New scenarios for classical and quantum mechanical systems with position-dependent mass

We analyze the dynamical equations obeyed by a classical system with position-dependent mass. It is shown that there is a non-conservative force quadratic in the velocity associated to the variable mass. We construct the Lagrangian and the Hamiltonian for this system and find the modifications required in the Euler-Lagrange and Hamilton's equations to reproduce the appropriate Newton's dynamical law. Since the Hamiltonian is not time invariant, we get a constant of motion suited to write the dynamical equations in the form of the Hamilton's ones. The time-dependent first integrals of motion are then obtained from the factorization of such a constant. A canonical transformation is found to map the variable mass equations to those of a constant mass. As particular cases, we recover some recent results for which the dependence of the mass on the position was already unnoticed, and find new solvable potentials of the P\"oschl-Teller form which seem to be new. The latter are associated to either the su(1,1) or the su(2) Lie algebras depending on the sign of the Hamiltonian.

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.

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.

We develop a combined machine learning (ML) and quantum mechanics approach that enables data-efficient reconstruction of flexible molecular force fields from high-level ab initio calculations, through the consideration of fundamental physical constraints. We discuss how such constraints are recovered and incorporated into ML models. Specifically, we use conservation of energy - a fundamental property of closed classical and quantum mechanical systems -- to derive an efficient gradient-domain machine learning (GDML) model. The challenge of constructing conservative force fields is accomplished by learning in a Hilbert space of vector-valued functions that obey the law of energy conservation. We proceed with the development of a multi-partite matching algorithm that enables a fully automated recovery of physically relevant point-group and fluxional symmetries from the training dataset into a symmetric variant of our model. The developed symmetric GDML (sGDML) approach is able to faithfully reproduce global force fields at the accuracy high-level ab initio methods, thus enabling sample intensive tasks like molecular dynamics simulations at that level of accuracy.

PDM-charged particles in PD-magnetic plus Aharonov–Bohm flux fields: Unconfined “almost-quasi-free” and confined in a Yukawa plus Kratzer exact solvability

We consider the problem of removal of ordering ambiguity in position dependent mass quantum systems characterized by a generalized position dependent mass Hamiltonian which generalizes a number of Hermitian as well as non-Hermitian ordered forms of the Hamiltonian. We implement point canonical transformation method to map one-dimensional time-independent position dependent mass Schrödinger equation endowed with potentials onto constant mass counterparts which are considered to be exactly solvable. We observe that a class of mass functions and the corresponding potentials give rise to solutions that do not depend on any particular ordering, leading to the removal of ambiguity in it. In this case, it is imperative that the ordering is Hermitian. For non-Hermitian ordering, we show that the class of systems can also be exactly solvable and is also shown to be iso-spectral using suitable similarity transformations. We also discuss the normalization of the eigenfunctions obtained from both Hermitian and non-Hermitian orderings. We illustrate the technique with the quadratic Liénard type nonlinear oscillators, which admit position dependent mass Hamiltonians.

Wilhelm Weber’s electrodynamics is an action-at-a-distance theory which has the property that equal charges inside a critical radius become attractive. Weber’s electrodynamics inside the critical radius can be interpreted as a classical Hamiltonian system whose kinetic energy is, however, expressed with respect to a Lorentzian metric. In this article we study the Schrödinger equation associated with this Hamiltonian system, and relate it to Weyl’s theory of singular Sturm–Liouville problems.

This study explores the influence of a Stark-like perturbative potential on a quantum particle confined to a cylindrical surface (QPCS) and its implications for extra-dimensional theories. The QPCS framework is particularly relevant to Kaluza-Klein (KK) theory, which postulates extra spatial dimensions to unify electromagnetism and gravity. In KK theory, these extra dimensions are typically hidden and require high-energy conditions for detection. Motivated by the challenge of uncovering these dimensions more feasibly, this research applies a perturbative potential of the form ĤSL = βzV₀z(θ) to a QPCS characterized by length L and radius R₀. This potential is inspired by the Stark effect in hydrogen atoms, where energy level splitting serves as an indicator of an external influence. The study demonstrates that, for a degenerate configuration (R₀ = L/π), the Stark-like perturbation effectively induces energy level splitting, which can be interpreted as a means of revealing hidden dimensions. The first-order energy correction in this scenario depends explicitly on the quantum numbers nz and nθ, highlighting the potential for this approach to probe extra-dimensional effects in lower-energy quantum systems.

In this work, we discuss the case of a non-relativistic quantum particle in the cosmic string space-time under the influence of the presence of external magnetic and exponential potential. It’s shown with the help of the Laplace transform method the second-order differential equation of the Schrodinger equation is solvable where it is reduced to a first-order differential equation. We obtain the eigenfunction and the non-relativistic energy spectrum. In addition, these results may be useful in the future study of thermodynamic, optic, and magnetic properties of the atomic interaction of a system particle. Keywords: Schrodinger equation; Laplace transform method; Position dependent mass; External magnetic field; Cosmic string

The behavior of any physical system is governed by its underlying dynamical equations. Much of physics is concerned with discovering these dynamical equations and understanding their consequences. In this Letter, we show that, remarkably, identifying the underlying dynamical equation from any amount of experimental data, however precise, is a provably computationally hard problem (it is NP hard), both for classical and quantum mechanical systems. As a by-product of this work, we give complexity-theoretic answers to both the quantum and classical embedding problems, two long-standing open problems in mathematics (the classical problem, in particular, dating back over 70years).

We formulate from first principles a theory of stochastic processes in configuration space. The fundamental equations of the theory are an equation of motion which generalizes Newton's second law and an equation which expresses the condition of conservation of matter. Two types of stochastic motion are possible, both described by the same general equations, but leading in one case to classical Brownian motion behavior and in the other to quantum mechanical behavior. The Schrodinger equation, which is derived here with no further assumption, is thus shown to describe a specific stochastic process. It is explicitly shown that only in the quantum mechanical process does the superposition of probability amplitudes give rise to interference phenomena; moreover, the presence of dissipative forces in the Brownian motion equations invalidates the superposition principle. At no point are any special assumptions made concerning the physical nature of the underlying stochastic medium, although some suggestions are discussed in the last section.

On a connection between quantum and classical mechanical systems

A simple $U(1)$ gauge transformation, which is equivalent to the inclusion of the Aharonov-Bohm type potential, is introduced and applied to superintegrable systems in classical and quantum mechanics. It produces an inverse-square potential with two- and three-particle interactions which is identical to the potential of the recently studied truncated Calogero model.

Quadratic algebras are generalizations of Lie algebras which include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by B\^ocher contractions of the conformal Lie algebra $\mathfrak{so}(4,\mathbb {C})$ to itself. In 2 dimensions there are two kinds of quadratic algebras, nondegenerate and degenerate. In the geometric case these correspond to 3 parameter and 1 parameter potentials, respectively. In a previous paper we classified all abstract parameter-free nondegenerate quadratic algebras in terms of canonical forms and determined which of these can be realized as quadratic algebras of 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces, and studied the relationship between B\^ocher contractions of these systems and abstract contractions of the free quadratic algebras. Here we carry out an analogous study of abstract parameter-free degenerate quadratic algebras and their possible geometric realizations. We show that the only free degenerate quadratic algebras that can be constructed in phase space are those that arise from superintegrability. We classify all B\^ocher contractions relating degenerate superintegrable systems and, separately, all abstract contractions relating free degenerate quadratic algebras. We point out the few exceptions where abstract contractions cannot be realized by the geometric B\^ocher contractions.

Using the well known position-dependent mass (PDM) von Roos Hamiltonian et al (2009 J. Phys. A: Math. Theor. 42 025304) have studied the problem of two-dimensional PDM particles in the presence of magnetic fields. They have reported exact solutions for the wavefunctions and energies. In the first part of their study ‘PDM-Shrödinger equation in two-dimensional Cartesian coordinates’, they have used the so called Zhu and Kroemer’s ordering and (Zhu and Kroemer 1983 Phys. Rev. B 27 3519). While their treatment for this part is correct beyond doubt, their treatment of the second part of ‘PDM in a magnetic field’ is improper. We address these improper treatments and report the correct presentation for minimal coupling under PDM settings.