Nonrelativistic Limit Research Articles

Relativistic effects in ionization of heavy hydrogen-like ions by short x-ray pulses

The probability of photoionization of a heavy hydrogen like ion is calculated within the relativistic perturbation theory. The Coulomb potential of a heavy nucleus is taken into account accurately by usage of the corresponding solutions of the Dirac equation. The obtained probability is compared with both nonrelativistic limit and with nonperturbative results based on the numerical solution of the time-dependent Dirac equation. Suppression of ionization is observed in the secondary ionization channel at photon frequency of about ℏ ω ≈ mc 2.

The Standard Model and quantum state reduction from Heim’s field theory

Abstract Core parts of the Standard Model are derived from B. Heim’s quantum field theory, whose poly-metric describes spacetime and matter in a unified formalism. Its non-linear eigenvalue equation transforms into the Einstein field equation in the macroscopic limit. The 6-dimensional Heim space can be determined as locally isomorphic to a SU(2) ⊗ SU(2) ⊗ U(1) ⊗ U(1) symmetry and thus to the SU(3), which allows to connect to the local gauge symmetries and boson fields of the Standard Model. The Fermion and Higgs field and their coupling are deduced from Heim’s basic equations, providing new insight into possible correlations of these fields. Furthermore, the derivation yields an additional imaginary coupling term which seems to account for quantum mechanical state reduction in the non-relativistic limit. The recently performed calculation of the mass spectrum of elementary particles in a new approach based on Heim’s theory (with average error to the data < 1 % ${< } 1\%$ ) appears as even more relevant, having now shown that the theory can connect to the achievements of the Standard Model.

Noncommutative Dirac and Schrödinger equations in the background of the new generalized Morse potential and a class of Yukawa potential with the improved Coulomb-like tensor potential as a tensor in 3D-ERQM and 3D-ENRQM symmetries

Relativistic and nonrelativistic quantum mechanics formulated in a noncommutative space–space have recently become the object of renewed interest. In the context of three-dimensional extended relativistic quantum mechanics (3D-ERQM) symmetries with arbitrary spin-orbit coupling quantum number [Formula: see text], we approximate to solve the deformed Dirac equation for a new suggested new generalized Morse potential and a class of Yukawa potential including improved Coulomb-like tensor interaction (N(GMP-CYP) plus ICLTI). In the framework of the spin and pseudospin (p-spin) symmetry, we obtain the global new energy eigenvalue which equals the energy eigenvalue in the usual relativistic QM as the main part plus three corrected parts produced from the effect of the spin-orbit interaction, the new modified Zeeman, and the rotational Fermi term, by using the parametric of the well-known Bopp’s shift method and standard perturbation theory using Greene–Aldrich approximation to nonlinear and exponential terms in the effective potential. The new values that we got appeared sensitive to the quantum numbers ([Formula: see text]), the mixed potential depths ([Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text]), the range of the potential [Formula: see text] and noncommutativity parameters ([Formula: see text],[Formula: see text],[Formula: see text]). We have studied the nonrelativistic limit of new spin symmetry under the N(GMP-ICYP) model, we will also treat some important special cases such as the new generalized Morse potential, the new class of Yukawa potential, the new Hellmann potential, the new inversely quadratic Yukawa potential, the new Hulthén potential and new Coulomb potential. Finally, we studied a case of composite systems.

Nonrelativistic limit of the Euler‐HMPN approximation models arising in radiation hydrodynamics

In this paper, we are concerned with the nonrelativistic limit of a class of computable approximation models for radiation hydrodynamics. The models consist of the compressible Euler equations coupled with moment closure approximations to the radiative transfer equation. They are first‐order partial differential equations with source terms. As hyperbolic relaxation systems, they are showed to satisfy the structural stability condition proposed by the second author. Based on this, we verify the nonrelativistic limit by combining an energy method with a formal asymptotic analysis.

Schrödinger Operators with Oblique Transmission Conditions in {mathbb R}^2

In this paper we study the spectrum of self-adjoint Schrödinger operators in L^2(mathbb {R}^2) with a new type of transmission conditions along a smooth closed curve Sigma subseteq mathbb {R}^2. Although these oblique transmission conditions are formally similar to delta '-conditions on Sigma (instead of the normal derivative here the Wirtinger derivative is used) the spectral properties are significantly different: it turns out that for attractive interaction strengths the discrete spectrum is always unbounded from below. Besides this unexpected spectral effect we also identify the essential spectrum, and we prove a Krein-type resolvent formula and a Birman-Schwinger principle. Furthermore, we show that these Schrödinger operators with oblique transmission conditions arise naturally as non-relativistic limits of Dirac operators with electrostatic and Lorentz scalar delta -interactions justifying their usage as models in quantum mechanics.

Quantum quenches in the sinh-Gordon and Lieb–Liniger models

The non-relativistic limit of integrable field theories at equilibrium has been intensively studied in the previous years; the simplest non-trivial case relates the sinh-Gordon model to the Lieb–Liniger model (LLM). Here we study this non-relativistic limit out of equilibrium, namely in the time evolution after a quantum quench. The obtained results agree with the known ones for the LLM, thus showing that the non-relativistic limit is applicable in this out-of-equilibrium setting.

Confinement time and ambipolar potential in a relativistic mirror-confined plasma

Advanced aneutronic fusion fuels such as proton-Boron11 tend to require much higher temperatures than conventional fuels like deuterium–tritium. For electrons, the bulk plasma temperature can approach a substantial fraction of the rest mass. In a mirror confinement system, where the electrons are confined by an ambipolar potential of at least five electron temperatures, the tail electrons which can escape the potential are fully relativistic, which must be taken into account in calculating their confinement. In this paper, simple estimates are employed to extend the scaling of the confinement time into the relativistic regime. By asymptotically matching this scaling to known solutions in the non-relativistic limit, accurate forms for the confinement time (and thus, the ambipolar potential) are obtained. These forms are verified using finite-element-based Fokker–Planck simulations over a wide range of parameters. Comparing relativistic and nonrelativistic mirror-confined plasmas with the same ratio of confining potential |eϕ| to electron temperature Te and the same mirror ratio R, the net result is a decrease in the confinement time due to relativistic effects by a factor of S≡(1+15Te/8mec2)/(1+2|eϕ|/mec2).

On a neutral Dirac particle interacting with a magnetic field in a topological defect space-time and its hidden supersymmetry

In this paper, we study the relativistic quantum dynamics of a neutral Dirac particle with a permanent magnetic dipole moment that interacts with an external magnetic field in the background space-time of a linear topological defect called spiral dislocation. The generalized Dirac wave equation is derived from the full action of that model involving the Lagrangian density of the Dirac spinor field in the background and the interaction model. The energy eigenvalues and corresponding wave functions are found in closed form by reducing the problem to that of a non-relativistic particle moving freely on a plane with a hole at the origin whose radius is determined by the defect parameter. In the limit of vanishing external magnetic field we are also able to establish a hidden SUSY structure of the underlying Dirac Hamiltonian allowing us to discuss the non-relativistic limit in some detail.

Frame dependence of the nonrelativistic limit of quantum fields

We study the nonrelativistic limit of quantum fields for an inertial and a noninertial observer. We show that nonrelativistic particle states appear as a superposition of relativistic and nonrelativistic particles in different frames. Hence, the nonrelativistic limit is frame dependent. We detail this result when the noninertial observer has uniform constant acceleration. Only for low accelerations, the accelerated observer agrees with the inertial frame about the nonrelativistic nature of particles locally. In such a quasi-inertial regime, both observers agree about the number of particles describing quantum field states. The same does not occur when the acceleration is arbitrarily large (e.g., the Unruh effect). We furthermore prove that wave functions of particles in the inertial and the quasi-inertial frame are identical up to the coordinate transformation relating the two frames.

Precision theoretical determination of electric-dipole matrix elements in atomic cesium

We compute the reduced electric-dipole matrix elements $\ensuremath{\langle}n{S}_{1/2}||D||{n}^{\ensuremath{'}}{P}_{J}\ensuremath{\rangle}$ with $n=6,7$ and ${n}^{\ensuremath{'}}=6,7,...,12$ in cesium using the most complete to date ab initio relativistic coupled-cluster method which includes singles, doubles, perturbative core triples, and valence triples. Our results agree with previous calculations at the linearized single double level but also show large contributions from nonlinear singles and doubles as well as valence triples. We also calculate the normalized ratio ${\ensuremath{\xi}}_{n,{n}^{\ensuremath{'}}}\ensuremath{\equiv}(1/\sqrt{2})\ensuremath{\langle}n{S}_{1/2}||D||{n}^{\ensuremath{'}}{P}_{1/2}\ensuremath{\rangle}/\ensuremath{\langle}n{S}_{1/2}||D||{n}^{\ensuremath{'}}{P}_{3/2}\ensuremath{\rangle}$ which is important for experimental determination of matrix elements. The ratios ${\ensuremath{\xi}}_{6,n}$ display large deviations from the nonrelativistic limit which we associate with Cooper-like minima. Several Appendices are provided where we document the procedure for constructing finite basis sets and our implementation of the random-phase approximation and Brueckner-orbitals method.

On numerical methods for the semi-nonrelativistic limit system of the nonlinear Dirac equation

Solving the nonlinear Dirac equation in the nonrelativistic limit regime numerically is difficult, because the solution oscillates in time with frequency of {mathscr {O}} ! left( varepsilon ^{-2}right) , where 0<varepsilon ll 1 is inversely proportional to the speed of light. Yongyong Cai and Yan Wang have shown, however, that such solutions can be approximated up to an error of {mathscr {O}} ! left( varepsilon ^2right) by solving the semi-nonrelativistic limit system, which is a non-oscillatory problem. For this system, we construct a two-step method, called the explicit exponential midpoint rule, and prove second-order convergence of the semi-discretization in time. Furthermore, we construct a benchmark method based on standard techniques and compare the efficiency of both methods. Numerical experiments show that the new integrator reduces the computational costs per time step to 40% and within a given runtime improves the accuracy significantly.

Two-dimensional transverse charge distributions of the Δ baryon: Interpolation between the nonrelativistic and ultrarelativistic limits

We investigate how the charge distributions of both the unpolarized and transversely polarized $\Delta$ baryon change as the longitudinal momentum~($P_{z}$) of the $\Delta$ baryon increases from $P_{z}=0$ to $P_{z}=\infty$ in a Wigner phase-space perspective. When the $\Delta$ baryon is longitudinally polarized, its two-dimensional charge distribution is kept to be spherically symmetric with $P_{z}$ varied, whereas when the $\Delta$ baryon is transversely polarized along the $x$-axis, the quadrupole contribution emerges at the rest frame ($P_{z}=0$). When $P_{z}$ grows, the electric dipole and octupole moments are induced. The induced dipole moment dominates over other higher multipole contributions and governs the deformation of the charge distribution of the $\Delta$ baryon.

A uniformly accurate method for the Klein-Gordon-Dirac system in the nonrelativistic regime

A multiscale time integrator Fourier pseudospectral (MTI-FP) method is presented for discretizing the massive Klein-Gordon-Dirac (KGD) system which involves a small dimensionless parameter 0<ε≤1. In the nonrelativistic limit regime, the KGD system admits rapid oscillations in time as ε→0+. In addition, the nonlinear Yukawa interaction and the indefinite Dirac operator bring other significant difficulties. The main idea of the MTI-FP method is to construct a precise multiscale decomposition by the frequency (MDF) to the solution of the KGD system at each time step and then employ the Fourier pseudospectral discretization for the spatial derivatives followed with the exponential wave integrator (EWI) for the time marching. This approach is explicit, easy to implement and preforms significantly better than the classical methods in the literature. More specifically, we rigorously establish the uniform error bounds at O(τ+hm0−1) for all ε∈(0,1] and optimal quadratic temporal error bounds at O(τ2) in the ε=O(1) regime, where τ is the time step size, h is mesh size and m0 depends on the regularity of the solution. Extensive numerical results demonstrate that our error bounds are optimal and sharp. Finally, we apply the MTI-FP method to numerically study the nonrelativistic limit behaviors of the KGD system when ε→0+.

I-SPin: an integrator for multicomponent Schrödinger-Poisson systems with self-interactions

We provide an algorithm and a publicly available code to numerically evolve multicomponent Schrödinger-Poisson (SP) systems with a SO(n) symmetry, including attractive or repulsive self-interactions in addition to gravity. Focusing on the case where the SP system represents the non-relativistic limit of a massive vector field, non-gravitational self-interactions (in particular spin-spin interactions) introduce complexities related to mass and spin conservation which are not present in purely gravitational systems. We address them with an analytical solution for the `kick' step in the algorithm, where we are able to decouple the multicomponent system completely. Equipped with this analytical solution, the full field evolution is second order accurate, preserves spin and mass to machine precision, and is reversible. Our algorithm allows for an expanding universe relevant for cosmology, and the inclusion of external potentials relevant for laboratory settings.

Massive color-kinematics duality and double-copy for Kaluza-Klein scattering amplitudes

We study the structure of scattering amplitudes of massive Kaluza-Klein (KK) states under toroidal compactification. We present a shifting method to quantitatively derive the scattering amplitudes of massive KK gauge bosons and KK gravitons from the corresponding massless amplitudes in the noncompactified higher dimensional theories. With these we construct the massive KK scattering amplitudes by extending the double-copy relations of massless scattering amplitudes within the field theory framework, including both the BCJ and CHY methods, and build up their connections to the massive KK KLT relations. We present the massive BCJ-type double-copy construction of the N-point KK gauge boson/graviton scattering amplitudes, and as the applications we derive explicitly the four-point KK scattering amplitudes as well as the five-point KK scattering amplitudes. We further study the nonrelativistic limit of these massive scattering amplitudes with the heavy external KK states and discuss the impact of the compactified extra dimensions on the low energy gravitational potential. Finally, we analyze the four-point and N-point mass spectral conditions and newly propose a novel group theory approach to prove that only the KK theories under toroidal compactification can satisfy these conditions for directly realizing massive double-copy in the field theory framework.

Relativistic energies and information entropy of the inversely quadratic Hellmann potential

The solutions to the Dirac equation are obtained in the spin and pseudospin symmetry limits are presented using the parametric Nikiforov-Uvarov (pNU) method with the inversely quadratic Hellman (IQH) potential. In the exact non-relativistic spin symmetry limit, the energy and wave function of the IQH potential are obtained and used to investigate the Shannon information entropy of the system. Numerical results of the relativistic energies of the spin and pseudospin symmetry limits of the Dirac equation with the IQH potential are presented and observed to exhibit degeneracy. Also, the results of the Shannon entropy for six states (n = 0, 1, 2, 3, 4, 5) show that the momentum space wave function and probability density are better localized than the position space wave function. Also, the Bialynicki-Birula and Mycielski (BBM) inequality is verified for the system. Our results are found to be consistent with those previously reported in the literature.

Obliquely propagating electron Bernstein wave in arbitrary degenerate plasma

Abstract On employing the linearized Vlasov–Maxwell model, the dispersion relation of obliquely propagating Bernstein wave in thermal electron gas including quantum effect of arbitrary/partial degeneracy in the presence of non-relativistic arbitrary magnetic field limits is derived. In particular, the results are obtained in the propagation range k z &gt; k x with k x ≠ 0 under high frequency (ω ≫ k.v) and weak propagation (Ω ≫ k.v or k → 0) limits. The propagation angle θ ′ $\left({\theta }^{\prime }\right)$ defines the obliqueness of wave such that for θ′ = 0° and 90° we obtain the perpendicular propagating pure Bernstein wave and parallel propagating Langmuir wave, respectively. The graphical analysis of newly reported results is made under the numerically evaluated values and the previously reported results are also retained. The possible applications of present results are found in partially/arbitrary dense astrophysical quantum plasma e.g., in brown dwarfs as well as they have wide ranging potential applications in modern technology e.g., in semi-conductors.

A transfer operator approach to relativistic quantum wavefunction * *Contribution to the Special Collection ‘Koopman Methods in Classical and Quantum-Classical Mechanics’.

The original intent of the Koopman–von Neumann formalism was to put classical and quantum mechanics on the same footing by introducing an operator formalism into classical mechanics. Here we pursue their path the opposite way and examine what transfer operators can say about quantum mechanical evolution. To that end, we introduce a physically motivated scalar wavefunction formalism for a velocity field on a 4-dimensional pseudo-Riemannian manifold, and obtain an evolution equation for the associated wavefunction, a generator for an associated weighted transfer operator. The generator of the scalar evolution is of first order in space and time. The probability interpretation of the formalism leads to recovery of the Schrödinger equation in the non-relativistic limit. In the special relativity limit, we show that the scalar wavefunction of Dirac spinors satisfies the new equation. A connection with string theoretic considerations for mass is provided.

Nonrelativistic limit of scalar and Dirac fields in curved spacetime

We give from first principles the non-relativistic limit of scalar and Dirac fields in curved spacetime. We aim to find general relativistic corrections to the quantum theory of particles affected by Newtonian gravity, a regime nowadays experimentally accessible. We believe that the ever-improving measurement accuracy and the theoretical interest in finding general relativistic effects in quantum systems require the introduction of corrections to the Schr\"{o}dinger-Newtonian theory. We rigorously determine these corrections by the non-relativistic limit of fully relativistic quantum theories in curved spacetime. For curved static spacetimes, we show how a non-inertial observer (equivalently, an observer in the presence of a gravitational field) can distinguish a scalar field from a Dirac field by particle-gravity interaction. We study the Rindler spacetime and discuss the difference between the resulting non-relativistic Hamiltonians. We find that for sufficiently large acceleration, the gravity-spin coupling dominates over the corrections for scalar fields, promoting Dirac particles as the best candidates for observing non-Newtonian gravity in quantum particle phenomenology.

Scrambling and entangling spinning particles

In this paper we revisit the gravitational eikonal amplitudes of two scattering spinning particles and inspect their scrambling power in the spin spaces that is quantified through the tripartite information. We found that in the non-relativistic limit and a special high-energy limit the leading contribution is a quantity that is universal and theory independent. The minimal coupling is singled out with minimal scrambling in a different high momenta limit. We also inspected the initial state dependence of entanglement generation and found that the spin coherent state with vanishing spin may not necessarily be the hardest to entangle. Interestingly, among a family of mixed states, the only P-rep state there known to be the best approximation of classical mixed states was singled out as one with minimal entanglement generated.