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2 pages, 147 KiB

Open AccessEditorial

Special Issue: “Symmetries in Quantum Mechanics and Statistical Physics”

by Georg Junker

Symmetry 2021, 13(11), 2027; https://doi.org/10.3390/sym13112027 - 26 Oct 2021

Cited by 1 | Viewed by 1420

Abstract

Symmetry is a fundamental concept in science and has played a significant role since the early days of quantum physics [...] Full article

(This article belongs to the Special Issue Symmetries in Quantum Mechanics and Statistical Physics)

Research

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19 pages, 800 KiB

Open AccessArticle

Time-Dependent Conformal Transformations and the Propagator for Quadratic Systems

by Qiliang Zhao, Pengming Zhang and Peter A. Horvathy

Symmetry 2021, 13(10), 1866; https://doi.org/10.3390/sym13101866 - 3 Oct 2021

Cited by 7 | Viewed by 1938

Abstract

The method proposed by Inomata and his collaborators allows us to transform a damped Caldirola–Kanai oscillator with a time-dependent frequency to one with a constant frequency and no friction by redefining the time variable, obtained by solving an Ermakov–Milne–Pinney equation. Their mapping “Eisenhart–Duval” [...] Read more.

The method proposed by Inomata and his collaborators allows us to transform a damped Caldirola–Kanai oscillator with a time-dependent frequency to one with a constant frequency and no friction by redefining the time variable, obtained by solving an Ermakov–Milne–Pinney equation. Their mapping “Eisenhart–Duval” lifts as a conformal transformation between two appropriate Bargmann spaces. The quantum propagator is calculated also by bringing the quadratic system to free form by another time-dependent Bargmann-conformal transformation, which generalizes the one introduced before by Niederer and is related to the mapping proposed by Arnold. Our approach allows us to extend the Maslov phase correction to an arbitrary time-dependent frequency. The method is illustrated by the Mathieu profile. Full article

(This article belongs to the Special Issue Symmetries in Quantum Mechanics and Statistical Physics)

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6 pages, 256 KiB

Open AccessArticle

On the Supersymmetry of the Klein–Gordon Oscillator

by Georg Junker

Symmetry 2021, 13(5), 835; https://doi.org/10.3390/sym13050835 - 10 May 2021

Cited by 6 | Viewed by 1873

Abstract

The three-dimensional Klein–Gordon oscillator exhibits an algebraic structure known from supersymmetric quantum mechanics. The supersymmetry is unbroken with a vanishing Witten index, and it is utilized to derive the spectral properties of the Klein–Gordon oscillator, which is closely related to that of the [...] Read more.

The three-dimensional Klein–Gordon oscillator exhibits an algebraic structure known from supersymmetric quantum mechanics. The supersymmetry is unbroken with a vanishing Witten index, and it is utilized to derive the spectral properties of the Klein–Gordon oscillator, which is closely related to that of the nonrelativistic harmonic oscillator in three dimensions. Supersymmetry also enables us to derive a closed-form expression for the energy-dependent Green’s function. Full article

(This article belongs to the Special Issue Symmetries in Quantum Mechanics and Statistical Physics)

6 pages, 304 KiB

Open AccessEditor’s ChoiceArticle

What Is the Size and Shape of a Wave Packet?

by Larry S. Schulman

Symmetry 2021, 13(4), 527; https://doi.org/10.3390/sym13040527 - 24 Mar 2021

Cited by 2 | Viewed by 2075

Abstract

Under pure quantum evolution, for a wave packet that diffuses (like a Gaussian), scattering can cause localization. Other forms of the wave function, spreading more rapidly than a Gaussian, are unlikely to localize. Full article

(This article belongs to the Special Issue Symmetries in Quantum Mechanics and Statistical Physics)

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50 pages, 639 KiB

Open AccessEditor’s ChoiceArticle

Power Law Duality in Classical and Quantum Mechanics

by Akira Inomata and Georg Junker

Symmetry 2021, 13(3), 409; https://doi.org/10.3390/sym13030409 - 3 Mar 2021

Cited by 5 | Viewed by 3151

Abstract

The Newton–Hooke duality and its generalization to arbitrary power laws in classical, semiclassical and quantum mechanics are discussed. We pursue a view that the power-law duality is a symmetry of the action under a set of duality operations. The power dual symmetry is [...] Read more.

The Newton–Hooke duality and its generalization to arbitrary power laws in classical, semiclassical and quantum mechanics are discussed. We pursue a view that the power-law duality is a symmetry of the action under a set of duality operations. The power dual symmetry is defined by invariance and reciprocity of the action in the form of Hamilton’s characteristic function. We find that the power-law duality is basically a classical notion and breaks down at the level of angular quantization. We propose an ad hoc procedure to preserve the dual symmetry in quantum mechanics. The energy-coupling exchange maps required as part of the duality operations that take one system to another lead to an energy formula that relates the new energy to the old energy. The transformation property of the Green function satisfying the radial Schrödinger equation yields a formula that relates the new Green function to the old one. The energy spectrum of the linear motion in a fractional power potential is semiclassically evaluated. We find a way to show the Coulomb–Hooke duality in the supersymmetric semiclassical action. We also study the confinement potential problem with the help of the dual structure of a two-term power potential. Full article

(This article belongs to the Special Issue Symmetries in Quantum Mechanics and Statistical Physics)

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19 pages, 3538 KiB

Open AccessArticle

Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian

by Manuel Gadella, José Hernández-Muñoz, Luis Miguel Nieto and Carlos San Millán

Symmetry 2021, 13(2), 350; https://doi.org/10.3390/sym13020350 - 21 Feb 2021

Cited by 3 | Viewed by 2779

Abstract

We find supersymmetric partners of a family of self-adjoint operators which are self-adjoint extensions of the differential operator

- d^{2} / d x^{2}

on

L^{2} [- a, a]

,

a > 0

, that is, the one [...] Read more.

We find supersymmetric partners of a family of self-adjoint operators which are self-adjoint extensions of the differential operator

- d^{2} / d x^{2}

on

L^{2} [- a, a]

,

a > 0

, that is, the one dimensional infinite square well. First of all, we classify these self-adjoint extensions in terms of several choices of the parameters determining each of the extensions. There are essentially two big groups of extensions. In one, the ground state has strictly positive energy. On the other, either the ground state has zero or negative energy. In the present paper, we show that each of the extensions belonging to the first group (energy of ground state strictly positive) has an infinite sequence of supersymmetric partners, such that the ℓ-th order partner differs in one energy level from both the

(ℓ - 1)

-th and the

(ℓ + 1)

-th order partners. In general, the eigenvalues for each of the self-adjoint extensions of

- d^{2} / d x^{2}

come from a transcendental equation and are all infinite. For the case under our study, we determine the eigenvalues, which are also infinite, all the extensions have a purely discrete spectrum, and their respective eigenfunctions for all of its ℓ-th supersymmetric partners of each extension. Full article

(This article belongs to the Special Issue Symmetries in Quantum Mechanics and Statistical Physics)

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15 pages, 283 KiB

Open AccessArticle

Deformed Shape Invariant Superpotentials in Quantum Mechanics and Expansions in Powers of ℏ

by Christiane Quesne

Symmetry 2020, 12(11), 1853; https://doi.org/10.3390/sym12111853 - 10 Nov 2020

Cited by 6 | Viewed by 1761

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 ℏ-independent superpotentials and to a superpotential explicitly depending on ℏ. Full article

(This article belongs to the Special Issue Symmetries in Quantum Mechanics and Statistical Physics)

14 pages, 314 KiB

Open AccessArticle

Supersymmetry of Relativistic Hamiltonians for Arbitrary Spin

by Georg Junker

Symmetry 2020, 12(10), 1590; https://doi.org/10.3390/sym12101590 - 24 Sep 2020

Cited by 6 | Viewed by 1799

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)

17 pages, 336 KiB

Open AccessFeature PaperArticle

Perturbation Theory Near Degenerate Exceptional Points

by Miloslav Znojil

Symmetry 2020, 12(8), 1309; https://doi.org/10.3390/sym12081309 - 5 Aug 2020

Cited by 6 | Viewed by 2973

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

L \geq 2

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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