Special Issue "Symmetries in Quantum Mechanics and Statistical Physics"

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Physics and Symmetry".

Deadline for manuscript submissions: 31 January 2021.

Special Issue Editor

Priv.-Doz. Dr. Georg Junker

Guest Editor
European Southern Observatory, Karl-Schwarzschild-Straße 2, D-85748 Garching, Germany; Institut für Theoretische Physik I, Universität Erlangen-Nürnberg, Staudtstraße 7, D-91058 Erlangen, Germany
Interests: supersymmetric quantum mechanics; classical N-vector models; group-theoretical and path-integral methods in physics

Special Issue Information

Dear Colleagues,

Symmetry is a fundamental concept in science and has played a significant role since the early days of quantum physics. For example, the rotational symmetry of Coulomb interactions is key in the group theoretic classification of atomic spectra, and its dynamical SO(4) symmetry accounts for the accidental degeneracy of the H atom spectrum. In physics, symmetry characterises the invariance of a system under certain transformations, being either discrete like mirror symmetry or continuous like rotational symmetry. In mathematics, symmetries are described by group theoretic means.

Symmetry methods are still powerful tools in contemporary problems of quantum mechanics and statistical physics, and they go beyond the classical Lie groups and algebras. Examples are the so-called supersymmetric quantum mechanics and the PT invariance of non-Hermitian Hamiltonians. In this Special Issue of Symmetry, we invite original contributions which utilise symmetry methods to understand and solve problems related to the keywords listed below. However, it is also open to other topics related to quantum mechanics and/or statistical physics where symmetry plays a key role.

Priv.-Doz. Dr. Georg Junker
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Schrödinger- and Pauli-Hamiltonians
  • relativistic wave equations
  • Feynman and Wiener path integrals
  • supersymmetric quantum mechanics
  • PT symmetry and complex potentials
  • group coherent states
  • Fokker–Planck and Langevin equation
  • Ising models and spin systems
  • classical vector models

Published Papers (3 papers)

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Research

Open AccessArticle
Deformed Shape Invariant Superpotentials in Quantum Mechanics and Expansions in Powers of
Symmetry 2020, 12(11), 1853; https://doi.org/10.3390/sym12111853 - 10 Nov 2020
Abstract
We show that the method developed by Gangopadhyaya, Mallow, and their coworkers to deal with (translational) shape invariant potentials in supersymmetric quantum mechanics and consisting in replacing the shape invariance condition, which is a difference-differential equation, which, by an infinite set of partial [...] Read more.
We show that the method developed by Gangopadhyaya, Mallow, and their coworkers to deal with (translational) shape invariant potentials in supersymmetric quantum mechanics and consisting in replacing the shape invariance condition, which is a difference-differential equation, which, by an infinite set of partial differential equations, can be generalized to deformed shape invariant potentials in deformed supersymmetric quantum mechanics. The extended method is illustrated by several examples, corresponding both to -independent superpotentials and to a superpotential explicitly depending on . Full article
(This article belongs to the Special Issue Symmetries in Quantum Mechanics and Statistical Physics)
Open AccessArticle
Supersymmetry of Relativistic Hamiltonians for Arbitrary Spin
Symmetry 2020, 12(10), 1590; https://doi.org/10.3390/sym12101590 - 24 Sep 2020
Abstract
Hamiltonians describing the relativistic quantum dynamics of a particle with an arbitrary but fixed spin are shown to exhibit a supersymmetric structure when the even and odd elements of the Hamiltonian commute. Here, the supercharges transform between energy eigenstates of positive and negative [...] Read more.
Hamiltonians describing the relativistic quantum dynamics of a particle with an arbitrary but fixed spin are shown to exhibit a supersymmetric structure when the even and odd elements of the Hamiltonian commute. Here, the supercharges transform between energy eigenstates of positive and negative energy. For such supersymmetric Hamiltonians, an exact Foldy–Wouthuysen transformation exists which brings it into a block-diagonal form separating the positive and negative energy subspaces. The relativistic dynamics of a charged particle in a magnetic field are considered for the case of a scalar (spin-zero) boson obeying the Klein–Gordon equation, a Dirac (spin one-half) fermion and a vector (spin-one) boson characterised by the Proca equation. In the latter case, supersymmetry implies for the Landé g-factor g=2. Full article
(This article belongs to the Special Issue Symmetries in Quantum Mechanics and Statistical Physics)
Open AccessFeature PaperArticle
Perturbation Theory Near Degenerate Exceptional Points
Symmetry 2020, 12(8), 1309; https://doi.org/10.3390/sym12081309 - 05 Aug 2020
Abstract
In an overall framework of quantum mechanics of unitary systems a rather sophisticated new version of perturbation theory is developed and described. The motivation of such an extension of the list of the currently available perturbation-approximation recipes was four-fold: (1) its need results [...] Read more.
In an overall framework of quantum mechanics of unitary systems a rather sophisticated new version of perturbation theory is developed and described. The motivation of such an extension of the list of the currently available perturbation-approximation recipes was four-fold: (1) its need results from the quick growth of interest in quantum systems exhibiting parity-time symmetry (PT-symmetry) and its generalizations; (2) in the context of physics, the necessity of a thorough update of perturbation theory became clear immediately after the identification of a class of quantum phase transitions with the non-Hermitian spectral degeneracies at the Kato’s exceptional points (EP); (3) in the dedicated literature, the EPs are only being studied in the special scenarios characterized by the spectral geometric multiplicity L equal to one; (4) apparently, one of the decisive reasons may be seen in the complicated nature of mathematics behind the L2 constructions. In our present paper we show how to overcome the latter, purely technical obstacle. The temporarily forgotten class of the L>1 models is shown accessible to a feasible perturbation-approximation analysis. In particular, an emergence of a counterintuitive connection between the value of L, the structure of the matrix elements of perturbations, and the possible loss of the stability and unitarity of the processes of the unfolding of the singularities is given a detailed explanation. Full article
(This article belongs to the Special Issue Symmetries in Quantum Mechanics and Statistical Physics)
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