arXiv subjects

Salvatore De Vincenzo

Publications and source records attributed to Salvatore De Vincenzo.

12 recordsLinked to original sources

arXiv2026

Constructing the Hamiltonian for a free 1D KFGM particle in an interval

We analyze the problem of a free 1D Klein-Fock-Gordon-Majorana (KFGM) particle in an interval. By free, we mean that there is no potential within the interval and that its walls are penetrable; hence, the pertinent energy current density does not vanish at the walls. Certainly, quantization in an interval is not trivial because certain restrictions imposed by the domains of the operators involved arise. Here, our objective is to obtain the Hamiltonian for these particles. In practice, the Feshbach-Villars (FV)--free Hamiltonian is the proper operator for characterizing them and is a function of the momentum operator. Additionally, a Majorana condition must also be imposed on the wavefunctions on which these two operators can act. Thus, we start by calculating the pseudo self-adjoint momentum operator. A three-parameter set of boundary conditions (BCs) constitutes its domain. Up to this point, the domain of the Hamiltonian is induced by the domain of the momentum operator; however, we ensure that only the BCs for which the energy current density has the same value at each end of the interval are in its domain. All these BCs essentially belong to a one-parameter set of BCs. Moreover, because the FV equation is invariant under the operation of parity, the parity-transformed wavefunction is also a solution of this equation, which further restricts the domain of the free FV Hamiltonian. Finally, knowing the most general three-parameter set of BCs for the pseudo self-adjoint FV Hamiltonian for a 1D KFGM particle in an interval, we find that only two BCs can remain within the domain of the FV--free Hamiltonian: the periodic BC and the antiperiodic BC. These BCs are satisfied by both the two-component FV wavefunction, with these components being related, and the one-component KFG wavefunction, which can be real or imaginary.

quant-ph↗

arXiv2025

Nontrivial local observables and impermeable and permeable boundary conditions for 1D KFGM particles

Real solutions of the 1D Klein-Fock-Gordon (KFG) equation automatically cancel out the usual two-vector current density; consequently, the respective continuity equation is trivially satisfied, and a globally conserved quantity cannot be obtained. Additionally, distinguishing between impermeable and permeable boundary conditions (BCs) at a given point is not possible. We address these first-quantized conflicts by using the simplest nontrivial local observables, i.e., an energy density and an energy current density, which allows us to characterize a strictly neutral 1D KFG particle, i.e., a 1D KFG-Majorana (KFGM) particle, when it is confined to an interval and when it is restricted, e.g., in an interval with transparent walls. All the BCs for this system are extracted from the pseudo self-adjointness of the Feshbach-Villars (FV) Hamiltonian plus two Majorana conditions. We show that these energy densities are components of an unusual energy-momentum tensor and can satisfy a continuity equation, leading to a conserved quantity for all available BCs. Moreover, the energy current density can characterize all the BCs as either impermeable or permeable. In contrast, the commonly used energy current density -- a component of the usual energy-momentum tensor -- cannot characterize all the BCs. Additionally, this quantity and its respective energy density -- another component of the usual energy-momentum tensor -- may not lead to a conserved quantity. We also obtain the BCs for which the abovementioned densities and those commonly used are equally satisfactory. In fact, this occurs only for four impermeable BCs and a one-parameter set of permeable BCs. Our results highlight the important role played by the BCs when they are imposed on a system in which particles occupy a finite region.

quant-ph↗

arXiv2025

On the locus formed by the maximum heights of an ultra-relativistic projectile

We consider the problem of relativistic projectiles in a uniform gravitational force field in an inertial frame. For the first time, we have found the curve that joins the points of maximum height of all trajectories followed by a projectile in the ultra-relativistic limit. The parametric equations of this curve produce an onion-like curve; in fact, it is one of the loops of a lemniscate-type curve. We also verify that the curve is an ellipse in the nonrelativistic approximation. These two limiting results are obtained by following two slightly distinct approaches. In addition, we calculate the nonrelativistic and ultra-relativistic approximations of the trajectory equation and parametric equations of the trajectory as functions of time. We also find the asymptotic behavior of the ultra-relativistic trajectory equations for points that are very distant from the launching point. All limiting cases in the article are studied in detail. Additionally, we discuss various physical issues related to some of our results. The content of the article is appropriate for advanced undergraduate students.

physics.class-ph↗

arXiv2023

Characterizing Klein-Fock-Gordon-Majorana particles in (1+1) dimensions

Theoretically, in (1+1) dimensions, one can have Klein-Fock-Gordon-Majorana (KFGM) particles. More precisely, these are one-dimensional (1D) Klein-Fock-Gordon (KFG) and Majorana particles at the same time. In principle, the wave equations considered to describe such first-quantized particles are the standard 1D KFG equation and/or the 1D Feshbach-Villars (FV) equation, each with a real Lorentz scalar potential and some kind of Majorana condition. The aim of this paper is to analyze the latter assumption fully and systematically; additionally, we introduce specific equations and boundary conditions to characterize these particles when they lie within an interval (or on a line with a tiny hole at a point). In fact, we write first-order equations in the time derivative that do not have a Hamiltonian form. We may refer to these equations as first-order 1D Majorana equations for 1D KFGM particles. Moreover, each of them leads to a second-order equation in time that becomes the standard 1D KFG equation when the scalar potential is independent of time. Additionally, we examine the nonrelativistic limit of one of the first-order 1D Majorana equations.

quant-ph↗

arXiv2023

On a paradox in quantum mechanics and its resolution

Consider a free Schrödinger particle inside an interval with walls characterized by the Dirichlet boundary condition. Choose a parabola as the normalized state of the particle that satisfies this boundary condition. To calculate the variance of the Hamiltonian in that state, one needs to calculate the mean value of the Hamiltonian and that of its square. If one uses the standard formula to calculate these mean values, one obtains both results without difficulty, but the variance unexpectedly takes an imaginary value. If one uses the same expression to calculate these mean values but first writes the Hamiltonian and its square in terms of their respective eigenfunctions and eigenvalues, one obtains the same result as above for the mean value of the Hamiltonian but a different value for its square (in fact, it is not zero); hence, the variance takes an acceptable value. From whence do these contradictory results arise? The latter paradox has been presented in the literature as an example of a problem that can only be properly solved by making use of certain fundamental concepts within the general theory of linear operators in Hilbert spaces. Here, we carefully review those concepts and apply them in a detailed way to resolve the paradox. Our results are formulated within the natural framework of wave mechanics, and to avoid inconveniences that the use of Dirac's symbolic formalism could bring, we avoid the use of that formalism throughout the article. In addition, we obtain a resolution of the paradox in an entirely formal way without addressing the restrictions imposed by the domains of the operators involved. We think that the content of this paper will be useful to undergraduate and graduate students as well as to their instructors.

quant-ph↗

arXiv2023

General pseudo self-adjoint boundary conditions for a 1D KFG particle in a box

We consider a 1D Klein-Fock-Gordon particle in a finite interval, or box. We construct for the first time the most general set of pseudo self-adjoint boundary conditions for the Hamiltonian operator that is present in the first order in time 1D Klein-Fock-Gordon wave equation, or the 1D Feshbach-Villars wave equation. We show that this set depends on four real parameters and can be written in terms of the one-component wavefunction for the second order in time 1D Klein-Fock-Gordon wave equation and its spatial derivative, both evaluated at the endpoints of the box. Certainly, we write the general set of pseudo self-adjoint boundary conditions also in terms of the two-component wavefunction for the 1D Feshbach-Villars wave equation and its spatial derivative, evaluated at the ends of the box; however, the set actually depends on these two column vectors each multiplied by the singular matrix that is present in the kinetic energy term of the Hamiltonian. As a consequence, we found that the two-component wavefunction for the 1D Feshbach-Villars equation and its spatial derivative do not necessarily satisfy the same boundary condition that these quantities satisfy when multiplied by the singular matrix. In any case, given a particular boundary condition for the one-component wavefunction of the standard 1D Klein-Fock-Gordon equation and using the pair of relations that arise from the very definition of the two-component wavefunction for the 1D Feshbach-Villars equation, the respective boundary condition for the latter wavefunction and its derivative can be obtained. Our results can be extended to the problem of a 1D Klein-Fock-Gordon particle moving on a real line with a point interaction (or a hole) at one point.

quant-ph↗

arXiv2023

The Dirac impenetrable barrier in the limit point of the Klein energy zone

We reanalyze the problem of a 1D Dirac single particle colliding with the electrostatic potential step of height $V_{0}$ with a positive incoming energy that tends to the limit point of the so-called Klein energy zone, i.e., $E\rightarrow V_{0}-\mathrm{m}c^{2}$, for a given $V_{0}$. In such a case, the particle is actually colliding with an impenetrable barrier. In fact, $V_{0}\rightarrow E+\mathrm{m}c^{2}$, for a given relativistic energy $E\,(<V_{0})$, is the maximum value that the height of the step can reach and that ensures the perfect impenetrability of the barrier. Nevertheless, we note that, unlike the nonrelativistic case, the entire eigensolution does not completely vanish, either at the barrier or in the region under the step, but its upper component does satisfy the Dirichlet boundary condition at the barrier. More importantly, by calculating the mean value of the force exerted by the impenetrable wall on the particle in this eigenstate and taking its nonrelativistic limit, we recover the required result. We use two different approaches to obtain the latter two results. In one of these approaches, the corresponding force on the particle is a type of boundary quantum force. Throughout the article, various issues related to the Klein energy zone, the transmitted solutions to this problem, and impenetrable barriers related to boundary conditions are also discussed. In particular, if the negative-energy transmitted solution is used, the lower component of the scattering solution satisfies the Dirichlet boundary condition at the barrier, but the mean value of the external force when $V_{0}\rightarrow E+\mathrm{m}c^{2}$ does not seem to be compatible with the existence of the impenetrable barrier.

quant-ph↗

arXiv2021

On the mean value of the force operator for 1D particles in the step potential

In the one-dimensional Klein-Fock-Gordon theory, the probability density is a discontinuous function at the point where the step potential is discontinuous. Thus, the mean value of the external classical force operator cannot be calculated from the corresponding formula of the mean value. To resolve this issue, we obtain this quantity directly from the Klein-Fock-Gordon equation in Hamiltonian form, or the Feshbach-Villars wave equation. Not without surprise, the result obtained is not proportional to the average of the discontinuity of the probability density but to the size of the discontinuity. In contrast, in the one-dimensional Schrödinger and Dirac theories this quantity is proportional to the value that the respective probability density takes at the point where the step potential is discontinuous. We examine these issues in detail in this paper. The presentation is suitable for the advanced undergraduate level.

quant-ph↗

arXiv2021

On wave equations for the Majorana particle in (3+1) and (1+1) dimensions

In general, the relativistic wave equation considered to mathematically describe the so-called Majorana particle is the Dirac equation with a real Lorentz scalar potential plus the so-called Majorana condition. Certainly, depending on the representation that one uses, the resulting differential equation changes. It could be a real or a complex system of coupled equations, or it could even be a single complex equation for a single component of the entire wave function. Any of these equations or systems of equations could be referred to as a Majorana equation or Majorana system of equations because it can be used to describe the Majorana particle. For example, in the Weyl representation, in (3+1) dimensions, we can have two non-equivalent explicitly covariant complex first-order equations; in contrast, in (1+1) dimensions, we have a complex system of coupled equations. In any case, whichever equation or system of equations is used, the wave function that describes the Majorana particle in (3+1) or (1+1) dimensions is determined by four or two real quantities. The aim of this paper is to study and discuss all these issues from an algebraic point of view, highlighting the similarities and differences that arise between these equations in the cases of (3+1) and (1+1) dimensions in the Dirac, Weyl, and Majorana representations. Additionally, to reinforce this task, we rederive and use results that come from a procedure already introduced by Case to obtain a two-component Majorana equation in (3+1) dimensions. Likewise, we introduce for the first time a somewhat analogous procedure in (1+1) dimensions and then use the results we obtain.

quant-ph↗

arXiv2020

On the boundary conditions for the 1D Weyl-Majorana particle in a box

In (1+1) space-time dimensions, we can have two particles that are Weyl and Majorana particles at the same time---1D Weyl-Majorana particles. That is, the right-chiral and left-chiral parts of the two-component Dirac wave function that satisfies the Majorana condition, in the Weyl representation, describe these particles, and each satisfies their own Majorana condition. Naturally, the nonzero component of each of these two two-component wave functions satisfies a Weyl equation. We investigate and discuss this issue and demonstrate that for a 1D Weyl-Majorana particle in a box, the nonzero components, and therefore the chiral wave functions, only admit the periodic and antiperiodic boundary conditions. From the latter two boundary conditions, we can only construct four boundary conditions for the entire Dirac wave function. Then, we demonstrate that these four boundary conditions are also included within the most general set of self-adjoint boundary conditions for a 1D Majorana particle in a box.

quant-ph↗

arXiv2020

On 3D and 1D Weyl particles in a 1D box

We construct the most general families of self-adjoint boundary conditions for three (equivalent) Weyl Hamiltonian operators, each describing a three-dimensional Weyl particle in a one-dimensional box situated along a Cartesian axis. These results are essentially obtained by using the most general family of self-adjoint boundary conditions for a Dirac Hamiltonian operator that describes a one-dimensional Dirac particle in a box, in the Weyl representation, and by applying simple changes of representation to this operator. Likewise, we present the most general family of self-adjoint boundary conditions for a Weyl Hamiltonian operator that describes a one-dimensional Weyl particle in a one-dimensional box. We also obtain and discuss throughout the article distinct results related to the Weyl equations in (3+1) and (1+1) dimensions, in addition to their respective wave functions, and present certain key results related to representations for the Dirac equation in (1+1) dimensions.

quant-ph↗

arXiv2020

On real solutions of the Dirac equation for a one-dimensional Majorana particle

We construct general solutions of the time-dependent Dirac equation in (1+1) dimensions with a Lorentz scalar potential, subject to the so-called Majorana condition, in the Majorana representation. In this situation, these solutions are real-valued and describe a one-dimensional Majorana single particle. We specifically obtain solutions for the following cases: a Majorana particle at rest inside a box, a free (i.e., in a penetrable box with the periodic boundary condition), in an impenetrable box with no potential (here we only have four boundary conditions), and in a linear potential. All these problems are treated in a very detailed and systematic way. In addition, we obtain and discuss various results related to real wave functions. Finally, we also wish to point out that, in choosing the Majorana representation, the solutions of the Dirac equation with a Lorentz scalar potential can be chosen to be real but do not need to be real. In fact, complex solutions for this equation can also be obtained. Thus, a Majorana particle cannot be described only with the Dirac equation in the Majorana representation without explicitly imposing the Majorana condition.

quant-ph↗