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Jaromir Tosiek

Publications and source records attributed to Jaromir Tosiek.

At least 19 recordsLinked to original sources
arXiv2026

Phase space quantum mechanics of a free particle moving in a plane

The eigenstates of a free quantum particle propagating in a plane are derived in the context of phase space quantum mechanics. Two possibilities are analysed. First, the case of a particle with fixed energy and angular momentum is considered. A special choice of coordinates on the four-dimensional phase space, suitable for representing the eigenstates of the particle under consideration, is presented. A phase-space counterpart of the product of operators, known as the Moyal product, is derived in these coordinates. The eigenvalue equations are solved, and their physically acceptable solutions, called Wigner functions, are identified. Second, a particle with fixed components of the Cartesian momentum is considered. A relationship is found between the Wigner function of the particle with fixed Cartesian momentum components and some functions characterising the particle with fixed energy and angular momentum.

quant-ph↗
arXiv2023

The continuity equation and its applications in phase space quantum mechanics

A quantum phase space version of the continuity equation for systems with internal degrees of freedom is derived. The $1$ -- D Dirac equation is introduced and its phase space counterpart is found. The phase space representation of free motion and of scattering in a nonrelativistic and relativistic case for setups with internal degrees of freedom is discussed and illustrated. Properties of Wigner functions of unbound states are analysed.

quant-ph↗
arXiv2023

Linear graviton as a quantum particle

Wave function of a single linear graviton and its interpretation are proposed. The evolution equation for this function is given. A Hermitian operator with mutually commuting components canonically conjugated to the momentum operator of the linear graviton is found.

gr-qc↗
arXiv2023

Construction of a photon position operator with commuting components from natural axioms

A general form of the photon position operator with commuting components fulfilling some natural axioms is obtained. This operator commutes with the photon helicity operator, is Hermitian with respect to the Bialynicki-Birula scalar product and defined up to a unitary transformation preserving the transversality condition. It is shown that, using the procedure analogous to the one introduced by T. T. Wu and C. N. Yang for the case of the Dirac magnetic monopole, the photon position operator can be defined by a flat connection in some trivial vector bundle over $\mathbb{R}^3 \setminus \{(0,0,0)\}$. This observation enables us to reformulate quantum mechanics of a~single photon on $(\mathbb{R}^{3} \setminus \{(0,0,0)\}) \times \mathbb{C}^2$.

quant-ph↗
arXiv2021

The geometrical interpretation of the photon position operator

It is shown that the photon position operator $\hat{\vec{X}}$ with commuting components can be written in the momentum representation as $\hat{\vec{X}}=i \hat{\vec{D}}$, where $\hat{\vec{D}}$ is a flat connection in the tangent bundle $T(\mathbb{R}^3 \setminus \{ (0,0,k_3) \in \mathbb{R}^3 : k_3 \geq 0\})$ over $\mathbb{R}^3 \setminus \{ (0,0,k_3) \in \mathbb{R}^3 : k_3 \geq 0\}$ equipped with the Cartesian structure. Moreover, $\hat{\vec{D}}$ is such that the tangent $2$-planes orthogonal to the momentum are parallelly propagated with respect to $\hat{\vec{D}}$ and, also, $\hat{\vec{D}}$ is an anti-Hermitian operator with respect to the scalar product $\langle \mathbfΨ | \hat{H}^{-2s} |\mathbfΦ \rangle$. The eigenfunctions $\mathbfΨ_{\vec{X}} (\vec{x})$ of the position operator $\hat{\vec{X}}$ are found.

quant-ph↗
arXiv2020

The Weyl -- Wigner -- Moyal Formalism on a Discrete Phase Space. II. The Photon Wigner Function

Classical model of light in helicity formalism is presented. Then quantum point of view at photons -- construction and interpretation of photon wave function is proposed. Quantum mechanics of photon is investigated. The Białynicki -- Birula scalar product $\langle {\bf Ψ}_{1}|{\bf Ψ}_{2}\rangle_{BB}$ and the generalized Hermitian conjugation $\widehatγ^{+\hspace{-0.45em}+}$ of linear operator $\widehatγ$ are discussed. Quantum description of light on a phase space is developed. A photon Wigner function is built.

quant-ph↗
arXiv2019

The Weyl-Wigner-Moyal formalism on a discrete phase space. I. A Wigner function for a nonrelativistic particle with spin

The Weyl-Wigner-Moyal formalism for quantum particle with discrete internal degrees of freedom is developed. A one to one correspondence between operators in the Hilbert space $L^{2}(\mathbb{R}^{3})\otimes{\mathcal{H}}^{(s+1)}$ and functions on the phase space $\mathbb{R}^{3}\times\mathbb{R}^{3}\times \{0,...,s\} \times\{0,...,s\}$ is found. The expressions for the Stratonovich-Weyl quantizer, star product and Wigner functions of such systems for arbitrary values of spin are obtained in detail. As examples the Landau levels and the corresponding Wigner functions for a spin $\frac{1}{2}$ nonrelativistic particle as well as the magnetic resonance for a spin $\frac{1}{2}$ nonrelativistic uncharged particle are analysed.

quant-ph↗
arXiv2019

Formal Series of Generalised Functions and Their Application to Deformation Quantisation

Foundations of the formal series $*$ -- calculus in deformation quantisation are discussed. Several classes of continuous linear functionals over algebras applied in classical and quantum physics are introduced. The notion of nonnegativity in formal series calculus is proposed. Problems with defining quantum states over the set of formal series are analysed.

quant-ph↗
arXiv2016

From the discrete Weyl -- Wigner formalism for symmetric ordering to a number -- phase Wigner function

The general Weyl -- Wigner formalism in finite dimensional phase spaces is investigated. Then this formalism is specified to the case of symmetric ordering of operators in an odd -- dimensional Hilbert space. A respective Wigner function on the discrete phase space is found and the limit, when the dimension of Hilbert space tends to infinity, is considered. It is shown that this limit gives the number -- phase Wigner function in quantum optics. Analogous results for the `almost' symmetric ordering in an even -- dimensional Hilbert space are obtained. Relations between the discrete Wigner functions introduced in our paper and some other discrete Wigner functions appearing in literature are studied.

quant-ph↗
arXiv2015

The Wentzel - Kramers - Brillouin approximation method applied to the Wigner function

An adaptation of the WKB method in the deformation quantization formalism is presented with the aim to obtain an approximate technique of solving the eigenvalue problem for energy in the phase space quantum approach. A relationship between the phase $σ(\vec{r})$ of a wave function $\exp \left(\frac{i}{\hbar} σ(\vec{r}) \right)$ and its respective Wigner function is derived. Formulas to calculate the Wigner function of a product and of a superposition of wave functions are proposed. Properties of a Wigner function of interfering states are also investigated. Examples of this quasi - classical approximation in deformation quantization are analysed. A strict form of the Wigner function for states represented by tempered generalised functions has been derived. Wigner functions of unbound states in the Poeschl - Teller potential have been found.

quant-ph↗
arXiv2015

Uncertainty relations in quantum optics. Is the photon intelligent?

The Robertson -- Schrödinger, Heisenberg -- Robertson and Trifonov uncertainty relations for arbitrary two functions $f_{1}$ and $f_{2}$ depending on the quantum phase and the number of photons respectively, are given. Intelligent states and states which minimize locally the product of uncertainties $(Δf_{1})^{2}\cdot (Δf_{2})^{2}$ or the sum $(Δf_{1})^{2}+(Δf_{2})^{2}$ are investigated for the cases $f_{1}=ϕ,\exp{(iϕ)}, \exp{(-iϕ)}, \cosϕ, \sinϕ$ and $f_{2}=n$.

quant-ph↗
arXiv2013

From the Weyl quantization of a particle on the circle to number-phase Wigner functions

A generalized Weyl quantization formalism for a particle on the circle investigated in \cite{1} is developed. A Wigner function for the state $\hat{\varrho}$ and the kernel $\mathcal{K}$ for a particle on the circle is defined and its properties are analyzed. Then it is shown how this Wigner function can be easily modified to give the number-phase Wigner function in quantum optics. Some examples of such number-phase Wigner function are considered.

math-ph↗
arXiv2012

Eigenvalue equation for a 1--D Hamilton function in deformation quantization

The eigenvalue equation has been found for a Hamilton function in a form independent of the choice of a potential. This paper proposes a modified Fedosov construction on a flat symplectic manifold. Necessary and sufficient conditions for solutions of an eigenvalue equation to be Wigner functions of pure states are presented. The 1--D harmonic oscillator eigenvalue equation in the coordinates time and energy is solved. A perturbation theory based on the variables time and energy is elaborated.

math-ph↗
arXiv2011

The Fedosov deformation quantization for some induced symplectic connection

The Fedosov deformation quantization on a cotangent bundle with a symplectic connection induced by some linear symmetric connection on the base space is considered. A global construction of the symplectic homogeneous connection on the cotangent bundle modelled on the linear symmetric connection from the base space is proposed. Examples of the induced symplectic connection are given. A detailed analysis of the Abelian connection and flat sections representing special types of functions for this kind of symplectic connection is presented. Some properties of the *-product determined by the induced symplectic connection are shown.

math-ph↗
arXiv2011

Notes on thermodynamics in special relativity

Foundations of thermodynamics in special theory of relativity are considered. We argue that from the phenomenological point of view the correct relativistic transformations of heat and absolute temperature are given by the formulae proposed by H. Ott, H. Arzeliès and C. M\oller. It is shown that the same transformation rules can be also found from the relativistic Gibbs distribution for ideal gas. This distribution has been recently verified by the computer simulations. Phenomenological and statistical thermometers in relativistic thermodynamics are analysed.

cond-mat.stat-mech↗
arXiv2008

The Fedosov *-product in Mathematica

The computer program `Fecom.nb' implementing the Fedosov *-product in Darboux coordinates is presented. It has been written in Mathematica 6.0 but it can be easily modified to be run in some earlier version of Mathematica. To optimize computations elements of the Weyl algebra are treated as polynomials. Several procedures which order the terms are included. The program is available at the web page http://cpc.cs.qub.ac.uk/summaries/AEBU_v1_0.html

math-ph↗
arXiv2005

The Weyl bundle as a differentiable manifold

Construction of an infinite dimensional differentiable manifold ${\mathbb R}^{\infty}$ not modelled on any Banach space is proposed. Definition, metric and differential structures of a Weyl algebra and a Weyl algebra bundle are presented. Continuity of the $\circ$-product in the Tichonov topology is proved. Construction of the $*$-product of the Fedosov type in terms of theory of connection in a fibre bundle is explained.

math-ph↗