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21 pages, 367 KiB

Open AccessArticle

Features, Paradoxes and Amendments of Perturbative Non-Hermitian Quantum Mechanics

by Miloslav Znojil

Symmetry 2024, 16(5), 629; https://doi.org/10.3390/sym16050629 - 19 May 2024

Viewed by 808

Abstract

Quantum mechanics of unitary systems is considered in quasi-Hermitian representation and in the dynamical regime in which one has to take into account the ubiquitous presence of perturbations, random or specific. In this paper, it is shown that multiple technical obstacles encountered in [...] Read more.

Quantum mechanics of unitary systems is considered in quasi-Hermitian representation and in the dynamical regime in which one has to take into account the ubiquitous presence of perturbations, random or specific. In this paper, it is shown that multiple technical obstacles encountered in such a context can be circumvented via just a mild amendment of the so-called Rayleigh–Schrödinger perturbation–expansion approach. In particular, the quasi-Hermitian formalism characterized by an enhancement of flexibility is shown to remain mathematically tractable while, on the phenomenological side, opening several new model-building horizons. It is emphasized that they include, i.a., the study of generic random perturbations and/or of multiple specific non-Hermitian toy models. In parallel, several paradoxes and open questions are shown to survive. Full article

(This article belongs to the Special Issue Quantum Mechanics: Concepts, Symmetries, and Recent Developments)

24 pages, 721 KiB

Open AccessArticle

The Quantum Ratio

by Kenichi Konishi and Hans-Thomas Elze

Symmetry 2024, 16(4), 427; https://doi.org/10.3390/sym16040427 - 4 Apr 2024

Cited by 2 | Viewed by 892

Abstract

The concept of quantum ratio has emerged from recent efforts to understand how Newton’s equations appear for the center of mass (CM) of an isolated macroscopic body at finite body temperatures as a first approximation of quantum mechanical equations. It is defined as [...] Read more.

The concept of quantum ratio has emerged from recent efforts to understand how Newton’s equations appear for the center of mass (CM) of an isolated macroscopic body at finite body temperatures as a first approximation of quantum mechanical equations. It is defined as

Q \equiv R_{q} / L_{0}

, where the quantum fluctuation range

R_{q}

is the spatial extension of the pure-state CM wave function, whereas

L_{0}

stands for the body’s linear size (the space support of the internal bound-state wave function). The two cases

R_{q} / L_{0} ≲ 1

and

R_{q} / L_{0} ≫ 1

roughly correspond to the body’s CM behaving classically or quantum mechanically, respectively. In the present note, we elaborate on this concept and illustrate it through several examples. An important notion following from introduction of the quantum ratio is that the elementary particles (thus, the electron and the photon) are quantum mechanical even when environment-induced decoherence places them into a mixed state. Thus, decoherence and classical state should not be identified. This simple observation, further illustrated by consideration of a few atomic and molecular processes, may have significant implications for the way that quantum mechanics works in biological systems. Full article

(This article belongs to the Special Issue Quantum Mechanics: Concepts, Symmetries, and Recent Developments)

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14 pages, 272 KiB

Open AccessArticle

A Charged Particle with Anisotropic Mass in a Perpendicular Magnetic Field–Landau Gauge

by Orion Ciftja

Symmetry 2024, 16(4), 414; https://doi.org/10.3390/sym16040414 - 2 Apr 2024

Viewed by 915

Abstract

The loss of any symmetry in a system leads to quantum problems that are typically very difficult to solve. Such a situation arises for particles with anisotropic mass, like electrons in various semiconductor host materials, where it is known that they may have [...] Read more.

The loss of any symmetry in a system leads to quantum problems that are typically very difficult to solve. Such a situation arises for particles with anisotropic mass, like electrons in various semiconductor host materials, where it is known that they may have an anisotropic effective mass. In this work, we consider the quantum problem of a spinless charged particle with anisotropic mass in two dimensions and study the resulting energy and eigenstate spectrum in a uniform constant perpendicular magnetic field when a Landau gauge is adopted. The exact analytic solution to the problem is obtained for arbitrary values of the anisotropic mass using a mathematical technique that relies on the scaling of the original coordinates. The characteristic features of the energy spectrum and corresponding eigenstate wave functions are analyzed. The results of this study are expected to be of interest to quantum Hall effect theory. Full article

(This article belongs to the Special Issue Quantum Mechanics: Concepts, Symmetries, and Recent Developments)

13 pages, 409 KiB

Open AccessArticle

The GHZ Theorem Revisited within the Framework of Gauge Theory

by David H. Oaknin

Symmetry 2023, 15(7), 1327; https://doi.org/10.3390/sym15071327 - 29 Jun 2023

Viewed by 1603

Abstract

The Greenberger-Horne-Zeilinger version of the Einstein-Podolsky-Rosen (EPR) paradox is widely regarded as a conclusive logical argument that rules out the possibility of reproducing the predictions of Quantum Mechanics within the framework of any physical theory sharing the notions of reality and relativistic causality [...] Read more.

The Greenberger-Horne-Zeilinger version of the Einstein-Podolsky-Rosen (EPR) paradox is widely regarded as a conclusive logical argument that rules out the possibility of reproducing the predictions of Quantum Mechanics within the framework of any physical theory sharing the notions of reality and relativistic causality that we acknowledge as a given in our classical descriptions of the macroscopic world. Thus, this renowned argument stands as a seemingly insurmountable roadblock on the path to a very desired, physically intuitive understanding of quantum phenomena and, in particular, quantum entanglement. In this paper, we notice, however, that the GHZ argument involves unaccounted spurious gauge degrees of freedom and that it can be overcome once these degrees are properly taken into account. It is then possible to explicitly build a successful statistical model for the GHZ experiment based on the usual notions of relativistic causality and physical reality. This model, thus, completes—in the EPR sense—the quantum description of the GHZ state and paves the way to a novel intuitive interpretation of the quantum formalism and a deeper understanding of the physical reality that it describes. Full article

(This article belongs to the Special Issue Quantum Mechanics: Concepts, Symmetries, and Recent Developments)

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18 pages, 874 KiB

Open AccessArticle

A New Solvable Generalized Trigonometric Tangent Potential Based on SUSYQM

by Lulin Xiong, Xin Tan, Shikun Zhong, Wei Cheng and Guang Luo

Symmetry 2022, 14(8), 1593; https://doi.org/10.3390/sym14081593 - 3 Aug 2022

Cited by 1 | Viewed by 1581

Abstract

Supersymmetric quantum mechanics has wide applications in physics. However, there are few potentials that can be solved exactly by supersymmetric quantum mechanics methods, so it is undoubtedly of great significance to find more potentials that can be solved exactly. This paper studies the [...] Read more.

Supersymmetric quantum mechanics has wide applications in physics. However, there are few potentials that can be solved exactly by supersymmetric quantum mechanics methods, so it is undoubtedly of great significance to find more potentials that can be solved exactly. This paper studies the supersymmetric quantum mechanics problems of the Schrödinger equation with a new kind of generalized trigonometric tangent superpotential:

A \tan n p x + B \tan m p x

. We will elaborate on this new potential in the following aspects. Firstly, the shape invariant relation of partner potential is generated by the generalized trigonometric tangent superpotential. We find three shape invariance forms that satisfy the additive condition. Secondly, the eigenvalues and the eigenwave functions of the potential are studied separately in these three cases. Thirdly, the potential algebra of such a superpotential is discussed, and the discussions are explored from two aspects: one parameter’s and two parameters’ potential algebra. Through the potential algebra, the eigenvalue spectrums are given separately which are consistent with those mentioned earlier. Finally, we summarize the paper and give an outlook on the two-parameter shape-invariant potential. Full article

(This article belongs to the Special Issue Quantum Mechanics: Concepts, Symmetries, and Recent Developments)

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8 pages, 2116 KiB

Open AccessArticle

A Time-Symmetric Resolution of the Einstein’s Boxes Paradox

by Michael B. Heaney

Symmetry 2022, 14(6), 1217; https://doi.org/10.3390/sym14061217 - 13 Jun 2022

Viewed by 1447

Abstract

The Einstein’s Boxes paradox was developed by Einstein, de Broglie, Heisenberg, and others to demonstrate the incompleteness of the Copenhagen Formulation of quantum mechanics. I explain the paradox using the Copenhagen Formulation. I then show how a time-symmetric formulation of quantum mechanics resolves [...] Read more.

The Einstein’s Boxes paradox was developed by Einstein, de Broglie, Heisenberg, and others to demonstrate the incompleteness of the Copenhagen Formulation of quantum mechanics. I explain the paradox using the Copenhagen Formulation. I then show how a time-symmetric formulation of quantum mechanics resolves the paradox in the way envisioned by Einstein and de Broglie. Finally, I describe an experiment that can distinguish between these two formulations. Full article

(This article belongs to the Special Issue Quantum Mechanics: Concepts, Symmetries, and Recent Developments)

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14 pages, 2357 KiB

Open AccessArticle

Global Quantum Information-Theoretic Measures in the Presence of Magnetic and Aharanov-Bohm (AB) Fields

by Collins Okon Edet, Emmanuel Benjamin Ettah, Syed Alwee Aljunid, Rosdisham Endut, Norshamsuri Ali, Akpan Ndem Ikot and Muhammad Asjad

Symmetry 2022, 14(5), 976; https://doi.org/10.3390/sym14050976 - 10 May 2022

Cited by 14 | Viewed by 1874

Abstract

The global quantum information-theoretical analysis of the class of Yukawa potential (CYP) in the presence of magnetic and Aharonov–Bohm (AB) fields has been examined both analytically and numerically in this research piece. The energy equation and wave function for the CYP are obtained [...] Read more.

The global quantum information-theoretical analysis of the class of Yukawa potential (CYP) in the presence of magnetic and Aharonov–Bohm (AB) fields has been examined both analytically and numerically in this research piece. The energy equation and wave function for the CYP are obtained by solving the Schrodinger equation in the presence of external magnetic and AB fields using the functional analysis technique. The probability density is used to calculate the Tsallis, Rényi, and Onicescu information energy entropies numerically. The influence of the screening parameter

(β)

, magnetic

(\vec{B})

, and AB

(ξ)

fields on the global information-theoretical measurements for the CYP is explored. Atomic and molecular physics, quantum chemistry, and physics are specific areas where these research findings will find application. Full article

(This article belongs to the Special Issue Quantum Mechanics: Concepts, Symmetries, and Recent Developments)

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21 pages, 345 KiB

Open AccessArticle

Dynamical Symmetries of the 2D Newtonian Free Fall Problem Revisited

by Tuong Trong Truong

Symmetry 2022, 14(1), 27; https://doi.org/10.3390/sym14010027 - 27 Dec 2021

Viewed by 2282

Abstract

Among the few exactly solvable problems in theoretical physics, the 2D (two-dimensional) Newtonian free fall problem in Euclidean space is perhaps the least known as compared to the harmonic oscillator or the Kepler–Coulomb problems. The aim of this article is to revisit this [...] Read more.

Among the few exactly solvable problems in theoretical physics, the 2D (two-dimensional) Newtonian free fall problem in Euclidean space is perhaps the least known as compared to the harmonic oscillator or the Kepler–Coulomb problems. The aim of this article is to revisit this problem at the classical level as well as the quantum level, with a focus on its dynamical symmetries. We show how these dynamical symmetries arise as a special limit of the dynamical symmetries of the Kepler–Coulomb problem, and how a connection to the quartic anharmonic oscillator problem, a long-standing unsolved problem in quantum mechanics, can be established. To this end, we construct the Hilbert space of states with free boundary conditions as a space of square integrable functions that have a special functional integral representation. In this functional space, the free fall dynamical symmetry algebra is shown to be isomorphic to the so-called Klink’s algebra of the quantum quartic anharmonic oscillator problem. Furthermore, this connection entails a remarkable integral identity for the quantum quartic anharmonic oscillator eigenfunctions, which implies that these eigenfunctions are in fact zonal functions of an underlying symmetry group representation. Thus, an appropriate representation theory for the 2D Newtonian free fall quantum symmetry group may potentially open the way to exactly solving the difficult quantization problem of the quartic anharmonic oscillator. Finally, the initial value problem of the acoustic Klein–Gordon equation for wave propagation in a sound duct with a varying circular section is solved as an illustration of the techniques developed here. Full article

(This article belongs to the Special Issue Quantum Mechanics: Concepts, Symmetries, and Recent Developments)

6 pages, 303 KiB

Open AccessArticle

The Surjective Mapping Conjecture and the Measurement Problem in Quantum Mechanics

by Fritz Wilhelm Bopp

Symmetry 2021, 13(11), 2155; https://doi.org/10.3390/sym13112155 - 11 Nov 2021

Viewed by 1400

Abstract

Accepting a time-symmetric quantum dynamical world with ontological wave functions or fields, we follow arguments that naturally lead to a two-boundary interpretation of quantum mechanics. The usual two boundary picture is a valid superdeterministic interpretation. It has, however, one unsatisfactory feature. The random [...] Read more.

Accepting a time-symmetric quantum dynamical world with ontological wave functions or fields, we follow arguments that naturally lead to a two-boundary interpretation of quantum mechanics. The usual two boundary picture is a valid superdeterministic interpretation. It has, however, one unsatisfactory feature. The random selection of a chosen measurement path of the universe is far too complicated. To avoid it, we propose an alternate two-boundary concept called surjective mapping conjecture. It takes as fundamental a quantum-time running forward like the usual time on the wave-function side and backward on the complex conjugate side. Unrelated fixed arbitrary boundary conditions at the initial and the final quantum times then determine the measurement path of the expanding and contracting quantum-time universe in the required way. Full article

(This article belongs to the Special Issue Quantum Mechanics: Concepts, Symmetries, and Recent Developments)

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12 pages, 331 KiB

Open AccessArticle

An Application of the Madelung Formalism for Dissipating and Decaying Systems

by Maedeh Mollai and Seyed Majid Saberi Fathi

Symmetry 2021, 13(5), 812; https://doi.org/10.3390/sym13050812 - 6 May 2021

Cited by 1 | Viewed by 1912

Abstract

This paper is concerned with the modeling and analysis of quantum dissipation and diffusion phenomena in the Schrödinger picture. We derive and investigate in detail the Schrödinger-type equations accounting for dissipation and diffusion effects. From a mathematical viewpoint, this equation allows one to [...] Read more.

This paper is concerned with the modeling and analysis of quantum dissipation and diffusion phenomena in the Schrödinger picture. We derive and investigate in detail the Schrödinger-type equations accounting for dissipation and diffusion effects. From a mathematical viewpoint, this equation allows one to achieve and analyze all aspects of the quantum dissipative systems, regarding the wave equation, Hamilton–Jacobi and continuity equations. This simplification requires the performance of “the Madelung decomposition” of “the wave function”, which is rigorously attained under the general Lagrangian justification for this modification of quantum mechanics. It is proved that most of the important equations of dissipative quantum physics, such as convection-diffusion, Fokker–Planck and quantum Boltzmann, have a common origin and can be unified in one equation. Full article

(This article belongs to the Special Issue Quantum Mechanics: Concepts, Symmetries, and Recent Developments)

Review

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35 pages, 1933 KiB

Open AccessFeature PaperReview

Quantum Spin-Wave Theory for Non-Collinear Spin Structures, a Review

by Hung T. Diep

Symmetry 2022, 14(8), 1716; https://doi.org/10.3390/sym14081716 - 17 Aug 2022

Cited by 1 | Viewed by 3293

Abstract

In this review, we trace the evolution of the quantum spin-wave theory treating non-collinear spin configurations. Non-collinear spin configurations are consequences of the frustration created by competing interactions. They include simple chiral magnets due to competing nearest-neighbor (NN) and next-NN interactions and systems [...] Read more.

In this review, we trace the evolution of the quantum spin-wave theory treating non-collinear spin configurations. Non-collinear spin configurations are consequences of the frustration created by competing interactions. They include simple chiral magnets due to competing nearest-neighbor (NN) and next-NN interactions and systems with geometry frustration such as the triangular antiferromagnet and the Kagomé lattice. We review here spin-wave results of such systems and also systems with the Dzyaloshinskii–Moriya interaction. Accent is put on these non-collinear ground states which have to be calculated before applying any spin-wave theory to determine the spectrum of the elementary excitations from the ground states. We mostly show results obtained by the use of a Green’s function method. These results include the spin-wave dispersion relation and the magnetizations, layer by layer, as functions of T in 2D, 3D and thin films. Some new unpublished results are also included. Technical details and discussion on the method are shown and discussed. Full article

(This article belongs to the Special Issue Quantum Mechanics: Concepts, Symmetries, and Recent Developments)

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