scholarly journals Wide Effectiveness of a Sine Basis for Quantum-Mechanical Problems indDimensions

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Richard L. Hall ◽  
Alexandra Lemus Rodríguez
Keyword(s):  
Quantum Mechanics ◽  
Quantum Systems ◽  
Quantum Mechanical ◽  
Natural Basis ◽  
Boson Systems ◽  
Spanning Set

It is shown that the spanning set forL2([0,1])provided by the eigenfunctions{2sin(nπx)}n=1∞of the particle in a box in quantum mechanics provides a very effective variational basis for more general problems. The basis is scaled to[a,b], whereaandbare then used as variational parameters. What is perhaps a natural basis for quantum systems confined to a spherical box inRdturns out to be appropriate also for problems that are softly confined byU-shaped potentials, including those with strong singularities atr=0. Specific examples are discussed in detail, along with some boundN-boson systems.

scholarly journals ${\mathcal N}$-FOLD SUPERSYMMETRIC QUANTUM MECHANICS WITH REFLECTIONS

2012 ◽  
Vol 27 (19) ◽  
pp. 1250102 ◽  
Author(s):  
TOSHIAKI TANAKA
Keyword(s):  
Quantum Mechanics ◽  
Hermitian Matrix ◽  
Mechanical Systems ◽  
Quantum Systems ◽  
Supersymmetric Quantum Mechanics ◽  
The Other ◽  
Quantum Mechanical ◽  
Quantum Mechanical Systems ◽  
Ordinary Quantum

We formulate [Formula: see text]-fold supersymmetry in quantum mechanical systems with reflection operators. As in the cases of other systems, they possess the two significant characters of [Formula: see text]-fold supersymmetry, namely, almost isospectrality and weak quasi-solvability. We construct explicitly the most general one- and two-fold supersymmetric quantum mechanical systems with reflections. In the case of [Formula: see text], we find that there are seven inequivalent such systems, three of which are characterized by three arbitrary functions having definite parity while the other four characterized by two arbitrary functions. In addition, four of the seven inequivalent systems do not reduce to ordinary quantum systems without reflections. Furthermore, in certain particular cases, they are essentially equivalent to the most general two-by-two Hermitian matrix two-fold supersymmetric quantum systems obtained previously by us.

scholarly journals Signatures of Quantum Mechanics in Chaotic Systems

Entropy ◽  
2019 ◽  
Vol 21 (6) ◽  
pp. 618 ◽  
Author(s):  
Kevin M. Short ◽  
Matthew A. Morena
Keyword(s):  
Quantum Mechanics ◽  
Quantum Entanglement ◽  
Chaotic Systems ◽  
Measurement Problem ◽  
Quantum Systems ◽  
Analytical Tool ◽  
Quantum Mechanical ◽  
Unstable Periodic Orbits ◽  
Classical Correspondence ◽  
Quantum Mechanical Systems

We examine the quantum-classical correspondence from a classical perspective by discussing the potential for chaotic systems to support behaviors normally associated with quantum mechanical systems. Our main analytical tool is a chaotic system’s set of cupolets, which are highly-accurate stabilizations of its unstable periodic orbits. Our discussion is motivated by the bound or entangled states that we have recently detected between interacting chaotic systems, wherein pairs of cupolets are induced into a state of mutually-sustaining stabilization that can be maintained without external controls. This state is known as chaotic entanglement as it has been shown to exhibit several properties consistent with quantum entanglement. For instance, should the interaction be disturbed, the chaotic entanglement would then be broken. In this paper, we further describe chaotic entanglement and go on to address the capacity for chaotic systems to exhibit other characteristics that are conventionally associated with quantum mechanics, namely analogs to wave function collapse, various entropy definitions, the superposition of states, and the measurement problem. In doing so, we argue that these characteristics need not be regarded exclusively as quantum mechanical. We also discuss several characteristics of quantum systems that are not fully compatible with chaotic entanglement and that make quantum entanglement unique.

scholarly journals Ontological states and dynamics of discrete (pre-) quantum systems

2017 ◽  
Vol 15 (08) ◽  
pp. 1740013 ◽  
Author(s):  
Hans-Thomas Elze
Keyword(s):  
Quantum Mechanics ◽  
Equations Of Motion ◽  
Superposition Principle ◽  
Quantum Systems ◽  
Quantum Mechanical ◽  
Interpretation Of Quantum Mechanics ◽  
First Order ◽  
Mechanical Models ◽  
Superposition States ◽  
Ising Spins

The notion of ontological states is introduced here with reference to the Cellular Automaton Interpretation of Quantum Mechanics (QM) proposed by ’t Hooft. A class of discrete deterministic “Hamiltonian” Cellular Automata (CA) is defined that has been shown to bear many features in common with continuum quantum mechanical models, however, deformed by the presence of a finite discreteness scale [Formula: see text], such that for [Formula: see text], the usual properties result e.g. concerning linearity, dispersion relations, multipartite systems, and Superposition Principle. We argue that within this class of models, only very primitive realizations of ontological states and their dynamics can exist, since the equations of motion tend to produce superposition states that are not ontological. The most interesting, if not the only way out seems to involve interacting multipartite systems composed of two-state “Ising spins”, which evolve by a unitary transfer matrix. Thus, quantum like and ontological models appear side by side here, but distinguished by second-order and first-order dynamics, respectively.

Modelling Flexible Protein-Ligand Binding in p38α MAP Kinase using the QUBE Force Field

2019 ◽  
Author(s):  
Joshua Horton ◽  
Alice Allen ◽  
Daniel Cole
Keyword(s):  
Quantum Mechanics ◽  
Force Field ◽  
Map Kinase ◽  
Binding Free Energy ◽  
Quantum Mechanical ◽  
Enhanced Sampling ◽  
Relative Binding ◽  
Stringent Test ◽  
Flexible Protein ◽  
P38α Map Kinase

<div><div><div><p>The quantum mechanical bespoke (QUBE) force field is used to retrospectively calculate the relative binding free energy of a series of 17 flexible inhibitors of p38α MAP kinase. The size and flexibility of the chosen molecules represent a stringent test of the derivation of force field parameters from quantum mechanics, and enhanced sampling is required to reduce the dependence of the results on the starting structure. Competitive accuracy with a widely-used biological force field is achieved, indicating that quantum mechanics derived force fields are approaching the accuracy required to provide guidance in prospective drug discovery campaigns.</p></div></div></div>

scholarly journals IMPLICATIONS OF A QUANTUM MECHANICAL TREATMENT OF THE UNIVERSE

1998 ◽  
Vol 13 (05) ◽  
pp. 347-351 ◽  
Author(s):  
MURAT ÖZER
Keyword(s):  
Quantum Mechanics ◽  
Wave Packet ◽  
Early Universe ◽  
Mechanical Treatment ◽  
Correspondence Principle ◽  
Scale Factor ◽  
Quantum Mechanical ◽  
Quantum Mechanical Treatment ◽  
The Standard Model ◽  
The Universe

We attempt to treat the very early Universe according to quantum mechanics. Identifying the scale factor of the Universe with the width of the wave packet associated with it, we show that there cannot be an initial singularity and that the Universe expands. Invoking the correspondence principle, we obtain the scale factor of the Universe and demonstrate that the causality problem of the standard model is solved.

scholarly journals QUANTUM SINGULARITIES IN STATIC SPACETIMES

2011 ◽  
Vol 20 (05) ◽  
pp. 729-743 ◽  
Author(s):  
JOÃO PAULO M. PITELLI ◽  
PATRICIO S. LETELIER
Keyword(s):  
Quantum Mechanics ◽  
Wave Operator ◽  
Point Of View ◽  
Quantum Mechanical ◽  
Mathematical Framework ◽  
Physical Content ◽  
Dirac Equations ◽  
Quantum Particles ◽  
Klein Gordon ◽  
Quantum Singularities

We review the mathematical framework necessary to understand the physical content of quantum singularities in static spacetimes. We present many examples of classical singular spacetimes and study their singularities by using wave packets satisfying Klein–Gordon and Dirac equations. We show that in many cases the classical singularities are excluded when tested by quantum particles but unfortunately there are other cases where the singularities remain from the quantum mechanical point of view. When it is possible we also find, for spacetimes where quantum mechanics does not exclude the singularities, the boundary conditions necessary to turn the spatial portion of the wave operator to be self-adjoint and emphasize their importance to the interpretation of quantum singularities.

On the error of approximations in quantum mechanics. I. General theory

1965 ◽  
Vol 257 (1081) ◽  
pp. 309-326 ◽  
Keyword(s):  
Quantum Mechanics ◽  
General Theory ◽  
Computer Applications ◽  
Quantum Mechanical ◽  
Quantum Mechanical Calculations ◽  
Density Matrices ◽  
Single Error

General formulas for estimating the errors in quantum-mechanical calculations are given in the formalism of density matrices. Some properties of the traces of matrices are used to simplify the estimating and to indicate a way of obtaining a better approximation. It is shown that the simultaneous correction of all the equations to be fulfilled leads in most cases to a faster convergence than the exact fulfilment of some of the equations and approximating stepwise to some of the others. The corrective formulas contain only direct operations of the matrices occurring and so they are advantageous in computer applications. In the last section a ‘subjective error’ definition is given and by taking into account the weight of the errors of the several equations a faster convergence and a single error quantity is obtained. Some special applications of the method will be published later.

Quanta of the Third Kind

2021 ◽  
Vol 6 (3) ◽  
Author(s):  
Frank Wilczek
Keyword(s):  
Quantum Mechanics ◽  
Experimental Results ◽  
Quantum Mechanical ◽  
The Third

Quantum mechanics is nearly one hundred years old; and yet the challenge it presents to the imagination is so great that scientists are still coming to terms with some of its most basic implications. Theoretical insights and recent experimental results in anyon physics are leading physicists to revise and expand their ideas about what quantum-mechanical particles are and how they behave.

scholarly journals Wigner's Problem and Alternative Commutation Relations for Quantum Mechanics

1997 ◽  
Vol 11 (10) ◽  
pp. 1281-1296 ◽  
Author(s):  
V. I. Man'ko ◽  
G. Marmo ◽  
F. Zaccaria ◽  
E. C. G. Sudarshan
Keyword(s):  
Quantum Mechanics ◽  
Vector Field ◽  
Equations Of Motion ◽  
Quantum Systems ◽  
Heisenberg Picture ◽  
Commutation Relations

It is shown that for quantum systems the vector field associated with the equations of motion may admit alternative Hamiltonian descriptions, both in the Schrödinger and Heisenberg picture. We illustrate these ambiguities in terms of simple examples.

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