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

As dynamical quantization of Einstein’s gravitational theory meets unsolved problems, it is worth considering the alternative method of quantization suggested by Fermi’s quantization of special-relativistic electrodynamics, which for that theory has been the starting point of most modern applications of quantum electrodynamics. This method avoids first-class constraints by an alteration of the Lagrangian. In physical formulas, this introduces unwanted terms, that are at the end equated weakly to zero by auxiliary conditions. By absence of first-class constraints, this theory could be quantized canonically, if it would not contain fermion fields. As shown by Dirac, in the presence of fermion fields the canonical commutation relations of the altered theory have to be replaced by modified commutation relations. The ultimate purpose of this and following papers is to prove the covariance of Dirac’s modified commutation relations, first under infinitesimal transformations, and thence under finite transformations. As usual, this requires the proof of existence of a conserved and invariant generator for the transformations admitted, here coordinate transformations and local Lorentz transformations of the tetrad field. For infinitesimal transformations, the conventional method of deriving from the Lagrangian density ℒ a generator T1 linear in the infinitesimal parameters that determine the transformations is used. However, for guaranteeing integrability of the procedure for generating transformations of the field variables, from infinitesimal transformations to finite transformations, it is necessary to show the existence of a more accurate generator T, no longer linear in the parameters, which will by eiTFe−iT−F generate the transformations δ̄F of the field variables F with second-order accuracy. While it is left to a following paper to discuss the exact form of Dirac’s modified commutation relations and to prove the conservation and covariance of the generator T and to prove that the conventional first-order generator T1 will, this time by Dirac’s modified commutation relations, generate the first-order substantial variations δ̄1F of the field variables, in the present paper formulas are derived for expressing the second-order-infinitesimal terms in the transformation formulas by means of the first-order terms, and a method is derived for obtaining from T1 the more accurate generator T. Finally, it is verified that the transformations F(x)→F′(x′) =F(x)+δ̄F generated by the generators T do satisfy the transformation group property with second-order accuracy.

We present a new approach to study the one-dimensional Dirac equation in the background of a position-dependent mass m. Taking the Fermi velocity v f to be a local variable, we explore the resulting structure of the coupled equations and arrive at an interesting constraint of m turning out to be the inverse square of v f. We address several solvable systems that include the free particle, shifted harmonic oscillator, Coulomb and nonpolynomial potentials. In particular, in the supersymmetric quantum mechanics context, the upper partner of the effective potential yields a new form for an inverse quadratic functional choice of the Fermi velocity.

Using a path integral approach, we consider a fractional Schrödinger equation with delta-perturbed infinite square well. The Lévy path integral, which is generalized from the Feynman path intergal for the propagator, is expanded into a perturbation series. From this, the energy-dependent Green's function is obtained.

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

A systematic classification of Feynman path integrals in quantum mechanics is presented and a table of solvable path integrals is given which reflects the progress made during the last 15 years, including, of course, the main contributions since the invention of the path integral by Feynman in 1942. An outline of the general theory is given which will serve as a quick reference for solving path integrals. Explicit formulas for the so-called basic path integrals are presented on which the general scheme to classify and calculate path integrals in quantum mechanics is based.

We extend the standard treatment of the asymmetric infinite square well to include solutions that have zero curvature over part of the well. This type of solution, both within the specific context of the asymmetric infinite square well and within the broader context of bound states of arbitrary piecewise-constant potential energy functions, is not often discussed as part of quantum mechanics texts at any level. We begin by outlining the general mathematical condition in one-dimensional time-independent quantum mechanics for a bound-state wavefunction to have zero curvature over an extended region of space and still be a valid wavefunction. We then briefly review the standard asymmetric infinite square well solutions, focusing on zero-curvature solutions as represented by energy eigenstates in position and momentum space.

The method of Path Integrals (PI’s) was developed by Richard Feynman in the 1940’s. It offers an alternate way to look at quantum mechanics (QM), which is equivalent to the Schrodinger formulation. As will be seen in this project work, many elementary problems are much more difficult to solve using path integrals than ordinary quantum mechanics. The benefits of path integrals tend to appear more clearly while using quantum field theory (QFT) and perturbation theory. However, one big advantage of Feynman’s formulation is a more intuitive way to interpret the basic equations than in ordinary quantum mechanics. Here we give a basic introduction to the path integral formulation, starting from the well known quantum mechanics as formulated by Schrodinger. We show that the two formulations are equivalent and discuss the quantum mechanical interpretations of the theory, as well as the classical limit. We also perform some explicit calculations by solving the free particle and the harmonic oscillator problems using path integrals. The energy eigenvalues of the harmonic oscillator is found by exploiting the connection between path integrals, statistical mechanics and imaginary time.

The Feynman path integral has revolutionized modern approaches to quantum physics. Although the path integral formalism has proven very successful and spawned several approximation schemes, the direct evaluation of real-time path integrals is still extremely expensive and numerically delicate due to its high-dimensional and oscillatory nature. We propose an efficient method for the numerical evaluation of the real-time world-line path integral for theories where the potential is dominated by a quadratic at infinity. This is done by rewriting the high-dimensional oscillatory integral in terms of a series of low-dimensional oscillatory integrals, that we efficiently evaluate with Picard-Lefschetz theory or approximate with the eikonal approximation. Subsequently, these integrals are stitched together with a series of fast Fourier transformations to recover the lattice regularized Feynman path integral. Our method directly applies to problems in quantum mechanics, the word-line quantization of quantum field theory, and quantum gravity. Published by the American Physical Society 2025

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.

We examine a deformed quantum mechanics in which the commutator between coordinates and momenta is a function of momenta. The Jacobi identity constraint on a two-parameter class of such modified commutation relations (MCR's) shows that they encode an intrinsic maximum momentum; a sub-class of which also imply a minimum position uncertainty. Maximum momentum causes the bound state spectrum of the one-dimensional harmonic oscillator to terminate at finite energy, whereby classical characteristics are observed for the studied cases. We then use a semi-classical analysis to discuss general concave potentials in one dimension and isotropic power-law potentials in higher dimensions. Among other conclusions, we find that in a subset of the studied MCR's, the leading order energy shifts of bound states are of opposite sign compared to those obtained using string-theory motivated MCR's, and thus these two cases are more easily distinguishable in potential experiments.

Feynman’s path integral approach is studied in the framework of the Wigner–Dunkl deformation of quantum mechanics. We start with reviewing some basics from Dunkl theory and investigate the time evolution of a Gaussian wave packet, which exhibits the same dispersion relation as observed in standard quantum mechanics. Feynman’s path integral approach is then extended to Wigner–Dunkl quantum mechanics. The harmonic oscillator problem is solved explicitly. We then look at the Euclidean time evolution and the related Dunkl process. This process, which exhibit jumps, can be represented by two continuous Bessel processes, one with reflection and one with absorption at the origin. The Feynman–Kac path integral for the harmonic oscillator problem is explicitly calculated.

The infinite square-well potential in one dimension is solved using Feynman path integration. This solution uses an image point method equivalent to a sum over classical paths.

The authors examine several topical subjects, commencing with a general introduction to path integrals in quantum mechanics and the group theoretical backgrounds for path integrals. Applications of harmonic analysis, polar coordinate formulation, various techniques and path integrals of SU(2) and SU(1,1) are discussed. Soluble examples presented include particle-flux system, a pulsed oscillator, magnetic monopole, the Coulomb problems in curved space and others. The second part deals with the SU(2) coherent states and their applications. Construction and generalization of the SU(2) coherent states, formulation of coherent path integrals for spin and unitary spin, and semiclassical quantization are presented. Applications are made to the study of quantum fluctuation, the nonlinear field model and phase holonomy. The final chapters present the theory of the SU(1,1) coherent states and their applications. The radial Coulomb problem, the Morse oscillator, and the large-N approximation are discussed. Applications to problems in quantum optics such as squeezed states, interaction with the squeezed vacuum states, and phase operator formalism are also included. This book is intended as an introduction to the subject as well as a work of reference.

A new higher order Schrödinger equation characterized by a position-dependent mass is introduced based on long-range spatial kernel effects and von Roos arguments. The extended Schrödinger equation depends on the sign of the moments M k , k = 0 , 1 , 2 , … and a stabilized quantum dynamic is realized for M 2 > 0 and M 4 > 0 . We have discussed its implications in several quantum mechanical systems where more than a few were raised, mainly the emergence of the quantum Pais-Uhlenbeck and the relativistic quantum harmonic oscillators besides the complex periodic potential characterized by a PT symmetry.

In this paper, we present a method for studying systems in the modified formulation of quantum mechanics known as Snyder space, proposed by Snyder (1947 Phys. Rev. 71 38–41). Snyder space predicts a modified commutation algebra for position and momentum operators. The method described in this paper introduces operators satisfying the canonical commutation relations and relates them to the position and momentum operators of Snyder space, effectively mapping a problem in Snyder space into a similar problem in standard quantum mechanics. The method is applied to the simple harmonic oscillator (SHO) in one and two dimensions as well as to the one-dimensional infinite square well. The energy spectra are calculated perturbatively for the SHO. We also find an exact spectrum for the one-dimensional infinite square well potential. These results are shown to agree with similar results found elsewhere in the literature.