Journal of Statistical Mechanics Theory and Experiment
Quantum mechanics used for modeling macroscopic structural systems: finite-element static analysis of a simple bar by the concepts in quantum mechanics
In this paper, the two following fundamental questions in solid mechanics are answered by extending quantum-mechanical rules to macro level: 1. Why cannot both kinetic and kinematic quantities be known on a boundary point? 2. How does kinematic boundary data transfer over the continuum? For this purpose, the simple two-node bar finite element is introduced as a two-level quantum system (qubit); hence, two bar elements which are attached in series constitute an entangled two-qubit system. When a nodal or inter-nodal kinetic/kinematic boundary condition (BC) is established (i.e. a quantum measurement is performed), the system collapses in a new state. A simple bar (discretized by finite element method) might thus be considered as a set of some entangled two-element systems, which successively collapse in new states when kinetic/kinematic BCs are consecutively applied. Through such a process, boundary kinetic/kinematic information transfer along the whole bar and the static analysis is implemented.
In this paper, the two following fundamental questions in solid mechanics are answered by extending quantum-mechanical rules to macro level: 1. Why cannot both kinetic and kinematic quantities be known on a boundary point? 2. How does kinematic boundary data transfer over the continuum? For this purpose, the simple two-node bar finite element is introduced as a two-level quantum system (qubit); hence, two bar elements which are attached in series constitute an entangled two-qubit system. When a nodal or inter-nodal kinetic/kinematic boundary condition (BC) is established (i.e. a quantum measurement is performed), the system collapses in a new state. A simple bar (discretized by finite element method) might thus be considered as a set of some entangled two-element systems, which successively collapse in new states when kinetic/kinematic BCs are consecutively applied. Through such a process, boundary kinetic/kinematic information transfer along the whole bar and the static analysis is implemented.
What is quantum mechanics and what does it do ? In very general terms, the basic problem that both classical Newtonian mechanics and quantum mechanics seek to address can be stated very simply: if the state of a dynamic system is known initially and something is done to it, how will the state of the system change with time in response ? In this chapter, we will give a brief overview of, first, how Newtonian mechanics goes about solving the problem for systems in the macroscopic world and, then, how quantum mechanics does it for systems on the atomic and subatomic scale. We will see qualitatively what the differences and similarities of the two schemes are and what the domain of applicability of each is. Brief overview of classical mechanics To answer the question posed above systematically, we must first give a more rigorous formulation of the problem and introduce the special language and terminology (in double quotation marks) that will be used in subsequent discussions. For the macroscopic world, common sense tells us that, to begin with, we should identify the “system” that we are dealing with in terms of a set of “static properties” that do not change with time in the context of the problem. For example, the mass of an object might be a static property. The change in the “state” of the system is characterized by a set of “dynamic variables.
We consider the following two questions. Suppose that a quantum system suffers a change of the boundary condition or the potential at a given space location. Then (1) when will the wavefunction shows a response to this change at another location? And (2) how does the wavefunction changes?The answer to question (1) could reveal how a quantum system gets information on the boundary condition or the potential. Here we show that if the response takes place immediately, then it can allow superluminal signal transfer. Else if the response propagates in space with a finite velocity, then it could give a simple explanation why our world shows classicality on the macroscopic scale. Furthermore, determining the exact value of this velocity can either clarify the doubts on static experiments for testing Bell’s inequality, or support the pilot‐wave interpretation of quantum mechanics. We propose a feasible experimental scheme for measuring this velocity, which can be implemented with state‐of‐art technology, e.g., single‐electron biprism interferometry.Question (2) is studied with a square‐well potential model, and we find a paradox between the impossibility of superluminal signal transfer and the normalization condition of wavefunctions. To solve the paradox, we predict that when a change of the potential occurs at a given space location, the system will show no response to this change at all, until after a certain time interval. Otherwise either special relativity or quantum mechanics will be violated. As a consequence, no physical process can actually happen within Planck time. Therefore it gives a simple proof that time is discrete, with Planck time being the smallest unit. Combining with the answer to question (1), systems with a larger size and a slower velocity could have a larger unit of time, making it possible to test the discreteness of time experimentally. Our result also sets a limit on the speed of computers, and gives instruction to the search of quantum gravity theories.
We present examples of macroscopic systems entailing a quantum mechanical structure. One of our examples has a structure which is isomorphic to the spin structure for a spin 1/2 and another system entails a structure isomorphic to the structure of two spin 1/2 in the entangled singlet state. We elaborate this system by showing that an arbitrary tensor product state representing two entangled qubits can be described in a complete way by a specific internal constraint between the ray or density states of the two qubits, which describes the behavior of the state of one of the spins if measurements are executed on the other spin. Since any n-qubit unitary operation can be decomposed into 2-qubit gates and unary operations, we argue that our representation of 2-qubit entanglement contributes to a better understanding of the role of n-qubit entanglement in quantum computation. We illustrate our approach on two 2-qubit algorithms proposed by Deutsch, respectively Arvind et al. One of the advantages of the 2-qubit case besides its relative simplicity is that it allows for a nice geometrical representation of entanglement, which contributes to a more intuitive grasp of what is going on in a 2-qubit quantum computation.
A generating functional formalism is developed to facilitate the derivation of coarse-grained dynamics of macroscopically relevant variables in various types of many-body problems. The relevant variables can be found either by using symmetry analysis of the action or experimental data. The outcome of the coarse-graining process is a set of coupled partial differential equations for the expectation values of the relevant observables. The set of the macroscopically important observables consists typically of conserved variables and order operators, which are related to the broken symmetries of the system. Symmetry breaking (spontaneous or explicit) in the formalism can be dynamically induced by coupling the system to a generalized heat bath.\n\nThe nonequilibrium generating functional formalism is applied to coarse-graining of generalized hydrodynamics of macroscopic quantum phenomena such as Bose fluids and superconductors, and finally classical fluids. The method is by no means restricted to fluids, but can be applied to condensed matter problems as well. High density systems are in general more problematic and it remains to be seen if one can come up with more powerful approximation methods within the current formalism in the future.\n\nThe generating functional formalism is shown to reproduce in the specific limits the well-known ground state, finite temperature and time-dependent density functional theories in the quantum regime. We enlarge the set of relevant variables to include also non-conserved order-operators and show how to form generalized density functional theories, which can be by construction linked to the phase-transition behaviour of the system. They are extensions of the traditional density functional theories that were not developed to describe phase changes but only facilitated the expression of the system's (free) energy in terms of the conserved density variable (e.g. number density of electrons).\n\nWe develop a generalized density functional theory (effective action of the generating functional) for superconductors and superfluids. For superconductors it is shown by simple symmetry analysis that the relevant order parameter is the pair field of two electrons. Coarse-graining out the phononic degrees of freedom is shown to lead to a retarded effective interaction between the electrons. In the local static limit (effective interaction between electrons static) the theory reduces to the time-dependent Ginzburg-Landau theory. Similarly, by symmetry arguments, the order parameter of a weakly interacting Bose fluid is identified and its dynamics in the tree level approximation is shown to follow the Gross-Pitaevskii equation. Higher order corrections can be systematically produced.\n\nClassical molecular dynamics and stochastic cellular automata types of higher level descriptions can be mapped into the same formalism using commonly known techniques. It is also shown how to connect the static and dynamic density functionals to phase-field models (models of critical dynamics), which are commonly used in materials science modelling. A specific example is provided for a liquid-gas-solid system. The phase-field modelling is then applied to the derivation of dynamics of the liquid-gas phase boundaries and triple lines for which a new projection operator technique is developed. We explain in detail how to go from microscopics to macroscopics and relate the effective parameters on each level to each other. For example, we express the macroscopic surface tensions from the phase-field model in terms of the molecular dynamics simulation parameters. We also discuss how to extract the relevant cross-over time and length scales from the theoretical description and relate them to the experimentally measurable ones. It is shown how the coupling of the triple line and the liquid-gas boundary leads to emergence of memory effects. For confined geometry we also derive a new form for the restoring force acting on the contact line. The phase-field model of the liquid-gas-solid system is first discussed without explicit hydrodynamics from the point of view of diffuse mass transfer and then hydrodynamic effects (momentum conservation) are added.\n\nEvolution equations are derived for the mass, momentum and energy densities representing the proper set of macro variables of simple fluids. The derivation is based on a macroscopic Poisson bracket formulation of phase-field dynamics. Navier-Stokes equations are obtained with extra force terms arising from wall-fluid interactions and capillary stresses. It is also shown how the hydrodynamics boundary conditions for stress tensor can be derived from the bulk fields alone in the sharp interface limit. Some mechanisms for generation of the no-slip boundary condition are also hypothesized. These results are important because they show how in general boundary conditions and corrections to them can be derived and how the numerical implementation can benefit from such a formulation of hydrodynamics, where all boundary conditions are expressed in terms of bulk fields alone. Again, these results have wider applicability than the setting where they have been derived.\n\nThe advantage of a systematic coarse-graining procedure enables us to clarify the role of the order parameter concept in the phase-field description of one and two component fluids. We also consider different ways of including gravitational effects and apply dimensional analysis to obtain a condition for the onset of capillary turbulence.\n\nFinally, we present some new formalism for stochastic processes, where the relevant field variables are embedded in an environment characterized by quenched disorder. Whenever some lower dimensional (e.g. defect) structures are projected out of the bulk field evolution equations containing either frozen of coarse-graining generated noise fields, a more complicated effective stochastic noise component will appear in the projected equation of motion of the defect. In particular, the effective noise can be long-ranged even if the quenched background disorder is not. Several condensed matter and hydrodynamical examples describing dynamics of interfaces and line-like objects such as polymers and domain wall defects are considered.
Continuous models of objective state reduction evade exact spatial-support trace limits by employing smooth localizations. Exploring whether a discrete substrate natively avoids continuous spreading provides a strict motivation for mapping candidate discrete spatial ontologies. Acting as a strict thermodynamic screen for macroscopic pointer stability, we evaluate the unitary recurrence-suppression resources available to candidate discrete baths via a continuous-time pure-dephasing surrogate. Because this surrogate synthetically imports orthodox continuous-time unitarity, our topological recurrence rankings are conditionally bounded by this imported structure; we explicitly defer the kinematic persistence of exact spatial voids to future discrete-time update rules. Delineating surrogate artifacts from analytical deliverables, we map two conditional limits. (1) Density of States Mappings—we frame a physical mesoscopic design tradeoff between Euclidean 3-dimensionality, exact spatial homogeneity, scalar mass-density point-couplings, and extensive variance suppression. Infinite discrete 3D cubic lattices satisfy Euclidean geometry and homogeneity but forfeit local depth due to unnormalized 4d bandwidth expansion. The fixed mass gap regulates the infrared Van Hove singularity from a divergence into a massive spectral concentration. Their finite mesoscopic counterparts suffer severe continuous-time revivals because exact crystallographic symmetry structurally collapses their distinct active frequencies (20 for the 7^3 symmetric torus vs 343). Breaking exact spatial homogeneity (disordered grids) or abandoning Euclidean 3-dimensionality entirely (non-Euclidean spectral comparators) is mathematically required to restore extensive participation and effectively delay mesoscopic revivals. A genuine discrete-time Grover quantum walk robustly confirms this continuous-time diagnostic in the cubic regime, bounding the heuristic's domain of validity to crystallographic degeneracy. (2) The Continuum Representational Limit—we chain Effective Field Theory (EFT) localization bounds to the functional-analytic no-gos of the companion taxonomy. By the data processing inequality for Fisher information (marginalization bound), the 3D Fisher information is bounded below by its 1D marginal. A 1D Cramer-Rao conditional localization estimate for a boundary transition layer of width a >= l yields an expected kinetic energy penalty. Approximating an exact discrete boundary at the non-relativistic crossover drives the local approximation into fundamental kinematic breakdown at the systemic rest-mass scale. Transitioning the continuum form domain to the fractional space H^{1/2}(R^3) reveals that sharp boundaries natively violate the relativistic form domain limit. To preserve non-relativistic EFT validity, the representation is forced to decouple from the fundamental lattice and apply mesoscopic transition mollifiers. Under standard continuous free evolution, applying a compactly-supported mollifier preserves a void instantly at synthesis but triggers instantaneous Paley-Wiener spreading for all t>0. Consequently, if foundational spatial exactness exists natively in a discrete substrate, the operational continuum representation strictly acts as an epistemic approximation (Path A); if strictly ontically reified by the theorist, foundational exactness is permanently forfeited to a For-All-Practical-Purposes (FAPP) approximation (Path C).
Quantum Mechanics is a radically new way of thinking about the Nature.It is used for explaining the behavior of an electron to the possible existence of parallel universes.Most of the times, it is just too difficult to get an intuitive grasp of Quantum Mechanics.Quantum Mechanics is a mathematical framework for describing atomic and subatomic systems.In this paper the basic principles of Quantum Mechanics have been presented.It has been shown that classical probabilities when added increase the overall probability while in quantum probabilities can cancel each other and hence lower the overall probability.A simple approach has been taken by modeling the systems as graphs.The states and dynamics of the systems have been modeled as matrices.The principle of Superposition and the phenomenon of Interference have been explained in a single simulation.Finally the BLAS functions of Julia programming language have been used for simulating the systems under study on a classical computer and a detailed discussion is presented to highlight the incapability of the classical explanation of quantum phenomenon.