Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
4.9
2.7
Journals
Symmetry
Special Issues
Symmetries in Quantum Mechanics
share announcement

Symmetries in Quantum Mechanics

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

Deadline for manuscript submissions: closed (31 October 2020) | Viewed by 69807

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Prof. Dr. G. Jordan Maclay

E-Mail Website
Guest Editor
Quantum Fields LLC, St. Charles, IL 60174, USA
Interests: quantum fields; symmetries and group theory; Casimir forces; vacuum fluctuations; foundations of quantum theory; history of science
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This special issue on Symmetries in Quantum Mechanics highlights two of the most important concepts forming the foundation of modern physics: symmetry and quantum mechanics.  Historically they were linked in the seminal work on the H atom and the enumeration of the periodic table.  Symmetry arguments were used to determine the transition probabilities for radiation predicted by quantum theory.  Since those early times the uses of symmetry in quantum theory have expanded enormously, but have not lost their roots.  Symmetry properties can elucidate fundamental behaviours of quantum or classical systems in a way that shows elegance and beauty, as well as providing insight to solve many problems and to facilitate calculations. This is a golden age for experiment and theory to explore the synergy between symmetry and quantum mechanics.

Today researchers use traditional quantum theory, relativistic quantum theory, and quantum field theory, as well alternative theories such as non-linear quantum mechanics, SUSY, time asymmetric and PT symmetric quantum mechanics, stochastic electrodynamics, Bohmian mechanics, conformal field theory, quantum gravity and string theory. 

Researchers are continuing to explore symmetries of the Dirac, Klein-Gordon and Schrodinger equations, the Coulomb and other potentials, van der Waals and Casimir forces, minimal coupling, nuclear structure models, and expanding their explorations to topological materials, cosmology, and systems that are dissipative, non-linear, stochastic, time dependent, non-equilibrium, or chaotic.

The symmetry principles used to analyse quantum systems include representations of symmetry groups, Lie groups, continuous and discrete symmetry transformations, symmetries for non traditional potentials and dissipative systems, geometrical, kinematical and dynamical symmetries, exact and approximate symmetries, uncertainty relations, spontaneously broken symmetry, conserved quantities that are generators of symmetry transformations, and commutation relations between symmetry operators.

The systems addressed by quantum mechanical approaches and symmetry principles have broadened to include the quantum to classical transition, for example between many-body localized and ergodic phases, and to include mesoscopic systems based on semiconductor structures or simulations at the physics-chemistry-biology interface, multiphoton entangled states, quantum crystals and quantum solids, with macroscopic quantum states and quantum sensing. With recent experimental progress and the appeal of quantum enhanced applications, the interest in macroscopic quantum effects has grown.

Quantum technologies represent a rapidly evolving field in which specific symmetry properties of quantum mechanical systems, such as superposition, entanglement and interference, are exploited to enhance the performance of various applications such as sensing, transmission, and processing of information. Some of the key approaches are based on the use of entanglement, quantum coherence, and squeezed states to revolutionize computing, measurements, and communications. This has stimulated research in different areas of physics, including quantum optics, quantum information, atomic and solid state physics, and nanoscale thermodynamics to engineer, to manipulate and to protect from environmental dephasing the fragile many-particle entangled states. Recent experiments have been done with ultracold atoms, trapped ions, superconducting qubits, spin systems, Rydberg atoms, quantum dots, atoms or photons in cavities, and quantum materials. Many of these technologies rely on the ability to control the states and degrees of freedom of atoms very precisely using electromagnetic fields, for example, using the quantum control of molecular rotation as means of encoding information.  Other methods explored include the use of more general quantum correlations such as quantum discord, quantum contextuality, non-equilibrium states, and the use of phase transitions.

Quantum metrology is based in the use of entangled states and quantum correlations, and promises to provide enhanced measurement sensitivity, for example it promises frequency measurement with uncertainly at the 10^−18 level, sufficiently precise to allow exploration of violations of Lorentz invariance.

I cordially invite you to submit a paper for this special issue on whatever aspect of symmetry in quantum mechanics excites you the most.

Prof. Dr. G. Jordan Maclay
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 submissions that pass pre-check are 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 2400 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

  • Symmetry
  • Lie groups
  • Groups
  • Lorentz group
  • Kepler symmetry
  • Phase space symmetry
  • Discrete symmetry
  • Hidden symmetry
  • Generators of symmetry transformations
  • Quantum correlations
  • Quantum measurement
  • Quantum sensing
  • Quantum computation
  • Quantum entanglement
  • EPR
  • Decoherence
  • Superposition
  • Dynamical symmetry
  • Geometrical symmetry
  • Kinematical symmetry
  • Mesoscopic system
  • Quantum-to-classical
  • Uncertainty relations

Published Papers (12 papers)

Download All Papers
Order results
Result details
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:

Editorial

Jump to: Research, Review

3 pages, 154 KiB  
Editorial
Special Issue: “Symmetries in Quantum Mechanics”
by G. Jordan Maclay
Symmetry 2021, 13(9), 1620; https://doi.org/10.3390/sym13091620 - 03 Sep 2021
Viewed by 1242
Abstract
This Special Issue “Symmetries in Quantum Mechanics” describes research using two of the most fundamental probes we have in nature [...] Full article
(This article belongs to the Special Issue Symmetries in Quantum Mechanics)

Research

Jump to: Editorial, Review

21 pages, 342 KiB  
Article
An Invariant Characterization of the Levi-Civita Spacetimes
by Cooper K. Watson, William Julius, Matthew Gorban, David D. McNutt, Eric W. Davis and Gerald B. Cleaver
Symmetry 2021, 13(8), 1469; https://doi.org/10.3390/sym13081469 - 11 Aug 2021
Cited by 2 | Viewed by 2282
Abstract
In the years 1917–1919 Tullio Levi-Civita published a number of papers presenting new solutions to Einstein’s equations. This work, while partially translated, remains largely inaccessible to English speaking researchers. In this paper we review these solutions, and present them in a modern readable [...] Read more.
In the years 1917–1919 Tullio Levi-Civita published a number of papers presenting new solutions to Einstein’s equations. This work, while partially translated, remains largely inaccessible to English speaking researchers. In this paper we review these solutions, and present them in a modern readable manner. We will also compute both Cartan–Karlhede and Carminati–Mclenaghan invariants such that these solutions are invariantly characterized by two distinct methods. These methods will allow for these solutions to be totally and invariantly characterized. Because of the variety of solutions considered here, this paper will also be a useful reference for those seeking to learn to apply the Cartan–Karlhede algorithm in practice. Full article
(This article belongs to the Special Issue Symmetries in Quantum Mechanics)
16 pages, 2148 KiB  
Article
Optical-Cavity-Induced Current
by Garret Moddel, Ayendra Weerakkody, David Doroski and Dylan Bartusiak
Symmetry 2021, 13(3), 517; https://doi.org/10.3390/sym13030517 - 22 Mar 2021
Cited by 7 | Viewed by 20930
Abstract
The formation of a submicron optical cavity on one side of a metal–insulator–metal (MIM) tunneling device induces a measurable electrical current between the two metal layers with no applied voltage. Reducing the cavity thickness increases the measured current. Eight types of tests were [...] Read more.
The formation of a submicron optical cavity on one side of a metal–insulator–metal (MIM) tunneling device induces a measurable electrical current between the two metal layers with no applied voltage. Reducing the cavity thickness increases the measured current. Eight types of tests were carried out to determine whether the output could be due to experimental artifacts. All gave negative results, supporting the conclusion that the observed electrical output is genuinely produced by the device. We interpret the results as being due to the suppression of vacuum optical modes by the optical cavity on one side of the MIM device, which upsets a balance in the injection of electrons excited by zero-point fluctuations. This interpretation is in accord with observed changes in the electrical output as other device parameters are varied. A feature of the MIM devices is their femtosecond-fast transport and scattering times for hot charge carriers. The fast capture in these devices is consistent with a model in which an energy ∆E may be accessed from zero-point fluctuations for a time ∆t, following a ∆E∆t uncertainty-principle-like relation governing the process. Full article
(This article belongs to the Special Issue Symmetries in Quantum Mechanics)
►▼ Show Figures

Figure 1

23 pages, 930 KiB  
Article
Phishing for (Quantum-Like) Phools—Theory and Experimental Evidence
by Ariane Lambert-Mogiliansky and Adrian Calmettes
Symmetry 2021, 13(2), 162; https://doi.org/10.3390/sym13020162 - 21 Jan 2021
Cited by 1 | Viewed by 1484
Abstract
Quantum-like decision theory is by now a theoretically well-developed field (see e.g., Danilov, Lambert-Mogiliansky & Vergopoulos, 2018). We provide a first test of the predictions of an application of this approach to persuasion. One remarkable result entails that, in contrast to Bayesian persuasion, [...] Read more.
Quantum-like decision theory is by now a theoretically well-developed field (see e.g., Danilov, Lambert-Mogiliansky & Vergopoulos, 2018). We provide a first test of the predictions of an application of this approach to persuasion. One remarkable result entails that, in contrast to Bayesian persuasion, distraction rather than relevant information has a powerful potential to influence decision-making. We first develop a quantum decision model of choice between two uncertain alternatives. We derive the impact of persuasion by means of distractive questions and contrast them with the predictions of the Bayesian model. Next, we provide the results from a first test of the theory. We conducted an experiment where respondents choose between supporting either one of two projects to save endangered species. We tested the impact of persuasion in the form of questions related to different aspects of the uncertain value of the two projects. The experiment involved 1253 respondents divided into three groups: a control group, a first treatment group and the distraction treatment group. Our main result is that, in accordance with the predictions of quantum persuasion but in violation with the Bayesian model, distraction significantly affects decision-making. Population variables play no role. Some significant variations between subgroups are exhibited and discussed. The results of the experiment provide support for the hypothesis that the manipulability of people’s decision-making can to some extent be explained by the quantum indeterminacy of their subjective representation of reality. Full article
(This article belongs to the Special Issue Symmetries in Quantum Mechanics)
►▼ Show Figures

Figure 1

26 pages, 3582 KiB  
Article
Permutation Symmetry in Coherent Electrons Scattering by Disordered Media
by Elena V. Orlenko and Fedor E. Orlenko
Symmetry 2020, 12(12), 1971; https://doi.org/10.3390/sym12121971 - 28 Nov 2020
Cited by 2 | Viewed by 1274
Abstract
A non-Anderson weak localization of an electron beam scattered from disordered matter is considered with respect to the principle of electron indistinguishability. A weak localization of electrons of a new type is essentially associated with inelastic processing. The origin of inelasticity is not [...] Read more.
A non-Anderson weak localization of an electron beam scattered from disordered matter is considered with respect to the principle of electron indistinguishability. A weak localization of electrons of a new type is essentially associated with inelastic processing. The origin of inelasticity is not essential. We take into account the identity principle for electron beam and electrons of the atom of the scatterer with an open shell. In spite of isotropic scattering by each individual scatterer, the electron exchange contribution has a hidden parameters effect on the resulting angular dependence of the scattering cross-section. In this case, the electrons of the open shell of an atomic scatterer can be in the s-state, that is, the atomic shell remains spherically symmetric. The methods of an invariant time-dependent exchange perturbation theory and a Green functions with exchange were applied. An additional angular dependence of the scattering cross-section appears during the coherent scattering process. It is shown exactly for the helium scatterer that the role of exchange effects in the case of a singlet is negligible, while for the triplet state, it is decisive, especially for those values of the energy of incident electrons when de Broglie’s waves are commensurate with the atomic. Full article
(This article belongs to the Special Issue Symmetries in Quantum Mechanics)
►▼ Show Figures

Figure 1

26 pages, 2718 KiB  
Article
Symmetries in Teleportation Assisted by N-Channels under Indefinite Causal Order and Post-Measurement
by Carlos Cardoso-Isidoro and Francisco Delgado
Symmetry 2020, 12(11), 1904; https://doi.org/10.3390/sym12111904 - 20 Nov 2020
Cited by 7 | Viewed by 1882
Abstract
Quantum teleportation has had notorious advances in the last decade, being successfully deployed in the experimental domain. In other terrains, the understanding of indefinite causal order has demonstrated a valuable enhancement in quantum communication to correct channel imperfections. In this work, we address [...] Read more.
Quantum teleportation has had notorious advances in the last decade, being successfully deployed in the experimental domain. In other terrains, the understanding of indefinite causal order has demonstrated a valuable enhancement in quantum communication to correct channel imperfections. In this work, we address the symmetries underlying imperfect teleportation when it is assisted by indefinite causal order to correct the use of noisy entangled resources. In the strategy being presented, indefinite causal order introduces a control state to address the causal ordering. Then, by using post-selection, it fulfills the teleportation enhancement to recover the teleported state by constructive interference. By analysing primarily sequential teleportation under definite causal order, we perform a comparison basis for notable outcomes derived from indefinite causal order. After, the analysis is conducted by increasing the number of teleportation processes, thus suggesting additional alternatives to exploit the most valuable outcomes in the process by adding weak measurement as a complementary strategy. Finally, we discuss the current affordability for an experimental implementation. Full article
(This article belongs to the Special Issue Symmetries in Quantum Mechanics)
►▼ Show Figures

Figure 1

21 pages, 532 KiB  
Article
Algebraic DVR Approaches Applied to Describe the Stark Effect
by Marisol Bermúdez-Montaña, Marisol Rodríguez-Arcos, Renato Lemus, José M. Arias, Joaquín Gómez-Camacho and Emilio Orgaz
Symmetry 2020, 12(10), 1719; https://doi.org/10.3390/sym12101719 - 19 Oct 2020
Cited by 7 | Viewed by 2236
Abstract
Two algebraic approaches based on a discrete variable representation are introduced and applied to describe the Stark effect in the non-relativistic Hydrogen atom. One approach consists of considering an algebraic representation of a cutoff 3D harmonic oscillator where the matrix representation of the [...] Read more.
Two algebraic approaches based on a discrete variable representation are introduced and applied to describe the Stark effect in the non-relativistic Hydrogen atom. One approach consists of considering an algebraic representation of a cutoff 3D harmonic oscillator where the matrix representation of the operators r2 and p2 are diagonalized to define useful bases to obtain the matrix representation of the Hamiltonian in a simple form in terms of diagonal matrices. The second approach is based on the U(4) dynamical algebra which consists of the addition of a scalar boson to the 3D harmonic oscillator space keeping constant the total number of bosons. This allows the kets associated with the different subgroup chains to be linked to energy, coordinate and momentum representations, whose involved branching rules define the discrete variable representation. Both methods, although originating from the harmonic oscillator basis, provide different convergence tests due to the fact that the associated discrete bases turn out to be different. These approaches provide powerful tools to obtain the matrix representation of 3D general Hamiltonians in a simple form. In particular, the Hydrogen atom interacting with a static electric field is described. To accomplish this task, the diagonalization of the exact matrix representation of the Hamiltonian is carried out. Particular attention is paid to the subspaces associated with the quantum numbers n=2,3 with m=0. Full article
(This article belongs to the Special Issue Symmetries in Quantum Mechanics)
►▼ Show Figures

Figure 1

8 pages, 466 KiB  
Article
The Heisenberg Uncertainty Principle as an Endogenous Equilibrium Property of Stochastic Optimal Control Systems in Quantum Mechanics
by Jussi Lindgren and Jukka Liukkonen
Symmetry 2020, 12(9), 1533; https://doi.org/10.3390/sym12091533 - 17 Sep 2020
Cited by 6 | Viewed by 21897
Abstract
We provide a natural derivation and interpretation for the uncertainty principle in quantum mechanics from the stochastic optimal control approach. We show that, in particular, the stochastic approach to quantum mechanics allows one to understand the uncertainty principle through the “thermodynamic equilibrium”. A [...] Read more.
We provide a natural derivation and interpretation for the uncertainty principle in quantum mechanics from the stochastic optimal control approach. We show that, in particular, the stochastic approach to quantum mechanics allows one to understand the uncertainty principle through the “thermodynamic equilibrium”. A stochastic process with a gradient structure is key in terms of understanding the uncertainty principle and such a framework comes naturally from the stochastic optimal control approach to quantum mechanics. The symmetry of the system is manifested in certain non-vanishing and invariant covariances between the four-position and the four-momentum. In terms of interpretation, the results allow one to understand the uncertainty principle through the lens of scientific realism, in accordance with empirical evidence, contesting the original interpretation given by Heisenberg. Full article
(This article belongs to the Special Issue Symmetries in Quantum Mechanics)
►▼ Show Figures

Figure 1

45 pages, 3176 KiB  
Article
Symmetry, Transactions, and the Mechanism of Wave Function Collapse
by John Gleason Cramer and Carver Andress Mead
Symmetry 2020, 12(8), 1373; https://doi.org/10.3390/sym12081373 - 18 Aug 2020
Cited by 3 | Viewed by 5143
Abstract
The Transactional Interpretation of quantum mechanics exploits the intrinsic time-symmetry of wave mechanics to interpret the ψ and ψ* wave functions present in all wave mechanics calculations as representing retarded and advanced waves moving in opposite time directions that form a quantum [...] Read more.
The Transactional Interpretation of quantum mechanics exploits the intrinsic time-symmetry of wave mechanics to interpret the ψ and ψ* wave functions present in all wave mechanics calculations as representing retarded and advanced waves moving in opposite time directions that form a quantum “handshake” or transaction. This handshake is a 4D standing-wave that builds up across space-time to transfer the conserved quantities of energy, momentum, and angular momentum in an interaction. Here, we derive a two-atom quantum formalism describing a transaction. We show that the bi-directional electromagnetic coupling between atoms can be factored into a matched pair of vector potential Green’s functions: one retarded and one advanced, and that this combination uniquely enforces the conservation of energy in a transaction. Thus factored, the single-electron wave functions of electromagnetically-coupled atoms can be analyzed using Schrödinger’s original wave mechanics. The technique generalizes to any number of electromagnetically coupled single-electron states—no higher-dimensional space is needed. Using this technique, we show a worked example of the transfer of energy from a hydrogen atom in an excited state to a nearby hydrogen atom in its ground state. It is seen that the initial exchange creates a dynamically unstable situation that avalanches to the completed transaction, demonstrating that wave function collapse, considered mysterious in the literature, can be implemented with solutions of Schrödinger’s original wave mechanics, coupled by this unique combination of retarded/advanced vector potentials, without the introduction of any additional mechanism or formalism. We also analyze a simplified version of the photon-splitting and Freedman–Clauser three-electron experiments and show that their results can be predicted by this formalism. Full article
(This article belongs to the Special Issue Symmetries in Quantum Mechanics)
►▼ Show Figures

Graphical abstract

31 pages, 937 KiB  
Article
Inflation with Scalar-Tensor Theory of Gravity
by Dalia Saha, Susmita Sanyal and Abhik Kumar Sanyal
Symmetry 2020, 12(8), 1267; https://doi.org/10.3390/sym12081267 - 01 Aug 2020
Cited by 3 | Viewed by 2149
Abstract
The latest released data from Planck in 2018 put up tighter constraints on inflationary parameters. In the present article, the in-built symmetry of the non-minimally coupled scalar-tensor theory of gravity is used to fix the coupling parameter, the functional Brans–Dicke parameter, and the [...] Read more.
The latest released data from Planck in 2018 put up tighter constraints on inflationary parameters. In the present article, the in-built symmetry of the non-minimally coupled scalar-tensor theory of gravity is used to fix the coupling parameter, the functional Brans–Dicke parameter, and the potential of the theory. It is found that all the three different power-law potentials and one exponential pass these constraints comfortably, and also gracefully exit from inflation. Full article
(This article belongs to the Special Issue Symmetries in Quantum Mechanics)
►▼ Show Figures

Figure 1

Review

Jump to: Editorial, Research

27 pages, 462 KiB  
Review
Gravitational Dispersion Forces and Gravity Quantization
by Fabrizio Pinto
Symmetry 2021, 13(1), 40; https://doi.org/10.3390/sym13010040 - 29 Dec 2020
Cited by 2 | Viewed by 2395
Abstract
The parallel development of the theories of electrodynamical and gravitational dispersion forces reveals important differences. The former arose earlier than the formulation of quantum electrodynamics so that expressions for the unretarded, van der Waals forces were obtained by treating the field as classical. [...] Read more.
The parallel development of the theories of electrodynamical and gravitational dispersion forces reveals important differences. The former arose earlier than the formulation of quantum electrodynamics so that expressions for the unretarded, van der Waals forces were obtained by treating the field as classical. Even after the derivation of quantum electrodynamics, semiclassical considerations continued to play a critical role in the interpretation of the full results, including in the retarded regime. On the other hand, recent predictions about the existence of gravitational dispersion forces were obtained without any consideration that the gravitational field might be fundamentally classical. This is an interesting contrast, as several semiclassical theories of electrodynamical dispersion forces exist although the electromagnetic field is well known to be quantized, whereas no semiclassical theory of gravitational dispersion forces was ever developed although a full quantum theory of gravity is lacking. In the first part of this paper, we explore this evolutionary process from a historical point of view, stressing that the existence of a Casimir effect is insufficient to demonstrate that a field is quantized. In the second part of the paper, we show that the recently published results about gravitational dispersion forces can be obtained without quantizing the gravitational field. This is done first in the unretarded regime by means of Margenau’s treatment of multipole dispersion forces, also obtaining mixed potentials. These results are extended to the retarded regime by generalizing to the gravitational field the approach originally proposed by McLachlan. The paper closes with a discussion of experimental challenges and philosophical implications connected to gravitational dispersion forces. Full article
(This article belongs to the Special Issue Symmetries in Quantum Mechanics)
66 pages, 4496 KiB  
Review
Dynamical Symmetries of the H Atom, One of the Most Important Tools of Modern Physics: SO(4) to SO(4,2), Background, Theory, and Use in Calculating Radiative Shifts
by G. Jordan Maclay
Symmetry 2020, 12(8), 1323; https://doi.org/10.3390/sym12081323 - 07 Aug 2020
Cited by 10 | Viewed by 5103
Abstract
Understanding the hydrogen atom has been at the heart of modern physics. Exploring the symmetry of the most fundamental two body system has led to advances in atomic physics, quantum mechanics, quantum electrodynamics, and elementary particle physics. In this pedagogic review, we present [...] Read more.
Understanding the hydrogen atom has been at the heart of modern physics. Exploring the symmetry of the most fundamental two body system has led to advances in atomic physics, quantum mechanics, quantum electrodynamics, and elementary particle physics. In this pedagogic review, we present an integrated treatment of the symmetries of the Schrodinger hydrogen atom, including the classical atom, the SO(4) degeneracy group, the non-invariance group or spectrum generating group SO(4,1), and the expanded group SO(4,2). After giving a brief history of these discoveries, most of which took place from 1935–1975, we focus on the physics of the hydrogen atom, providing a background discussion of the symmetries, providing explicit expressions for all of the manifestly Hermitian generators in terms of position and momenta operators in a Cartesian space, explaining the action of the generators on the basis states, and giving a unified treatment of the bound and continuum states in terms of eigenfunctions that have the same quantum numbers as the ordinary bound states. We present some new results from SO(4,2) group theory that are useful in a practical application, the computation of the first order Lamb shift in the hydrogen atom. By using SO(4,2) methods, we are able to obtain a generating function for the radiative shift for all levels. Students, non-experts, and the new generation of scientists may find the clearer, integrated presentation of the symmetries of the hydrogen atom helpful and illuminating. Experts will find new perspectives, even some surprises. Full article
(This article belongs to the Special Issue Symmetries in Quantum Mechanics)
►▼ Show Figures

Figure 1

Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
 
Displaying articles 1-12
Back to TopTop