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Radosław Szmytkowski

Publications and source records attributed to Radosław Szmytkowski.

15 recordsLinked to original sources
arXiv2026

Dynamical quasi-symmetry of the classical Kepler-Coulomb problem and the conservation of the Laplace-Runge-Lenz vector

The paper studies a classical non-relativistic particle moving in $\mathbb{R}^{N}$, $(N\geqslant2)$, in the presence of the Kepler-Coulomb force $\boldsymbol{F}(\boldsymbol{r})=α\boldsymbol{r}/r^{3}$, $(α\neq0)$. It is shown that the extended-configuration-space transformation involving the projective transformation of the radius vector $\boldsymbol{r}\to\boldsymbol{R}=\boldsymbol{r}/(1+\boldsymbol{c}\cdot\boldsymbol{r})$ and the differential time scaling $\mathrm{d}t\to\mathrm{d}T=\mathrm{d}t/(1+\boldsymbol{c}\cdot\boldsymbol{r})^{2}$, where $\boldsymbol{c}$ is a real vector-valued parameter with dimensions of inverse length, constitutes a dynamical quasi-symmetry in the sense that it maps Newton's equation of motion $m(\mathrm{d}^{2}\boldsymbol{r}/\mathrm{d}t^{2})=α\boldsymbol{r}/r^{3}$ into the equation $m(\mathrm{d}^{2}\boldsymbol{R}/\mathrm{d}T^{2})=(α\boldsymbol{R}/R^{3}) \operatorname{sgn}(1-\boldsymbol{c}\cdot\boldsymbol{R})$. This quasi-symmetry is independent of, yet complementary to, the well-known Kepler-Coulomb dynamical symmetry, which is SO($N+1$) for negative energies and SO($N,1$) for positive energies. A detailed analytical and graphical analysis of induced mappings between non-degenerate conic trajectories associated with the original and transformed systems is presented. Mappings between velocity hodographs are also briefly discussed. It is then proved that the conservation of energy in the transformed motion implies the conservation of the Laplace-Runge-Lenz vector in the original dynamics.

math-ph↗
arXiv2024

An analogue of the Pöschl-Teller anharmonic oscillator on an $N$-dimensional sphere

A Schrödinger particle on an $N$-dimensional ($N\geqslant2$) hypersphere of radius $R$ is considered. The particle is subjected to the action of a force characterized by the potential $V(θ)=2mω_{1}^{2}R^{2}\tan^{2}(θ/2)+2mω_{2}^{2}R^{2}\cot^{2}(θ/2)$, where $0\leqslantθ\leqslantπ$ is the hyperlatitude angular coordinate. In the general case when $ω_{1}\neqω_{2}$, this is a model of a hyperspherical analogue of the Pöschl-Teller anharmonic oscillator. Energy eigenvalues and normalized eigenfunctions for this system are found in closed analytical forms. For $N=2$, our results reproduce those obtained by Kazaryan et al. [Physica E 52 (2013) 122]. For $N\geqslant2$ arbitrary and for $ω_{2}=0$, the results of Mardoyan and Petrosyan [J. Contemp. Phys. 48 (2013) 70] for their model of an isotropic hyperspherical harmonic oscillator are recovered. The Euclidean limit for the anharmonic oscillator in question is also discussed.

quant-ph↗
arXiv2022

Zero-range potentials for Dirac particles: Bound-state problems

A model in which a Dirac particle in $\mathbb{R}^{3}$ is bound by $N\geqslant1$ spatially distributed zero-range potentials is presented. Interactions between the particle and the potentials are modeled by subjecting a particle's bispinor wave function to certain limiting conditions at the potential centers. Each of these conditions is parametrized by a $2\times2$ Hermitian matrix (or, equivalently, a real scalar and a real vector) and mixes the upper and the lower components of the wave function. The problem of determining particle's bound-state eigenenergies is reduced to the problem of finding real zeroes of a determinant of a certain $2N\times2N$ matrix. As the lower component of the particle's wave function is inverse-square singular at each of the potential centers, the wave function itself is not square-integrable. Nevertheless, one can define a scalar pseudo-product with the property that wave functions belonging to different eigenenergies are orthogonal with respect to it. The wave functions may then be normalized so that their self-pseudo-products are plus one, minus one or zero. An auxiliary set of Sturmian functions is constructed and used to derive an explicit representation of particle's matrix Green's function. For illustration purposes, two particular systems are studied in detail: 1) a particle bound in a field of a single zero-range potential, 2) a particle bound in a field of two identical zero-range potentials.

quant-ph↗
arXiv2019

Relativistic two-dimensional hydrogen-like atom in a weak magnetic field

A two-dimensional (2D) hydrogen-like atom with a relativistic Dirac electron, placed in a weak, static, uniform magnetic field perpendicular to the atomic plane, is considered. Closed forms of the first- and second-order Zeeman corrections to energy levels are calculated analytically, within the framework of the Rayleigh-Schrödinger perturbation theory, for an arbitrary electronic bound state. The second-order calculations are carried out with the use of the Sturmian expansion of the two-dimensional generalized radial Dirac-Coulomb Green function derived in the paper. It is found that, in contrast to the case of the three-dimensional atom [P. Stefańska, Phys. Rev. A 92 (2015) 032504], in two spatial dimensions atomic magnetizabilities (magnetic susceptibilities) are expressible in terms of elementary algebraic functions of a nuclear charge and electron quantum numbers. The problem considered here is related to the Coulomb impurity problem for graphene in a weak magnetic field.

quant-ph↗
arXiv2018

Second-order Stark effect and polarizability of a relativistic two-dimensional hydrogen-like atom in the ground state

The second-order Stark effect for a planar Dirac one-electron atom in the ground state is analyzed within the framework of the Rayleigh-Schrödinger perturbation theory, with the use of the Sturmian series expansion of the generalized Dirac-Coulomb Green function. A closed-form analytical expression for the static dipole polarizability of that system is found. The formula involves a generalized hypergeometric function ${}_{3}F_{2}$ with the unit argument. Numerical values of the polarizabilities for relativistic planar hydrogenic atoms with atomic numbers $1\leqslant Z\leqslant68$ are provided in a tabular form. A simple formula for the polarizability of a nonrelativistic two-dimensional hydrogenic atom, reported previously by several other authors, is recovered from our result in the nonrelativistic limit.

quant-ph↗
arXiv2018

Two-dimensional hydrogen-like atom in a weak magnetic field

We consider a non-relativistic two-dimensional (2D) hydrogen-like atom in a weak, static, uniform magnetic field perpendicular to the atomic plane. Within the framework of the Rayleigh-Schrödinger perturbation theory, using the Sturmian expansion of the generalized radial Coulomb Green function, we derive explicit analytical expressions for corrections to an arbitrary planar hydrogenic bound-state energy level, up to the fourth order in the strength of the perturbing magnetic field. In the case of the ground state, we correct an expression for the fourth-order correction to energy available in the literature.

quant-ph↗
arXiv2017

On the derivatives $\partial^{2}P_ν(z)/\partialν^{2}$ and $\partial Q_ν(z)/\partialν$ of the Legendre functions with respect to their degrees

We provide closed-form expressions for the degree-derivatives $[\partial^{2}P_ν(z)/\partialν^{2}]_{ν=n}$ and $[\partial Q_ν(z)/\partialν]_{ν=n}$, with $z\in\mathbb{C}$ and $n\in\mathbb{N}_{0}$, where $P_ν(z)$ and $Q_ν(z)$ are the Legendre functions of the first and the second kind, respectively. For $[\partial^{2}P_ν(z)/\partialν^{2}]_{ν=n}$, we find that $$\displaystyle\frac{\partial^{2}P_ν(z)}{\partialν^{2}}\bigg|_{ν=n}=-2P_{n}(z)\textrm{Li}_{2}\frac{1-z}{2}+B_{n}(z)\ln\frac{z+1}{2}+C_{n}(z),$$ where $\textrm{Li}_{2}[(1-z)/2]$ is the dilogarithm function, $P_{n}(z)$ is the Legendre polynomial, while $B_{n}(z)$ and $C_{n}(z)$ are certain polynomials in $z$ of degree $n$. For $[\partial Q_ν(z)/\partialν]_{ν=n}$ and $z\in\mathbb{C}\setminus[-1,1]$, we derive $$\displaystyle \frac{\partial Q_ν(z)}{\partialν}\bigg|_{ν=n}=-P_{n}(z)\textrm{Li}_{2}\frac{1-z}{2}-\frac{1}{2}P_{n}(z)\ln\frac{z+1}{2}\ln\frac{z-1}{2} +\frac{1}{4}B_{n}(z)\ln\frac{z+1}{2}-\frac{(-1)^{n}}{4}B_{n}(-z)\ln\frac{z-1}{2}-\frac{π^{2}}{6}P_{n}(z) +\frac{1}{4}C_{n}(z)-\frac{(-1)^{n}}{4}C_{n}(-z).$$ A counterpart expression for $[\partial Q_ν(x)/\partialν]_{ν=n}$, applicable when $x\in(-1,1)$, is also presented. Explicit representations of the polynomials $B_{n}(z)$ and $C_{n}(z)$ as linear combinations of the Legendre polynomials are given.

math.CA↗
arXiv2016

Static electric multipole susceptibilities of the relativistic hydrogen-like atom in the ground state: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

The ground state of the Dirac one-electron atom, placed in a weak, static electric field of definite $2^{L}$-polarity, is studied within the framework of the first-order perturbation theory. The Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30 (1997) 825, erratum: 30 (1997) 2747] is used to derive closed-form analytical expressions for various far-field and near-nucleus static electric multipole susceptibilities of the atom. The far-field multipole susceptibilities --- the polarizabilities $α_{L}$, electric-to-magnetic cross-susceptibilities $α_{\mathrm{E}L\to\mathrm{M}(L\mp1)}$ and electric-to-toroidal-magnetic cross-susceptibilities $α_{\mathrm{E}L\to\mathrm{T}L}$ --- are found to be expressible in terms of one or two non-terminating generalized hypergeometric functions ${}_{3}F_{2}$ with the unit argument. Counterpart formulas for the near-nucleus multipole susceptibilities --- the electric nuclear shielding constants $σ_{\mathrm{E}L\to\mathrm{E}L}$, near-nucleus electric-to-magnetic cross-susceptibilities $σ_{\mathrm{E}L\to\mathrm{M}(L\mp1)}$ and near-nucleus electric-to-toroidal-magnetic cross-susceptibilities $σ_{\mathrm{E}L\to\mathrm{T}L}$ --- involve terminating ${}_{3}F_{2}(1)$ series and for each $L$ may be rewritten in terms of elementary functions. Exact numerical values of the far-field dipole, quadrupole, octupole and hexadecapole susceptibilities are provided for selected hydrogenic ions. Analytical quasi-relativistic approximations, valid to the second order in $αZ$, where $α$ is the fine-structure constant and $Z$ is the nuclear charge number, are derived for all types of the far-field and near-nucleus susceptibilities considered in the paper.

physics.atom-ph↗
arXiv2013

The parameter derivatives $[\partial^{2}P_ν(z)/\partialν^{2}]_{ν=0}$ and $[\partial^{3}P_ν(z)/\partialν^{3}]_{ν=0}$, where $P_ν(z)$ is the Legendre function of the first kind

We derive explicit expressions for the parameter derivatives $[\partial^{2}P_ν(z)/\partialν^{2}]_{ν=0}$ and $[\partial^{3}P_ν(z)/\partialν^{3}]_{ν=0}$, where $P_ν(z)$ is the Legendre function of the first kind. It is found that {displaymath} \frac{\partial^{2}P_ν(z)}{\partialν^{2}}\bigg|_{ν=0} =-2\Li_{2}\frac{1-z}{2}, {displaymath} where $\Li_{2}z$ is the dilogarithm (this formula has been recently arrived at by Schramkowski using \emph{Mathematica}), and that {displaymath} \frac{\partial^{3}P_ν(z)}{\partialν^{3}}\bigg|_{ν=0} =12\Li_{3}\frac{z+1}{2}-6\ln\frac{z+1}{2}\Li_{2}\frac{z+1}{2} -π^{2}\ln\frac{z+1}{2}-12ζ(3), {displaymath} where $\Li_{3}z$ is the polylogarithm of order 3 and $ζ(s)$ is the Riemann zeta function.

math.CA↗
arXiv2011

Solution of the momentum-space Schrödinger equation for bound states of the N-dimensional Coulomb problem (revisited)

The Schrödinger-Coulomb Sturmian problem in $\mathbb{R}^{N}$, $N\geqslant2$, is considered in the momentum representation. An integral formula for the Gegenbauer polynomials, found recently by Cohl [arXiv:1105.2735], is used to separate out angular variables and reduce an integral Sturmian eigenvalue equation in $\mathbb{R}^{N}$ to a Fredholm one on $\mathbb{R}_{+}$. A kernel of the latter equation contains the Legendre function of the second kind. A symmetric Poisson-type series expansion of that function into products of the Gegenbauer polynomials, established by Ossicini [Boll. Un. Mat. Ital. 7 (1952) 315], is then used to determine the Schrödinger-Coulomb Sturmian eigenvalues and associated momentum-space eigenfunctions. Finally, a relationship existing between solutions to the Sturmian problem and solutions to a (physically more interesting) energy eigenvalue problem is exploited to find the Schrödinger-Coulomb bound-state energy levels in $\mathbb{R}^{N}$, together with explicit representations of the associated normalized momentum-space Schrödinger-Coulomb Hamiltonian eigenfunctions.

quant-ph↗
arXiv2011

Magnetic field-induced electric quadrupole moment in the ground state of the relativistic hydrogen-like atom: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

We consider a Dirac one-electron atom placed in a weak, static, uniform magnetic field. We show that, to the first order in the strength $B$ of the perturbing field, the only electric multipole moment induced by the field in the ground state of the atom is the quadrupole one. Using the Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30 (1997) 825; erratum 30 (1997) 2747], we derive a closed-form expression for the induced electric quadrupole moment. The result contains the generalized hypergeometric function $_{3}F_{2}$ of the unit argument. Earlier calculations by other authors, based on the non-relativistic model of the atom, predicted in the low-field region the quadratic dependence of the induced electric quadrupole moment on $B$.

physics.atom-ph↗
arXiv2011

Alternative construction of the closed form of the Green's function for the wavized Maxwell fish-eye problem

In the recent paper [J.\ Phys.\ A 44 (2011) 065203], we have arrived at the closed-form expression for the Green's function for the partial differential operator describing propagation of a scalar wave in an $N$-dimensional ($N\geqslant2$) Maxwell fish-eye medium. The derivation has been based on unique transformation properties of the fish-eye wave equation under the hyperspherical inversion. In this communication, we arrive at the same expression for the fish-eye Green's function following a different route. The alternative derivation we present here exploits the fact that there is a close mathematical relationship, through the stereographic projection, between the wavized fish-eye problem in $\mathbb{R}^{N}$ and the problem of propagation of scalar waves over the surface of the $N$-dimensional hypersphere.

math-ph↗
arXiv2011

Green's function for the wavized Maxwell fish-eye problem

Unique transformation properties under the hyperspherical inversion of a partial differential equation describing a stationary scalar wave in an $N$-dimensional ($N\geqslant2$) Maxwell fish-eye medium are exploited to construct a closed form of the Green's function for that equation. For those wave numbers for which the Green's function fails to exist, the generalized Green's function is derived. Prospective physical applications are mentioned.

math-ph↗
arXiv2010

Recurrence and differential relations for spherical spinors

We present a comprehensive table of recurrence and differential relations obeyed by spin one-half spherical spinors (spinor spherical harmonics) $Ω_{κμ}(\mathbf{n})$ used in relativistic atomic, molecular, and solid state physics, as well as in relativistic quantum chemistry. First, we list finite expansions in the spherical spinor basis of the expressions $\mathbf{A}\cdot\mathbf{B}\,Ω_{κμ}(\mathbf{n})$ and {$\mathbf{A}\cdot(\mathbf{B}\times\mathbf{C})\, Ω_{κμ}(\mathbf{n})$}, where $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$ are either of the following vectors or vector operators: $\mathbf{n}=\mathbf{r}/r$ (the radial unit vector), $\mathbf{e}_{0}$, $\mathbf{e}_{\pm1}$ (the spherical, or cyclic, versors), $\boldsymbolσ$ (the $2\times2$ Pauli matrix vector), $\hat{\mathbf{L}}=-i\mathbf{r}\times\boldsymbol{\nabla}I$ (the dimensionless orbital angular momentum operator; $I$ is the $2\times2$ unit matrix), $\hat{\mathbf{J}}=\hat{\mathbf{L}}+1/2\boldsymbolσ$ (the dimensionless total angular momentum operator). Then, we list finite expansions in the spherical spinor basis of the expressions $\mathbf{A}\cdot\mathbf{B}\,F(r)Ω_{κμ}(\mathbf{n})$ and $\mathbf{A}\cdot(\mathbf{B}\times\mathbf{C})\, F(r)Ω_{κμ}(\mathbf{n})$, where at least one of the objects $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ is the nabla operator $\boldsymbol{\nabla}$, while the remaining ones are chosen from the set $\mathbf{n}$, $\mathbf{e}_{0}$, $\mathbf{e}_{\pm1}$, $\boldsymbolσ$, $\hat{\mathbf{L}}$, $\hat{\mathbf{J}}$.

math-ph↗