A class of exactly solvable Schrödinger equation with moving boundary condition

Using first and second order supersymmetry formalism we obtain a class of exactly solvable potentials subject to moving boundary condition.     

Al-Salam-Chihara models of quantum oscillator

Models of the quantum oscillator, based on Al-Salam-Chihara orthogonal q-polynomials, are constructed. The position and momentum operators have the continuous simple spectra, covering a finite interval. Eigenfunctions of these operators are explicitly defined. The evolution operator is an integral operator with a kernel, whose explicit form is also derived.     

Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey–Wilson polynomials

Two sets of infinitely many exceptional orthogonal polynomials related to the Wilson and Askey–Wilson polynomials are presented. They are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical Hamiltonians, which are deformations of those for the Wilson and Askey–Wilson polynomials in terms of a degree ?   (?=1,2,…$?=1,2,\dots$) eigenpolynomial. These polynomials are exceptional in the sense that they start from degree ??1$??1$ and thus not constrained by any generalisation of Bochner's theorem.     

On the solutions of 3-particle spin elliptic Calogero-Moser systems

We describe a class of the singular solutions to the multicomponent analogs of the Lamé equation, arising as equations of motion of the elliptic Calogero-Moser systems of particles carrying spin 1/2. At special value of the coupling constant we propose the ansatz which allows one to get meromorphic solutions with two arbitrary parameters. They are quantized upon the requirement of the regularity of the wave function on the hyperplanes at which particles meet and imposing periodic boundary conditions. We find also the extra integrals of motion for three-particle systems which commute with the Hamiltonian for arbitrary values of the coupling constant.     

Feynman operator calculus and singular quantum oscillator

New applications of Feynman disentangling method in quantum mechanics are studied and the time-dependent singular oscillator problem is solved in this approach. The important role of representation group theory is discussed in this context.     

Exactly solvable energy-dependent potentials

We introduce a method for constructing exactly-solvable Schrödinger equations with energy-dependent potentials. Our method is based on converting a general linear differential equation of second order into a Schrödinger equation with energy-dependent potential. Particular examples presented here include harmonic oscillator, Coulomb and Morse potentials with various types of energy dependence.     

Generating functions and sum rules for quantum oscillator

Generating functions and sum rules are discussed for transition probabilities between quantum oscillator eigenstates with time-dependent parameters.     

Exactly solvable associated Lamé potentials and supersymmetric transformations

A systematic procedure to derive exact solutions of the associated Lamé equation for an arbitrary value of the energy is presented. Supersymmetric transformations in which the seed solutions have factorization energies inside the gaps are used to generate new exactly solvable potentials; some of them exhibit an interesting property of periodicity defects.     

Number-phase entropic uncertainty relations and Wigner functions for solvable quantum systems with discrete spectra

In this Letter, the “number-phase entropic uncertainty relation” and the “number-phase Wigner function” of generalized coherent states associated to a few solvable quantum systems with non-degenerate spectra are studied. We also investigate time evolution of “number-phase entropic uncertainty” and “Wigner function” of the considered physical systems with the help of temporally stable Gazeau-Klauder coherent states.     

Curvature-induced quantum behaviour on a helical nanotube

We investigate the effect of curvature on the behaviour of a quantum particle bound to move on a surface shaped as a helical tube. We derive and discuss the governing Schrödinger equation and the corresponding quantum effective potential which is periodic and points to the helical configuration as more energetically favorable as compared to the straight tube. The exhibited periodicity also leads to energy band structure of pure geometrical origin.     

New approach for calculating the energy level of quantum harmonic oscillator with invariable energy gap

To overcome the difficulty of calculating the energy level of quantum harmonic oscillator via traditional method, we derive a new approach by using the normal ordering of the operator and the invariant eigen-operator. Based on this method, we find the energy level of singular harmonic oscillator.     

Infinite quantum well: A coherent state approach

A new family of 2-component vector-valued coherent states for the quantum particle motion in an infinite square well potential is presented. They allow a consistent quantization of the classical phase space and observables for a particle in this potential. We then study the resulting position and (well-defined) momentum operators. We also consider their mean values in coherent states and their quantum dispersions.     

Analytical approximation to the solution of the Dirac equation with the Eckart potential including the spin-orbit coupling term

By using the supersymmetric WKB approximation approach and the functional analysis method, we solve approximately the Dirac equation with the Eckart potential for the arbitrary spin-orbit quantum number κ. The bound state energy eigenvalues and the associated two-component spinors of the Dirac particles are obtained approximately.     

The time-dependent quantum harmonic oscillator revisited: Applications to quantum field theory

In this article, we formulate the study of the unitary time evolution of systems consisting of an infinite number of uncoupled time-dependent harmonic oscillators in mathematically rigorous terms. We base this analysis on the theory of a single one-dimensional time-dependent oscillator, for which we first summarize some basic results concerning the unitary implementability of the dynamics. This is done by employing techniques different from those used so far to derive the Feynman propagator. In particular, we calculate the transition amplitudes for the usual harmonic oscillator eigenstates and define suitable semiclassical states for some physically relevant models. We then explore the possible extension of this study to infinite dimensional dynamical systems. Specifically, we construct Schrödinger functional representations in terms of appropriate probability spaces, analyze the unitarity of the time evolution, and probe the existence of semiclassical states for a wide range of physical systems, particularly, the well-known Minkowskian free scalar fields and Gowdy cosmological models.     

Simple unified derivation and solution of Coulomb, Eckart and Rosen-Morse potentials in prepotential approach

The four exactly solvable models related to non-sinusoidal coordinates, namely, the Coulomb, Eckart, Rosen-Morse type I and II models are normally being treated separately, despite the similarity of the functional forms of the potentials, their eigenvalues and eigenfunctions. Based on an extension of the prepotential approach to exactly and quasi-exactly solvable models proposed previously, we show how these models can be derived and solved in a simple and unified way.     

Prepotential approach to exact and quasi-exact solvabilities

Exact and quasi-exact solvabilities of the one-dimensional Schrödinger equation are discussed from a unified viewpoint based on the prepotential together with Bethe ansatz equations. This is a constructive approach which gives the potential as well as the eigenfunctions and eigenvalues simultaneously. The novel feature of the present work is the realization that both exact and quasi-exact solvabilities can be solely classified by two integers, the degrees of two polynomials which determine the change of variable and the zeroth order prepotential. Most of the well-known exactly and quasi-exactly solvable models, and many new quasi-exactly solvable ones, can be generated by appropriately choosing the two polynomials. This approach can be easily extended to the constructions of exactly and quasi-exactly solvable Dirac, Pauli, and Fokker-Planck equations.     

Exact solution of the Dirac equation with the Mie-type potential under the pseudospin and spin symmetry limit

We investigate the exact solution of the Dirac equation for the Mie-type potentials under the conditions of pseudospin and spin symmetry limits. The bound state energy equations and the corresponding two-component spinor wave functions of the Dirac particles for the Mie-type potentials with pseudospin and spin symmetry are obtained. We use the asymptotic iteration method in the calculations. Closed forms of the energy eigenvalues are obtained for any spin-orbit coupling term κ. We also investigate the energy eigenvalues of the Dirac particles for the well-known Kratzer-Fues and modified Kratzer potentials which are Mie-type potentials.     

Symplectic capacities and the geometry of uncertainty: The irruption of symplectic topology in classical and quantum mechanics

This paper aims at conducting an analysis of various uncertainty principles from a topological point of view where the notion of symplectic capacity plays a key role. The existence of symplectic capacities follows from a deep theorem of symplectic topology, Gromov’s non-squeezing theorem, which was discovered in the mid 1980’s, and which has led to numerous developments whose applications to Physics are not fully understood or exploited at the time of writing. We will show that the notion of symplectic non-squeezing contains, as a watermark, not only the Robertson–Schrödinger uncertainty relations (and a classical version thereof), but also Hardy’s uncertainty principle for a function and its Fourier transform. This observation will allow us to formulate the characterization of positivity for density matrices in a topological way. We also address some open questions and conjectures, whose solution cannot be given at the present time due to the lack of a sufficiently developed mathematical theory.     

Probability sum rules and consistent quantum histories

An example shows that weak decoherence is more restrictive than the minimal logical decoherence structure that allows probabilities to be used consistently for quantum histories. The probabilities in the sum rules that define minimal decoherence are all calculated by using a projection operator to describe each possibility for the state at each time. Weak decoherence requires more sum rules. They bring in additional variables, that require different measurements and a different way to calculate probabilities, and raise questions of operational meaning. The example shows that extending the linearly positive probability formula from weak to minimal decoherence gives probabilities that are different from those calculated in the usual way using the Born and von Neumann rules and a projection operator at each time.