, and induces a polarity dependence of the spectra. Using model calculations of the spin, orbital, and vibrational degrees of freedom we show that a dynamical Jahn-Teller effect reproduces the main experimental observations.

Perfect quantum protractors

In this paper we introduce and investigate the concept of a perfect quantum protractor, a pure quantum state |ψ⟩∈H that generates three different orthogonal bases of H under rotations around each of the three perpendicular axes. Such states can be understood as pure states of maximal uncertainty with regards to the three components of the angular momentum operator, as we prove that they maximise various entropic and variance-based measures of such uncertainty. We argue that perfect quantum protractors can only exist for systems with a well-defined total angular momentum j, and we prove that they do not exist for j∈{1/2,2,5/2}, but they do exist for j∈{1,3/2,3} (with numerical evidence for their existence when j=7/2). We also explain that perfect quantum protractors form an optimal resource for a metrological task of estimating the angle of rotation around (or the strength of magnetic field along) one of the three perpendicular axes, when the axis is not a priori known. Finally, we demonstrate this metrological utility by performing an experiment with warm atomic vapours of rubidium-87, where we prepare a perfect quantum protractor for a spin-1 system, let it precess around x, y or z axis, and then employ it to optimally estimate the rotation angle.

Non-Hermitian momentum operator for the particle in a box

We construct a discrete non-Hermitian momentum operator, which implements faithfully the non-self-adjoint nature of momentum for a particle in a box. Its eigenfunctions are strictly limited to the interior of the box in the continuum limit, with the quarter wave as first nontrivial eigenstate. We show how to construct the corresponding Hermitian Hamiltonian for the infinite well as a concrete example to realize unitary dynamics. The resulting Hilbert space can be decomposed into a physical and unphysical subspace, which are mutually orthogonal. The physical subspace in the continuum limit reproduces that of the continuum theory and we give numerical evidence that the correct probability distributions for momentum and energy are recovered. Published by the American Physical Society 2024

Massive Dirac equation in static Bumblebee black hole space-times, detailed derivations and novel exact solutions

In this work, we construct and investigate the relativistic spin-12 fermionic fields quantum dynamics in static spherically symmetric Bumblebee black hole background. The derivation of the Dirac equation in a general static spherically symmetric black hole space-time is carried out in detail via tetrad formalism. With the help of total angular momentum operator, the angular equation can be separated from the radial part where the solution is given in terms of the spinor harmonics. The radial Dirac equation has a well-known problem due to the presence of the square root terms appearing simultaneously with the rest mass that prevents us to find exact solutions. In this work, we present exact solutions of light mass fermion's wave function and energy levels bound in the static Bumblebee black hole. We discover the exact solutions of the massive Dirac's radial equation in terms of the Confluent Heun functions. Moreover, thanks to the well-known polynomial condition of the Confluent Heun functions, we also derive the energy quantization.

On quantum states for angular position and angular momentum of light

In the present paper we construct a properly defined quantum state ψ(θ) expressed in terms of elliptic Jacobi theta functions for the self-adjoint observables angular position θ and the corresponding angular momentum operator L=−id/dθ in units of ℏ=1 . The quantum uncertainties Δθ and ΔL for the state are well-defined and are shown to give a lower value of the uncertainty product ΔθΔL in contrast to the so called minimal uncertainty states as discussed in Franke-Arnold et al (2004 New J. Phys. 6 103-1-8). The mean value ⟨L⟩ of the state ψ(θ) is not required to be an integer. In the case of any half-integer mean value ⟨L⟩ the state constructed exhibits a remarkable critical behavior with upper and lower bounds Δθ=π2/3−2 and ΔL=1/2 .

Bessel vortices in spin-1 Bose–Einstein condensates with Zeeman splitting and spin–orbit coupling

Abstract We investigate the ground states of spin–orbit coupled spin-1 Bose–Einstein condensates in the presence of Zeeman splitting. By introducing the generalized momentum operator, the linear version of the system is solved exactly, yielding a set of Bessel vortices. Additionally, based on linear solution and using variational approximation, the solutions for the full nonlinear system and their ground state phase diagrams are derived, including the vortex states with quantum numbers m = 0, 1, as well as mixed states. In this work, mixed states in spin-1 spin–orbit coupling (SOC) BEC are interpreted for the first time as weighted superpositions of three vortex states. Furthermore, the results also indicate that under strong Zeeman splitting, the system cannot form localized states. The variational solutions align well with numerical simulations, showing stable evolution and meeting the criteria for long-term observation in experiments.

Quantization of resistivity as consequence of symmetry invariance

The Schrödinger equation for a spinless electron under the influence of a constant electromagnetic field is analyzed based on the conserved operators of the system when the magnetic field is described by Landau’s gauge. For this specific situation, the Lorentz force can be recovered if two conserved generalized momentum operators are considered: one along the x-axis and the second along the y-axis. From the general solution obtained, a time-dependent ground state is constructed characterized by quantized resistivity proportional to integer multiples of von Klitzing’s constant when an invariance condition under a unitary transform is satisfied.

Quantum gravity signatures in gravitational wave detectors placed inside a harmonic trap potential

In this work, we consider a general gravitational wave detector of gravitational waves interacting with an incoming gravitational wave carrying plus polarization only placed inside a harmonic trap. This model can be well acquainted with the description of a resonant detector of gravitational waves as well. The well-known detector-gravitational wave interaction scenario uses the method of a semiclassical approach where the detector is treated quantum mechanically but the gravitational wave is considered at a classical level. In our analysis, we use a discrete mode decomposition of the gravitational wave perturbation which results in a Hamiltonian involving the position and momentum operators corresponding to the gravitational wave and the harmonic oscillator. We have then calculated the transition probability for the harmonic oscillator-gravitational wave tensor product state for going from an initial state to some unknown final state. Using the energy flux relation of the gravitational waves, we observe that if we consider the total energy as a combination of the number of gravitons in the initial state of the detector then the transition probability for the resonant absorption case scenario takes the analytical form which is exactly similar to the semiclassical absorption case. In the case of the emission scenario, we observe a spontaneous emission of a single graviton which was completely absent in the semiclassical analog of this model. This therefore gives a direct signature of linearized quantum gravity. Published by the American Physical Society 2024

Abstract ladder operators for non self-adjoint Hamiltonians, with applications

Ladder operators are useful, if not essential, in the analysis of some given physical system since they can be used to find easily eigenvalues and eigenvectors of its Hamiltonian. In this paper we extend our previous results on abstract ladder operators considering in many details what happens if the Hamiltonian of the system is not self-adjoint. Among other results, we give an existence criterion for coherent states constructed as eigenstates of our lowering operators. In the second part of the paper we discuss two different examples of our framework: pseudo-quons and a deformed generalized Heisenberg algebra. Incidentally, and interestingly enough, we show that pseudo-quons can be used to diagonalize an oscillator-like Hamiltonian written in terms of (non self-adjoint) position and momentum operators which obey a deformed commutation rule of the kind often considered in minimal length quantum mechanics.

Empirical adequacy of the time operator canonically conjugate to a Hamiltonian generating translations

To admit a canonically conjugate time operator, the Hamiltonian has to be a generator of translations (like the momentum operator generates translations in space), so its spectrum must be unbounded. But the Hamiltonian governing our world is thought to be bounded from below. Also, judging by the number of fields and parameters of the Standard Model, the Hamiltonian seems much more complicated. In this article I give examples of worlds governed by Hamiltonians generating translations. They can be expressed as a partial derivative operator just like the momentum operator, but when expressed in function of other observables they can exhibit any level of complexity. The examples include any quantum world realizing a standard ideal measurement, any quantum world containing a clock or a free massless fermion, the quantum representation of any deterministic time-reversible dynamical system without time loops, and any quantum world that cannot return to a past state. Such worlds are as sophisticated as our world, but they admit a time operator. I show that, despite having unbounded Hamiltonian, they do not decay to infinite negative energy any more than any quantum or classical world. Since two such quantum systems of the same Hilbert space dimension are unitarily equivalent even if the physical content of their observables is very different, they are concrete counterexamples to Hilbert Space Fundamentalism (HSF). Taking the observables into account removes the ambiguity of HSF and the clock ambiguity problem attributed to the Page-Wootters formalism, also caused by assuming HSF. These results provide additional motivations to restore the spacetime symmetry in the formulation of Quantum Mechanics and for the Page-Wootters formalism.

Hamiltonian, geometric momentum and force operators for a spin zero particle on a curve: physical approach

The Hamiltonian for a spin zero particle that is confined to a 1D curve embedded in the 3D space is constructed. Confinement is achieved by starting with the particle living in a small tube surrounding the curve, and assuming an infinitely strong normal force that squeezes the thickness of the tube to zero, eventually pinning the particle to the curve. We follow the new approach that we applied to confine a particle to a surface, in that we start with an expression for the 3D momentum operators whose components along and normal to the curve directions are separately Hermitian. The kinetic energy operator expressed in terms of the momentum operator in the normal direction is then a Hermitian operator in this case. When this operator is dropped and the thickness of the tube surrounding the curve is set to zero, one automatically gets the Hermitian curve Hamiltonian that contains the geometric potential term as expected. It is demonstrated that the origin of this potential lies in the ordering or symmetrization of the original 3D momentum operators in order to render them Hermitian. The Hermitian momentum operator for the particle as it is confined to the curve is also constructed and is seen to be similar to what is known as the geometric momentum of a particle confined to a surface in that it has a term proportional to the curvature that is along the normal to the curve. The force operator of the particle on the curve is also derived, and is shown to reduce, for a curve with a constant curvature and torsion, to a -apparently- single component normal to the curve that is a symmetrization of the classical expression plus a quantum term. All the above quantities are then derived for the specific case of a particle confined to a cylindrical helix embedded in 3D space.

Regularized quantum motion in a bounded set: Hilbertian aspects

It is known that the momentum operator canonically conjugated to the position operator for a particle moving in some bounded interval of the line (with Dirichlet boundary conditions) is not essentially self-adjoint: it has a continuous set of self-adjoint extensions. We prove that essential self-adjointness can be recovered by symmetrically weighting the momentum operator with a positive bounded function approximating the indicator function of the considered interval. This weighted momentum operator is consistently obtained from a similarly weighted classical momentum through the so-called Weyl-Heisenberg covariant integral quantization of functions or distributions.

A new variable-order fractional momentum operator for wave absorption when solving Schrödinger equations

Absorbing boundary conditions are often needed for solving Schrödinger equations when the analytical solution is unknown. This article presents a new wave-absorption method by unifying the Schrödinger equations with associated one-way wave equations via a fractional-order momentum operator pˆα+1 where 0≤α≤1. By gradually varying the fractional order α, the computational boundary is made to be transparent. This method allows for incident waves to propagate freely and forces reflected waves to decay exponentially with time. The unified equation is then solved using the Finite-Difference Time-Domain (FDTD) method so that the scheme is iteratively explicit and can be parallelized easily. Finally, the obtained fractional FDTD scheme is tested to simulate soliton and particle propagations and is compared with the existing Perfectly Matched Layer (PML) method. Results show that the new method is promising.

Conserved spin operator of Dirac’s theory in spatially flat FLRW space-times

New conserved spin and orbital angular momentum operators of Dirac’s theory on spatially flat FLRW space-times are proposed generalizing thus the recent results concerning the role of Pryce’s spin operator in the flat case (Cotăescu in Eur Phys J C, 82, 1073, 2022). These operators split the conserved total angular momentum generating the new spin and orbital symmetries that form the rotations of the isometry groups. The new spin operator is defined and studied in active mode with the help of a suitable spectral representation giving its Fourier transform. Moreover, in the same manner is defined the operator of the fermion polarization. The orbital angular momentum is derived in passive mode using a new method, inspired by Wigner’s theory of induced representations, but working properly only for global rotations. In this approach the quantization is performed finding that the one-particle spin and orbital angular momentum operators have the same form in any FLRW spacetime regardless their concrete geometries given by various scale factors.

Q-Deformed quantum mechanics related to the Tamm-Dancoff oscillator algebra and some physical applications

In this paper, we consider a system of the q-deformed bosonic Tamm-Dancoff oscillators, whose spectrum has some exponential cutoff factors at high energies. We first investigate the q-calculus in the Tamm-Dancoff (TD) boson algebra, and within this framework, the q-derivative, q-integral and q-exponential function are introduced. Using these properties, we construct a new formalism for the q-deformed quantum mechanics, which accordingly involve the q-adjoint operator and the q-Hermitian operator properties. We then derive the q-deformed Heisenberg relation, and develop the q-Hermitian momentum operator. The q-deformed Schrödinger equation is introduced, and as applications, we study the momentum eigenfunction and one-dimensional box problem. Another application of the TD type deformation onto lattice oscillations is also discussed through a model of the q-deformed Debye solid. Finally, other potential applications of the TD-oscillators gas model are concisely pointed out.

Linear graviton as a quantum particle

Wave function of a single linear graviton and its interpretation are proposed. The evolution equation for this function is given. A Hermitian operator with mutually commuting components canonically conjugated to the momentum operator of the linear graviton is found. Second quantization of the linear graviton quantum mechanics as well as quantization of the classical free linear graviton field are investigated.

Self-adjoint momentum operator for a particle confined in a multi-dimensional cavity

Based on the recent construction of a self-adjoint momentum operator for a particle confined in a one-dimensional interval, we extend the construction to arbitrarily shaped regions in any number of dimensions. Different components of the momentum vector do not commute with each other unless very special conditions are met. As such, momentum measurements should be considered one direction at a time. We also extend other results, such as the Ehrenfest theorem and the interpretation of the Heisenberg uncertainty relation to higher dimensions.

Theoretical Investigation on the Conservation Principles of an Extended Davey–Stewartson System with Riesz Space Fractional Derivatives of Different Orders

In this article, a generalized form of the Davey–Stewartson system, consisting of three nonlinear coupled partial differential equations, will be studied. The system considers the presence of fractional spatial partial derivatives of the Riesz type, and extensions of the classical mass, energy, and momentum operators will be proposed in the fractional-case scenario. In this work, we will prove rigorously that these functionals are conserved throughout time using some functional properties of the Riesz fractional operators. This study is intended to serve as a stepping stone for further exploration of the generalized Davey–Stewartson system and its wide-ranging applications.

Interference effects in Compton scattering of keV photons at H 2+ : a nonrelativistic analytical approach

In this paper we study interference effects in Compton scattering at molecule by keV photons. As a test case we investigate the molecular system H 2+ at the equilibrium internuclear distance and in the dissociative channel. The initial molecular wavefunction is described as a linear combination of atomic orbitals centered on each nucleus. The final continuum state is a two-centre continuum Coulomb wavefunction. Using these approximations, we obtain an analytical form for the Compton scattering fully differential cross section. We investigate the molecular effects at various internuclear distance for an incident photon energy close to 2.1 keV. The relative contribution of the A⋅P and A 2 coupling terms in the Compton scattering magnitude is discussed (where P is the electron momentum operator and A the vector potential of the field). We focus on Cohen-Fano type interference effects, as the ones observed in the photoionization of molecules.