Quantum mechanics with spatial non-local effects and position-dependent mass

PDM creation and annihilation operators of the harmonic oscillators and the emergence of an alternative PDM-Hamiltonian

Quantum mechanical systems with position dependent masses (PDMs) admitting two parametric Lie symmetry groups are classified. Namely, all PDM systems are specified which, in addition to their invariance w.r.t. a two parametric Lie group, admit at least one second order integral of motion. The presented classification is partially extended to the more generic systems which do not accept any Lie group.

An inhomogeneous Kaluza-Klein compactification to four dimensions, followed by a conformal transformation, results in a system with position dependent mass (PDM). This origin of a PDM is quite different from the condensed matter one. A substantial generalization of a previously studied nonlinear oscillator with variable mass is obtained, wherein the position dependence of the mass of a nonrelativistic particle is due to a dilatonic coupling function emerging from the extra dimension. Previously obtained solutions for such systems can be extended and reinterpreted as nonrelativistic particles interacting with dilaton fields, which, themselves, can have interesting structures. An application is presented for the nonlinear oscillator, where within the new scenario the particle is coupled to a dilatonic string.

Information entropy has played a key role in a wide range of disciplines, for instance, classical and quantum information processing, quantum computing, quantum dynamics and quantum metrology. Here, we develop an information theoretic formalism using Shannon entropy, to investigate the quantum dynamics of Hamiltonian systems with position-dependent mass. Such systems are of fundamental interest in many areas, for instance, condensed matter, mathematical physics and foundations of quantum mechanics. We explore the phenomenon of fractional revivals for the temporal evolution of wave-packet solutions of Schrödinger equation with position-dependent mass by studying, analytically and numerically, the time-development of Shannon information entropy in position and momentum spaces. It is shown by our numerical results that the effect of spatially varying mass on the fractional revivals can not be fully harnessed using conventional measures, for instance, autocorrelation function. However, based on our numerical analysis it is concluded that information entropy is not only more sensitive to identify the fractional revivals but it also better elucidates the effect of position-dependent mass on the structure of fractional revivals in the form of symmetry breaking.

Chapter 6 - Quantum Solitons in Solid Matter

A universal practical formula is given for calculating an integral which includes two confluent hypergeometric functions, power and exponential functions; then by means of this formula, the expressions of the radial matrix elements for a relativistic harmonic oscillator are given.

Pseudospin symmetry in the relativistic harmonic oscillator

Stability of few-charge systems in quantum mechanics

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.

In principio erat verbum. It seems appropriate to add the first few words of St. John's gospel to the Latin quotations that were bandied about at the beginning of the recent Workshop on Cosmology, which was held at the Fermi National Accelerator Laboratory in Batavia, Ill. This outburst of Latinity and classical scholarship, which even included a dispute over whether the word apparatus belongs to the second or the fifth declension, seemed an appropriate beginning for a conference on a subject the origins of the universe that has fascinated scholars ever since classical times. In the beginning was the word, en arche hen ho logos. One of the meanings of logos in this context is a word descriptive of the basic structure of the universe and, more than descriptive, a dynamical word that outlines and perhaps even determines the course of the history that follows the beginning. From the philosophers of ancient Hellas to the professors of physics in modern universities, most have expected that this word would be mathematical. It is the last generation or two that has expected it to be quantum mechanical. Quantum mechanics has to describe everything, said Murray Gell-Mann of Caltech in Pasadena in the opening talk of the workshop. And it became clear that by everything he meant the work of historians and crime detectives and the songs of birds as well as the motions of galaxies. But he added: questions are all murky and border on the philosophical. The main reason for the dominance of quantum mechanics was cited by James Hartle of the University of California at Santa Barbara: laws of physics are quantum mechanical; quantum cosmology is the proper framework. Nevertheless, the large objects that dominate the universe as we now see it operate according to classical mechanics. The serious problem for all of physics, and for quantum cosmology in particular, is to find some kind of linkage by which classical physics can be generated out of quantum mechanics. A similar question occurs historically: If we trace the expansion of the universe backwards we eventually come to a point in history when the universe was so small that it had to behave as a whole as a quantum mechanical system. Figuring out how the present came out of the past is the problem.

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 generalized uncertainty relation is constructed based on Li-Ostoja-Starzewski fractional gradient operator of order 0 < α ≤ 1 introduced recently in literature which is motivated from dimensional regularization method. The new generalized uncertainty relation leads to a new form of Schrödinger equation and emergent position-dependent mass. Special forms of position-dependent mass were studied and the problem of a particle in a box was explored. Dissimilar forms of allowed quantum energies were obtained which lead to different scenarios. We have confronted the theory with observations by evaluating the maximum wavelength obtained in the 1,3-butadiene molecule. Several features were discussed accordingly.

The ordering problem in quantum systems with position-dependent mass (PDM) is treated by inclusion of the classically fictitious similarity transformation into the kinetic term. This provides a generation of supersymmetry with the first order supercharges from the kinetic term alone, while inclusion of the potential term allows also to generate nonlinear supersymmetry with higher order supercharges. A broad class of finite-gap systems with PDM is obtained by different reduction procedures, and general results on supersymmetry generation are applied to them. We show that elliptic finite-gap systems of Lame and Darboux-Treibich-Verdier types can be obtained by reduction to Seiffert's spherical spiral and Bernoulli lemniscate in the presence of Calogero-like or harmonic oscillator potentials, or by angular momentum reduction of a free motion on some AdS_2-related surfaces in the presence of Aharonov-Bohm flux. The limiting cases include the Higgs and Mathews-Lakshmanan oscillator models as well as a reflectionless model with PDM exploited recently in the discussion of cosmological inflationary scenarios.

In this article, we look at the stochastic Wu–Zhang system (SWZS) forced by multiplicative Brownian motion in the Itô sense. The mapping method, which is an effective analytical method, is employed to investigate the exact wave solutions of the aforementioned equation. The proposed scheme provides new types of exact solutions including periodic solitons, kink solitons, singular solitons and so on, to describe the wave propagation in quantum mechanics and analyze a wide range of essential physical phenomena. In the absence of noise, we obtain some previously found solutions of SWZS. Additionally, using the MATLAB program, the impacts of the noise term on the analytical solution of the SWZS were demonstrated.

The relativistic quantum dynamics of the generalized Klein–Gordon (KG) oscillator having position-dependent mass in the Gödel-type space–time is investigated. We have presented the generalized KG oscillator in this space–time, and discussed the effect of Cornell potential and linear potential for our considered system. The modification from the parameters of position-dependent mass and characterizing the space–time for the energy spectrums are presented.