Foldy-Wouthuysen Transformation Research Articles

Foldy–Wouthuysen transformation of Dirac equation in the context of conformable fractional derivative

Foldy–Wouthuysen transformation of Dirac equation in the context of conformable fractional derivative

Foldy–Wouthuysen transformation of Dirac equation in the context of Magueijo–Smolin model of the doubly special relativity

In this paper, in the context of the Magueijo–Smolin (MS) model and using the Foldy–Wouthuysen transformation for relativistic spin-1/2 particles, we study the nonrelativistic limit (NRL) of the Dirac equation within the doubly special relativity. This leads to obtaining a deformed Schrödinger–Pauli equation; there, we test the influence of the MS model on the NRL of the Dirac equation. However, we also examine the efficacy and behavior of the Foldy–Wouthuysen transformation in deriving the NRL in the presence of MS model. Besides, we compute that the energies shift in the context of MS model and check if the deformed Schrödinger–Pauli equation form still shows explicitly the g-factor 2.

Foldy-Wouthuysen transformation and multiwave states of a graphene electron in external fields and free (2+1)-space

The relativistic Foldy-Wouthuysen transformation is used for an advanced description of planar graphene electrons in external fields and free (2+1)-space. It is shown that the initial Dirac equation should by based on the usual (4 × 4) Dirac matrices but not on the reduction of matrix dimensions and the use of (2 × 2) Pauli matrices. Nevertheless, the both approaches agree with the experimental data on graphene electrons in a uniform magnetic field. The pseudospin of graphene electrons is not the one-value spin and takes the values ±1/2. The exact Foldy-Wouthuysen Hamiltonian of a graphene electron in uniform and nonuniform magnetic fields is derived. The exact energy spectrum agreeing with the experiment and exact Foldy-Wouthuysen wave eigenfunctions are obtained. These eigenfunctions describe multiwave (structured) states in the (2+1)-space. It is proven that the Hermite-Gauss beams exist even in the free space. In the multiwave Hermite-Gauss states, graphene electrons acquire nonzero effective masses dependent on a quantum number and move with group velocities which are less than the Fermi velocity. Graphene electrons in a static electric field also can exist in the multiwave Hermite-Gauss states defining non-spreading coherent beams. These beams can be accelerated and decelerated.

Exact Foldy–Wouthuysen transformation for a Dirac equation describing the interaction of spin-1/2 relativistic particles with an external electromagnetic field

The Thomas–Bargmann–Michel–Telegdi (TBMT) equation is derived using the Exact Foldy–Wouthuysen transformation. Extra new terms were found, and we discuss their possible physical applications. The main point of this work is to detail the procedure to get the general result. We explicitly present the choice of parametrization we used on the initial Hamiltonian and the motivations to take it. We emphasize that the final equations can depend on this choice, and it is possible to prevent the manipulations of the quadratic Hamiltonian that become extremely cumbersome. More importantly, it is done in such a way that the transformed equations allow the direct separation into mass, kinetic, and interaction correction terms to the original TBMT equation.

Generalized Foldy-Wouthuysen transformation for relativistic two-component methods: Systematic analysis of two-component Hamiltonians.

The generalized Foldy-Wouthuysen (GFW) transformation was proposed as a generic form that unifies four types of transformations in relativistic two-component methods: unnormalized GFW(UN), and normalized form 1, form 2, and form 3 (GFW(N1), GFW(N2), and GFW(N3)). The GFW transformation covers a wide range of transformations beyond the simple unitary transformation of the Dirac Hamiltonian, allowing for the systematic classification of all existing two-component methods. New two-component methods were also systematically derived based on the GFW transformation. These various two-component methods were applied to hydrogen-like and helium-like ions. Numerical errors in energy were evaluated and classified into four types: the one-electron Hamiltonian approximation, the two-electron operator approximation, the newly defined "picture difference error (PDE)," and the error in determining the transformation, and errors in multi-electron systems were discussed based on this classification.

The effects of gravitational waves on a hydrogen atom

We investigate the influence of gravitational waves on a freely falling hydrogen atom by analyzing the dynamics of the bound electron described by the Dirac equation in the curved spacetime of a gravitational wave. From this, we derive the corresponding Dirac Hamiltonian in the local inertial frame of the atom, assuming gravitational waves are described by the linearized theory of general relativity. To maintain meaningful physical interpretations while obtaining a non-relativistic description, we employ the Foldy–Wouthuysen transformation. Through the analysis of resulting interaction terms and comparison with flat spacetime counterparts, valuable insights into the effects of gravitational waves (GWs) on the hydrogen atom are gained. Additionally, we explore selection rules governing the coupling between GWs and the atom and utilize first-order perturbation theory to quantify the induced energy shifts and spectral line splitting. This investigation contributes to our understanding of the interplay between quantum systems and gravitational waves, which could lead to alternative method of GWs indirect detection. However, measuring such tiny energy shifts would require a telescope with very high spectral resolution.

Foldy–Wouthuysen Transformation and Structured States of a Graphene Electron in External Fields and Free (2 + 1)-Space

Foldy–Wouthuysen Transformation and Structured States of a Graphene Electron in External Fields and Free (2 + 1)-Space

Non-Hermitian fermions with effective mass

In this work, we readdress the Dirac equation in the position-dependent mass (PDM) scenario. Here, one investigates the quantum dynamics of non-Hermitian fermionic particles with effective mass assuming a $(1+1)$-dimension flat spacetime. In seeking a Schr\"{o}dinger-like theory with PT symmetry is appropriate to assume a complex potential. This imaginary interaction produces an effective potential purely dependent on mass distribution. Furthermore, we study the non-relativistic limit by adopting the Foldy-Wouthuysen transformation. As a result, that limit leads to an ordering equivalent to the ordering that describes abrupt heterojunctions. Posteriorly, particular cases of mass distribution also were analyzed. Thus, interesting results arise for the case of a linear PDM, which produces an effective harmonic oscillator and induces the emergence of bound states. For a hyperbolic PDM, an effective potential barrier emerges. However, in this case, the fermions behave as free particles with positive-defined energy.

Foldy–Wouthuysen Transformation of Noncommutative Dirac Equation in the Presence of Minimal Uncertainty in Momentum

Foldy–Wouthuysen Transformation of Noncommutative Dirac Equation in the Presence of Minimal Uncertainty in Momentum

Douglas-Kroll and infinite order two-component transformations of Dirac-Fock operator.

We extended the conventional Douglas-Kroll (DK) and infinite order two-component (IOTC) methods to a technique applicable to Fock matrices, called extended DK (EDK) and extended IOTC (EIOTC), respectively. First, we defined a strategy to divide the Dirac-Fock operator into zero- and first-order terms. We then demonstrated that the first-order extended DK transformation, which is the Foldy-Wouthuysen transformation for the zero-order term, as well as the second- and third-order EDK and EIOTC, could be well defined. The EDK- and EIOTC-transformed Fock matrix, kinetic energy operator, nuclear attraction operator, and density matrix were derived. These equations were numerically evaluated, and it was found that these methods were accurate. In particular, EIOTC was consistent with the four-component approach. Four-component and extended two-component calculations are more expensive than non-relativistic calculations due to small-component-type two-electron integrals. We developed a new approximation formula, RIS-V, for small-component-type two-electron integrals, including the spin-orbit interaction between electrons. These results suggest that the RIS-V formula effectively accelerates the four-component and extended two-component methods.

High-order Hamiltonian obtained by Foldy–Wouthuysen transformation up to the order of mα 8

Complete relativistic corrections of an effective Hamiltonian for a single-particle system in an external electromagnetic field and their unitary equivalent form up to the order of mα 8 are obtained. The derivation is based on two approaches applying Foldy–Wouthuysen (FW) transformation to the Dirac Hamiltonian for a particle in an external electromagnetic field. The results are consistent with the previous work at the mα 6 and mα 8 order correction [Phys. Rev. A 71 012503 (2005); Phys. Rev. A 100 012513 (2019)]. We also further consider the effect of anomalous magnetic moments, namely, the Dirac–Pauli equation, and obtain FW-Hamiltonians at the same order. The results obtained can be used for the subsequent calculation of relativistic and radiation effects in simple atomic and molecular systems.

Foldy-Wouthuysen transformation in strong magnetic fields and relativistic corrections for quantum cyclotron energy levels

We carry out a direct, iterative Foldy--Wouthuysen transformation of a general Dirac Hamiltonian coupled to an electromagnetic field, including the anomalous magnetic moment. The transformation is carried out through an iterative disentangling of the particle and antiparticle Hamiltonians, in the expansion for higher orders of the momenta. The time-derivative term from the unitary transformation is found to be crucial in supplementing the transverse component of the electric field in higher orders. Final expressions are obtained for general combined electric and magnetic fields, including strong magnetic fields. The time-derivative of the electric field is shown to enter only in the seventh order of the fine-structure constant if the transformation is carried out in the standard fashion. We put special emphasis on the case of strong fields, which are important for a number of applications, such as electrons bound in Penning traps.

On the choice of relativistic wave equation for the particle having spin s = 3/2

Relativistic wave equation of motion without redundant components for the particle having spin 3/2 has been considered. In order to show the newness a comparison with the known equations for the spin s = 3/2 field is given. Therefore, the brief review of the relativistic wave equations for the particle with spin s = 3/2 is suggested. In our equation the wave function for the particle-antiparticle doublet contains only 8 components. The consideration is carried out both at the level of relativistic quantum mechanics and at the level of local field theory. The extended Foldy–Wouthuysen transformation, which gives the operator link between these two levels is suggested.

Many Body Current Density from Foldy–Wouthuysen Transformation of the Dirac–Coulomb Hamiltonian

This paper analyzes how special relativity changes the equation for the many-body-induced current density starting from the Foldy–Wouthuysen diagonalization of the Dirac–Coulomb Hamiltonian. This current density differs from that obtained with the Gordon decomposition due to the presence of a spin-orbit coupling contribution not considered before for many-body molecular systems. This contribution diverges on atomic nuclei due to the nature of the point charges considered in the nonrelativistic approach, demonstrating that conventionally used nonrelativistic methods are not suitable for dealing with spin effects such as spin-orbit coupling or effects smaller than α2, with α the fine structure constant, and that a fully relativistic approach with a finite charge should be used. Despite the singularity, the spin-orbit coupling current becomes an important contribution to the total current in open-shell systems with high-spin multiplicity and a high atomic number in the nuclear proximity. On long ranges, this contribution is overcome by the Coulomb potential and the derived electric field which decays very quickly for small distances from nuclear charges. An evaluation of this spin-orbit current has been performed in the linear response approach at the HF/DFT level of theory.

Expansion of the Two-particle Dirac Equation in Powers of 1/c to Higher Orders

Using an extension of the Foldy–Wouthuysen transformation to two-particle wave equations, we consider the problem of expansion of the two-body Dirac Hamiltonian in powers of 1/c up to the fourth order. The transformed Hamiltonian in an even-even form toorder 1/c4 is obtained. It is shown that the extra terms which can be eliminated by additional unitary transformations appear in the expansion in higher orders. As an example, the Breit equation for Coulomb particles is considered, and all the terms of order 1/c4 in its reduction are calculated. The obtained results can be used for the expansion of relativistic and quasirelativistic two-particle wave equations to higher orders.

Electron vortices in spin-orbit coupling system

There exists an electron vortex solution with orbital angular momentum quantum in a non-spin-orbit coupling system which has nonconservative orbital angular momentum. We discuss the system with spin-orbit coupling and nonconservative orbital angular momentum, and we can find that the electrons with the total angular momentum numbers also have vortex beam solutions. And the vortex beam is expressed as an entangled wave function of the spin wave function and the vortex wave function. Taking the electrons in the central force field for example, in this paper constructed is a spinor vortex structure which is caused by the propagation of electrons carrying a fixed quantum number of total angular momentum along the <i>z</i>-axis. The spinor vortex structure is under the condition that the orbital angular momentum caused by spin-orbit coupling is non-conserved but the total angular momentum is conserved. The corresponding electron vortex beams in spin-vortex entanglement are solved by perturbation method, and the Foldy-Wouthuysen transformation is utilized to show that the vortex solution of the four-component spinor does exist in the case of relativity, when the electron with a fixed total angular momentum quantum number propagates along the <i>z</i>-axis in the central force field. The spinor provides theoretical support for the existence of the vortex structure for the system where the orbital angular momentum is not conserved but the total angular momentum is conserved due to spin-orbit coupling.

Quantum Hydrodynamics of Spinning Particles in Electromagnetic and Torsion Fields

We develop a many-particle quantum-hydrodynamical model of fermion matter interacting with the external classical electromagnetic and gravitational/inertial and torsion fields. The consistent hydrodynamical formulation is constructed for the many-particle quantum system of Dirac fermions on the basis of the nonrelativistic Pauli-like equation obtained via the Foldy–Wouthuysen transformation. With the help of the Madelung decomposition approach, the explicit relations between the microscopic and macroscopic fluid variables are derived. The closed system of equations of quantum hydrodynamics encompasses the continuity equation, and the dynamical equations of the momentum balance and the spin density evolution. The possible experimental manifestations of the torsion in the dynamics of spin waves is discussed.

Energy of a pointlike neutron in an external electromagnetic field

A point-like neutron in an external electromagnetic field experiences a shift in energy that mimicks the effect of an actual structural deformation of an extended neutron, i.e., a proper polarizability. In order to be able to differentiate between the former and the latter, a Foldy-Wouthuysen transformation is constructed which yields the energy shift of a point-like neutron quadratic in the external field in a derivative expansion, generalizing a long-known result for the dipole electric polarizability due to Foldy. The ten leading Foldy contributions to the energy are determined for a zero-momentum neutron. In addition, eliminating the momentum operator in favor of the velocity operator, analogous results are derived for a zero-velocity neutron. In this case, operator ordering ambiguities are encountered that permit only a determination of eight of the ten Foldy terms.

Effective theories and non-minimal couplings in low-dimensional systems

Dimensionality aspects of non-minimal electromagnetic couplings are investigated. By means of the Foldy–Wouthuysen transformation, we attain (non-)relativistic interactions related to the non-minimal coupling in three-dimensional spacetime, for both the bosonic and fermionic fields. Next, we establish some comparisons and analyze particular situations in which the external electromagnetic fields are described either by Maxwell or Maxwell–Chern–Simons Electrodynamics. In addition, we consider the situation of a non-minimal coupling for the fermionic field in four dimensions, carry out its dimensional reduction to three dimensions and show that the three-dimensional scenario previously worked out can be recovered as a particular case. Finally, we discuss a number of structural aspects of both procedures.

Statistical properties of the two-dimensional Dirac oscillator with spin–orbit coupling

We investigate the statistical properties of a two-dimensional Dirac oscillator (2D DO) in the presence of a spin–orbit coupling, breaking its supersymmetry. The Hamiltonian of the problem is shown to admit an exact Foldy–Wouthuysen transformation, which allowed us to discard negative-energy states in carrying out our study. The main thermodynamical functions: the Helmholtz free energy, the mean energy, the entropy and the specific heat, are obtained from the canonical partition function, and their behaviours are analysed. The high temperature limit of different thermodynamical functions is probed analytically by utilizing the Euler–Maclaurin formula.