scholarly journals Quantum Mechanics Needs No Interpretation

2005 ◽  
Vol 70 (5) ◽  
pp. 621-637 ◽  
Author(s):  
Lubomír Skála ◽  
Vojtěch Kapsa
Keyword(s):  
Quantum Mechanics ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Mathematical Structure ◽  
Uncertainty Relations ◽  
Mean Values ◽  
Vector Potentials ◽  
Probability Densities ◽  
Klein Gordon ◽  
Definition Of

Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that the basic mathematical structure of quantum mechanics like the probability amplitudes, Born rule, probability density current, commutation and uncertainty relations, momentum operator, rules for including scalar and vector potentials and antiparticles can be derived from the definition of the mean values of powers of space coordinates and time. Equations of motion of quantum mechanics, the Klein-Gordon equation, Schrödinger equation and Dirac equation are obtained from the requirement of the relativistic invariance of the theory. The limit case of localized probability densities leads to the Hamilton-Jacobi equation of classical mechanics. Many-particle systems are also discussed.

scholarly journals Nonlinear Generalisation of Quantum Mechanics

2021 ◽  
Author(s):  
Alireza Jamali
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Lorentz Invariance ◽  
Current Theory ◽  
Quantum Mechanical ◽  
Dynamical Condition ◽  
Current Definition ◽  
Klein Gordon ◽  
Definition Of ◽  
Mechanical Momentum

It is known since Madelung that the Schrödinger equation can be thought of as governing the evolution of an incompressible fluid, but the current theory fails to mathematically express this incompressibility in terms of the wavefunction without facing problem. In this paper after showing that the current definition of quantum-mechanical momentum as a linear operator is neither the most general nor a necessary result of the de Broglie hypothesis, a new definition is proposed that can yield both a meaningful mathematical condition for the incompressibility of the Madelung fluid, and nonlinear generalisations of Schrödinger and Klein-Gordon equations. The derived equations satisfy all conditions that are expected from a proper generalisation: simplification to their linear counterparts by a well-defined dynamical condition; Galilean and Lorentz invariance (respectively); and signifying only rays in the Hilbert space.

scholarly journals A new quantum operator for distance

2019 ◽  
Vol 34 (06n07) ◽  
pp. 1950033
Author(s):  
Daniel Katz
Keyword(s):  
Quantum Mechanics ◽  
Equivalence Principle ◽  
Classical Mechanics ◽  
Proof Of Concept ◽  
Logical Relationship ◽  
Weak Equivalence Principle ◽  
Quantum Operator ◽  
Operator Means ◽  
Definition Of ◽  
Future Work

We introduce a new semirelativistic quantum operator for the length of the worldline a particle traces out as it moves. In this article the operator is constructed in a heuristic way and some of its elementary properties are explored. The operator ends up depending in a very complicated way on the potential of the system it is to act on so as a proof of concept we use it to analyze the expected distance traveled by a free Gaussian wave packet with some initial momentum. It is shown in this case that the distance such a particle travels becomes light-like as its mass vanishes and agrees with the classical result for macroscopic masses. This preliminary result has minor implications for the Weak Equivalence Principle (WEP) in quantum mechanics. In particular it shows that the logical relationship between two formulations of the WEP in classical mechanics extends to quantum mechanics. That our result is qualitatively consistent with the work of others emboldens us to start the task of evaluating the new operator in nonzero potentials. However, we readily acknowledge that the looseness in the definition of our operator means that all of our so-called results are highly speculative. Plans for future work with the new operator are discussed in the last section.

scholarly journals Classical Mechanics as a Fundamental Law of Quantum Mechanics

2020 ◽  
Author(s):  
Yehuda Roth
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Classical Mechanics ◽  
Second Law ◽  
Operator Form ◽  
Fundamental Law ◽  
Third Law ◽  
Definition Of ◽  
Force Operator ◽  
Physical Quantities

n our previous paper, we showed that the so-called quantum entanglement also exists in classical mechanics. The inability to measure this classical entanglement was rationalized with the definition of a classical observer which collapses all entanglement into distinguishable states. It was shown that evidence for this primary coherence is Newton’s third law. However, in reformulating a "classical entanglement theory" we assumed the existence of Newton’s second law as an operator form where a force operator was introduced through a Hilbert space of force states. In this paper, we derive all related physical quantities and laws from basic quantum principles. We not only define a force operator but also derive the classical mechanic's laws and prove the necessity of entanglement to obtain Newton’s third law.

scholarly journals QUANTUM MECHANICS AT PLANCK'S SCALE AND DENSITY MATRIX

2003 ◽  
Vol 12 (07) ◽  
pp. 1265-1278 ◽  
Author(s):  
A. E. SHALYT-MARGOLIN ◽  
J. G. SUAREZ
Keyword(s):  
Quantum Mechanics ◽  
Density Matrix ◽  
Semiclassical Approximation ◽  
Black Hole Entropy ◽  
Uncertainty Relations ◽  
Statistical Entropy ◽  
Fundamental Length ◽  
Definition Of ◽  
Liouville’S Equation ◽  
Natural Way

In this paper Quantum Mechanics with Fundamental Length is chosen as Quantum Mechanics at Planck's scale. This is possible due to the theory of General Uncertainty Relations. Here Quantum Mechanics with Fundamental Length is obtained as a deformation of Quantum Mechanics. The distinguishing feature of the proposed approach in comparison with previous ones, lies in the fact that here the density matrix are subjected to deformation, whereas in the previous approaches only commutators are deformed. The density matrix obtained by deforming the quantum-mechanical one is named the density pro-matrix throughout this paper. Within our approach two main features of Quantum Mechanics are conserved: the probabilistic interpretation of the theory and the well-known measuring procedure corresponding to that interpretation. The proposed approach allows a description of the dynamics. In particular, the explicit form of the deformed Liouville's equation and the deformed Shrödinger's picture are given. Some implications of obtained results are discussed. In particular, the problem of singularity, the hypothesis of cosmic censorship, a possible improvement of the definition of statistical entropy and the problem of information loss in black holes are considered. It is shown that the results obtained here allow one to deduce in a simple and natural way the Bekenstein–Hawking's formula for black hole entropy in semiclassical approximation.

scholarly journals Neutrino Mixing and Oscillations in Quantum Field Theory: A Comprehensive Introduction

Universe ◽  
2021 ◽  
Vol 7 (12) ◽  
pp. 504
Author(s):  
Luca Smaldone ◽  
Giuseppe Vitiello
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Mechanics ◽  
Dynamical Symmetry ◽  
Neutrino Energy ◽  
Uncertainty Relations ◽  
Quantum Field ◽  
Neutrino Mixing ◽  
Relativistic Limit ◽  
Definition Of

We review some of the main results of the quantum field theoretical approach to neutrino mixing and oscillations. We show that the quantum field theoretical framework, where flavor vacuum is defined, permits giving a precise definition of flavor states as eigenstates of (non-conserved) lepton charges. We obtain the exact oscillation formula, which in the relativistic limit reproduces the Pontecorvo oscillation formula and illustrates some of the contradictions arising in the quantum mechanics approximation. We show that the gauge theory structure underlies the neutrino mixing phenomenon and that there exists entanglement between mixed neutrinos. The flavor vacuum is found to be an entangled generalized coherent state of SU(2). We also discuss flavor energy uncertainty relations, which impose a lower bound on the precision of neutrino energy measurements, and we show that the flavor vacuum inescapably emerges in certain classes of models with dynamical symmetry breaking.

scholarly journals On the Stochastic Mechanics Foundation of Quantum Mechanics

Universe ◽  
2021 ◽  
Vol 7 (6) ◽  
pp. 166
Author(s):  
Michael Beyer ◽  
Wolfgang Paul
Keyword(s):  
Quantum Mechanics ◽  
Diffusion Processes ◽  
Equations Of Motion ◽  
Kinetic Equations ◽  
Classical Mechanics ◽  
Stochastic Mechanics ◽  
Foundation Of Quantum Mechanics ◽  
Recent Developments ◽  
Quantum Phenomena ◽  
Quantum Mechanical Systems

Among the famous formulations of quantum mechanics, the stochastic picture developed since the middle of the last century remains one of the less known ones. It is possible to describe quantum mechanical systems with kinetic equations of motion in configuration space based on conservative diffusion processes. This leads to the representation of physical observables through stochastic processes instead of self-adjoint operators. The mathematical foundations of this approach were laid by Edward Nelson in 1966. It allows a different perspective on quantum phenomena without necessarily using the wave-function. This article recaps the development of stochastic mechanics with a focus on variational and extremal principles. Furthermore, based on recent developments of optimal control theory, the derivation of generalized canonical equations of motion for quantum systems within the stochastic picture are discussed. These so-called quantum Hamilton equations add another layer to the different formalisms from classical mechanics that find their counterpart in quantum mechanics.

scholarly journals Shannon entropy and Fisher information of the one-dimensional Klein–Gordon oscillator with energy-dependent potential

2018 ◽  
Vol 33 (06) ◽  
pp. 1850033 ◽  
Author(s):  
Abdelmalek Boumali ◽  
Malika Labidi
Keyword(s):  
Quantum Mechanics ◽  
Fisher Information ◽  
Shannon Entropy ◽  
One Dimensional ◽  
Probability Densities ◽  
Klein Gordon ◽  
Dependent Potential ◽  
The One ◽  
Energy Dependent ◽  
Ordinary Quantum

In this paper, we studied, at first, the influence of the energy-dependent potentials on the one-dimensionless Klein–Gordon oscillator. Then, the Shannon entropy and Fisher information of this system are investigated. The position and momentum information entropies for the low-lying states n = 0, 1, 2 are calculated. Some interesting features of both Fisher and Shannon densities, as well as the probability densities, are demonstrated. Finally, the Stam, Cramer–Rao and Bialynicki–Birula–Mycielski (BBM) inequalities have been checked, and their comparison with the regarding results have been reported. We showed that the BBM inequality is still valid in the form [Formula: see text], as well as in ordinary quantum mechanics.

scholarly journals The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories

2016 ◽  
Vol 24 (2) ◽  
pp. 173-193
Author(s):  
Jana Musilová ◽  
Stanislav Hronek
Keyword(s):  
Quantum Mechanics ◽  
Variational Principle ◽  
Conservation Laws ◽  
Calculus Of Variations ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Stationary Points ◽  
General Description ◽  
Lagrange Equations ◽  
Jet Bundles

Abstract As widely accepted, justified by the historical developments of physics, the background for standard formulation of postulates of physical theories leading to equations of motion, or even the form of equations of motion themselves, come from empirical experience. Equations of motion are then a starting point for obtaining specific conservation laws, as, for example, the well-known conservation laws of momenta and mechanical energy in mechanics. On the other hand, there are numerous examples of physical laws or equations of motion which can be obtained from a certain variational principle as Euler-Lagrange equations and their solutions, meaning that the \true trajectories" of the physical systems represent stationary points of the corresponding functionals.It turns out that equations of motion in most of the fundamental theories of physics (as e.g. classical mechanics, mechanics of continuous media or fluids, electrodynamics, quantum mechanics, string theory, etc.), are Euler-Lagrange equations of an appropriately formulated variational principle. There are several well established geometrical theories providing a general description of variational problems of different kinds. One of the most universal and comprehensive is the calculus of variations on fibred manifolds and their jet prolongations. Among others, it includes a complete general solution of the so-called strong inverse variational problem allowing one not only to decide whether a concrete equation of motion can be obtained from a variational principle, but also to construct a corresponding variational functional. Moreover, conservation laws can be derived from symmetries of the Lagrangian defining this functional, or directly from symmetries of the equations.In this paper we apply the variational theory on jet bundles to tackle some fundamental problems of physics, namely the questions on existence of a Lagrangian and the problem of conservation laws. The aim is to demonstrate that the methods are universal, and easily applicable to distinct physical disciplines: from classical mechanics, through special relativity, waves, classical electrodynamics, to quantum mechanics.

Time in Quantum Mechanics

2011 ◽  
Author(s):  
Jan Hilgevoord ◽  
David Atkinson
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Uncertainty Relation ◽  
Classical Mechanics ◽  
Special Problem ◽  
Uncertainty Relations ◽  
Operator Formalism ◽  
Heisenberg Uncertainty ◽  
Time In Quantum Mechanics ◽  
Time Operators

Unlike classical mechanics, quantum mechanics assumes the famous Heisenberg uncertainty relations. One of these concerns time: the energy–time uncertainty relation. Unlike the canonical position–momentum uncertainty relation, the energy–time relation is not reflected in the operator formalism of quantum theory. Indeed, it is often said and taken as problematic that there is not a so-called “time operator” in quantum theory. This chapter sheds light on these questions and others, including the absorbing matter of whether quantum mechanics allows for the existence of ideal clocks. The second section notes that quantum mechanics does not involve a special problem for time, and that there is no fundamental asymmetry between space and time in quantum mechanics over and above the asymmetry which already exists in classical physics. The third section studies time operators in detail. The fourth section discusses various uncertainty relations involving time.

scholarly journals Modelling Wave-Particle Duality of Classical Particles and Waves

2021 ◽  
Author(s):  
Chen Yang ◽  
S. Olutunde Oyadiji
Keyword(s):  
Quantum Mechanics ◽  
Classical Mechanics ◽  
Mathematical Structure ◽  
Momentum Exchange ◽  
The Past ◽  
Wave Particle Duality ◽  
Classical Waves ◽  
Fundamental Phenomenon ◽  
Energy And Momentum ◽  
Classical Particles

Abstract Wave-particle duality is the fundamental phenomenon of particles and fields in quantum mechanics. In the past, the trajectory-like (particle-like) behaviour and wave-like behaviour has been considered separately. In this article, a superimposed model is derived to characterise wave-particle duality of classical particles. The superimposed model reflects an invariant mathematical structure (analogous variables and differential equations) from classical mechanics, classical field theories and quantum mechanics. Its analytical solution carries trajectory-like property (phase-independent) and wave-like property (phase-dependent) of particles that is consistent with to Schrodinger’s picture. Subsequently, the presented model is applied to model duality of classical waves in electromagnetism, acoustics and elasticity. The analysis implies the existence of quantum effects of classical waves at macroscopic scale. It predicts quantum picture on energy and momentum exchange between classical particles and waves. As seen in the model, wave-particle duality reflects inherent and indispensable characteristics of classical objects.

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