Position-dependent Mass Particle Research Articles

Deformed Fokker-Planck equation: Inhomogeneous medium with a position-dependent mass

We present the Fokker-Planck equation (FPE) for an inhomogeneous medium with a position-dependent mass particle by making use of the Langevin equation, in the context of a generalized deformed derivative for an arbitrary deformation space where the linear (nonlinear) character of the FPE is associated with the employed deformed linear (nonlinear) derivative. The FPE for an inhomogeneous medium with a position-dependent diffusion coefficient is equivalent to a deformed FPE within a deformed space, described by generalized derivatives, and constant diffusion coefficient. The deformed FPE is consistent with the diffusion equation for inhomogeneous media when the temperature and the mobility have the same position-dependent functional form as well as with the nonlinear Langevin approach. The deformed version of the H-theorem permits to express the Boltzmann-Gibbs entropic functional as a sum of two contributions, one from the particles and the other from the inhomogeneous medium. The formalism is illustrated with the infinite square well and the confining potential with linear drift coefficient. Connections between superstatistics and position-dependent Langevin equations are also discussed.

Path integral for quantum dynamics with position-dependent mass within the displacement operator approach

In the context of quantum mechanics reformulated in a modified Hilbert space, we can formulate the Feynman’s path integral approach for the quantum systems with position-dependent mass particle using the formulation of position-dependent infinitesimal translation operator. Which is similar a deformed quantum mechanics based on modified commutation relations. An illustration of calculation is given in the case of the harmonic oscillator, the infinite square well and the inverse square plus Coulomb potentials.

A new approach to the schrodinger equation with position-dependent mass and its implications in quantum dots and semiconductors

In this study, we introduce a new approach to construct the Schrödinger equation with position-dependent mass characterized by an emerging particle's effective mass, in perfect analogy with the problems involving a position-dependent mass particle in semiconductor heterostructures. We have discussed several of its properties and some of its implications in quantum dots and semiconductors. It was observed that the PDM approach affects only semiconductors with large interatomic scaling such as non-crystalline semiconductors and its effect on conventional semiconductors is insignificant.

A position-dependent mass harmonic oscillator and deformed space

We consider canonically conjugated generalized space and linear momentum operators x^q and p^q in quantum mechanics, associated with a generalized translation operator which produces infinitesimal deformed displacements controlled by a deformation parameter q. A canonical transformation (x^,p^)→(x^q,p^q) leads the Hamiltonian of a position-dependent mass particle in usual space to another Hamiltonian of a particle with constant mass in a conservative force field of the deformed space. The equation of motion for the classical phase space (x, p) may be expressed in terms of the deformed (dual) q-derivative. We revisit the problem of a q-deformed oscillator in both classical and quantum formalisms. Particularly, this canonical transformation leads a particle with position-dependent mass in a harmonic potential to a particle with constant mass in a Morse potential. The trajectories in phase spaces (x, p) and (xq, pq) are analyzed for different values of the deformation parameter. Finally, we compare the results of the problem in classical and quantum formalisms through the principle of correspondence and the WKB approximation.

Poisson brackets formulation for the dynamics of a position-dependent mass particle

Following the history of mechanics (Dugas in A history of mechanics, Dover, New York, 1988), we read that Poisson’s theorem figured in the 1811 edition of the celebrated Mecanique Analytique. Indeed, as it can be observed in the classical textbooks, Poisson brackets formulation is one of the cardinal chapters of analytical mechanics. Given the natural motivation to accommodate variable-mass systems at the level of classical analytical mechanics, we will herein direct our attention toward Poisson brackets formulation. To wit, considering the case of a position-dependent mass particle, in which we will assume that the absolute velocity of mass ejection or aggregation is a linear function of the generalized velocity, we will endeavor to provide such position-dependent mass problems with an appropriate Poisson brackets formulation. To our very best knowledge, this means an original contribution to the research field of the analytical mechanics of variable-mass systems. We will start establishing the Poisson brackets definition for the dynamics of a position-dependent mass particle, which will be posited in harmony with the classical mathematical portrait of analytical mechanics. Therefrom, we will demonstrate consequent results which give rise to the required formulation, namely, Jacobi’s identity, canonical equations expressed by means of such Poisson brackets and Poisson’s theorem. Last, we will apply Poisson’s theorem to evaluate the relationship between the conservation laws which are at our disposal in the domain of position-dependent mass problems.

Theorem on a new conservation law for the dynamics of a position-dependent mass particle

In this Note, we aim at proving a theorem on a new conservation law for the dynamics of a position-dependent mass particle. This new conservation law has the significant particularity of being concisely written in terms of the total energy of the problem. Here, we will consider the special case in which the absolute velocity of mass ejection or aggregation is a linear function of the generalized velocity. Our result is an original contribution in the traditional research field of variable-mass systems.

On a relativistic particle and a relativistic position-dependent mass particle subject to the Klein–Gordon oscillator and the Coulomb potential

The relativistic quantum dynamics of an electrically charged particle subject to the Klein–Gordon oscillator and the Coulomb potential is investigated. By searching for relativistic bound states, a particular quantum effect can be observed: a dependence of the angular frequency of the Klein–Gordon oscillator on the quantum numbers of the system. The meaning of this behaviour of the angular frequency is that only some specific values of the angular frequency of the Klein–Gordon oscillator are permitted in order to obtain bound state solutions. As an example, we obtain both the angular frequency and the energy level associated with the ground state of the relativistic system. Further, we analyse the behaviour of a relativistic position-dependent mass particle subject to the Klein–Gordon oscillator and the Coulomb potential.

Geometric theory on the dynamics of a position-dependent mass particle

The connection between geometry and dynamics is a canonical subject of analytical mechanics. A very traditional issue of this topic is the transformation of the mechanical problem at hand into a shortest-path problem. This means the mathematical translation of the dynamical problem into a problem of finding the geodesic of a certain space. In the classical domain of conservative systems, especially following the famous book of Lanczos, this translating bridge is established by the usual condition of constant total energy. By nature, the motion of a particle with position-dependent mass is not a conservative problem. Therefore, the classical geometrical theory of mechanics is not straightforwardly applicable. Given that, we here aim at developing the geometrical theory for the mechanics of a position-dependent mass particle. This is our intended contribution. To our best knowledge, the content of our single investigation is original within this variable mass context. Our theory will be developed in the light of the inverse problem of Lagrangian mechanics, which will accordingly sets the variational framework. From that, we will demonstrate the proper generalization of Euler-Maupertuis’ principle and the following generalization of Jacobi’s principle, which, analogously to the classical procedure, can be seen as intermediate steps to enter geometrical arguments. Then, the corresponding geodesic will appear. Finally, as a closing result, a theorem on the mathematical equivalence between such geodesic and the equation of motion of a position-dependent mass particle will be proved. Our investigation aims at providing the reader with a fundamental contribution to the geometry of variable mass mechanics.

A brief note on the analytical solution of Meshchersky’s equation within the inverse problem of Lagrangian mechanics

Meshchersky’s equation is a basic differential equation in the mechanics of variable-mass particles. This note particularly considers the case in which a one-dimensional and position-dependent mass particle is under the action of a potential force. The absolute velocity of mass ejection (or accretion) is supposed to be a linear function of the particle velocity. Within the formulation of the inverse problem of Lagrangian mechanics, an analytical solution of Meshchersky’s equation is here derived. The solution method follows from applying the concept of constant of motion of an extremum problem, which is a fundamental ground in the theory of invariant variational principles.

Generalized space and linear momentum operators in quantum mechanics

We propose a modification of a recently introduced generalized translation operator, by including a q-exponential factor, which implies in the definition of a Hermitian deformed linear momentum operator \documentclass[12pt]{minimal}\begin{document}$\hat{p}_q$\end{document}p̂q, and its canonically conjugate deformed position operator \documentclass[12pt]{minimal}\begin{document}$\hat{x}_q$\end{document}x̂q. A canonical transformation leads the Hamiltonian of a position-dependent mass particle to another Hamiltonian of a particle with constant mass in a conservative force field of a deformed phase space. The equation of motion for the classical phase space may be expressed in terms of the generalized dual q-derivative. A position-dependent mass confined in an infinite square potential well is shown as an instance. Uncertainty and correspondence principles are analyzed.

Bound states of spatially dependent mass Dirac equation with the Eckart potential including Coulomb tensor interaction

We investigate the approximate solutions of the Dirac equation with the position-dependent mass particle in the Eckart potential field including the Coulomb tensor interaction by using the parametric Nikiforov-Uvarov method. Taking an appropriate approximation to deal with the centrifugal term, the Dirac energy states and the corresponding normalized two-spinor components of the wave function are obtained in closed form. Some special cases of our solution are investigated. Furthermore, we present the correct solutions obtained via the asymptotic iteration method which are in agreement with the parametric Nikiforov-Uvarov method results.

Approximate -state solutions of the Dirac equation in spatially dependent mass for the Eckart potential including the Yukawa tensor interaction

We investigate the analytical approximation to arbitrary -state solutions of the Dirac equation with the position-dependent mass particle in the Eckart potential including the Yukawa tensor interaction in the framework of a parametric Nikiforov–Uvarov method. By taking a proper approximation to deal with the centrifugal term, the analytical relativistic energy eigenvalues and the corresponding normalized two-spinor components of the wave function are obtained in closed form. The relativistic and non-relativistic bound state solutions for some special cases, such as the Hulthén potential and the generalized Morse potential, are easily obtained from our general solution.

Displacement operator for quantum systems with position-dependent mass

A translation operator is introduced to describe the quantum dynamics of a position-dependent mass particle in a null or constant potential. From this operator, we obtain a generalized form of the momentum operator as well as a unique commutation relation for $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{\ensuremath{\gamma}}$. Such a formalism naturally leads to a Schr\"odinger-like equation that is reminiscent of wave equations typically used to model electrons with position-dependent (effective) masses propagating through abrupt interfaces in semiconductor heterostructures. The distinctive features of our approach are demonstrated through analytical solutions calculated for particles under null and constant potentials like infinite wells in one and two dimensions and potential barriers.

The bound energy spectrum of position-dependent mass particle in the null potential

The bound energy spectrum of a position-dependent mass particle which is trapped in a null potential is obtained by combining the point canonical transformation (PCT) method and the analytical transfer matrix (ATM) method. With the calculation for the transmission of a position-dependent mass particle passing through the null potential, the one-to-one correspondence between transmission resonance peak and eigenenergy is found and explained.

A singular position-dependent mass particle in an infinite potential well

An unusual singular position-dependent-mass particle in an infinite potential well is considered. The corresponding Hamiltonian is mapped through a point-canonical-transformation and an explicit correspondence between the target Hamiltonian and a Pöschl–Teller type reference Hamiltonian is obtained. New ordering ambiguity parametric setting are suggested.