Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?

The Everett-type interpretations of quantum mechanics and quantum cosmology proposed independently by Hartle, Geroch, and Wada are discussed. They essentially involve regarding a strong peak in the wave function as a definite prediction. Wave functions in quantum cosmology are usually peaked about correlations between coordinates and momenta, and methods for identifying such correlations are introduced. The first method involves Wigner's function, a quantum-mechanical analogue of the classical phase-space distribution. The properties of this distribution are discussed and it is shown how it can be of use in describing the emergence of classical behavior from quantum systems. The second method involves a suitably chosen canonical transformation. These methods are applied to harmonic-oscillator examples, which are of relevance to scalar field fluctuations in inflationary universe models. These methods are also applied to WKB wave functions in quantum mechanics and quantum cosmology. The manner in which the wave function becomes peaked about sets of classical solutions is elucidated. This is extended to include inhomogeneous perturbations about minisuperspace in quantum cosmology, and the derivation of the semiclassical Einstein equations, ${G}_{\ensuremath{\mu}\ensuremath{\nu}}$=8\ensuremath{\pi}G〈${T}_{\ensuremath{\mu}\ensuremath{\nu}}$〉, from the Wheeler-DeWitt equation is considered. A condition under which they are valid is derived. It is essentially the requirement that the distribution of ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$, as a function of the matter modes, is strongly peaked about its average value. Some situations in which this condition is satisfied are discussed.

This study, , solution of Gaussian Integral with differential equation [18,19]; By using the partial integral method, the indefinite integral solution by taking x under the differential [29,30,31] and the indefinite integral solution by taking the Gaussian integral under the differential [33,38,39] both harmonic series and function solutions are found. Indefinite integral solution of Gaussian Integral [38] In Quantum Physics, the wave function solution f(x,α) in terms of α variable in x position and k space by substituting it in wave function [44,45] and in one-dimensional time dependent Schrödinger equation, wave function equation [66, 67] is found. When the wave function in direct space is differentiated by by the partial integral method without using the approximate value of w(k)[50] in space k, the general wave equation in k space by position [74] and approximately the wave function f(x,α) [80] is found. min in space k depending on location and time the exact solution of the wave equations [83,89] and the Taylor Series solution give the wave equation [87,90] in k space according to position and time.

CHAPTER 13 - DENSITY MATRICES

15 - The Principles of Quantum Mechanics. II. The Postulates of Quantum Mechanics

In this book we will develop the quantum-theoretical description of systems which can be described in terms of a field theory. Starting from the framework of classical physics, the concept of a field at first might evoke ideas about macroscopic systems, for example velocity fields or temperature fields in fluids and gases, etc. Fields of this kind will not concern us, however; they can be viewed as derived quantities which arise from an averaging of microscopic particle densities. Our subjects are the fundamental fields that describe matter on a microscopic level: it is the quantum-mechanical wave function ψ(x, t) of a system which can be viewed as a field from which the observable quantities can be deduced. In quantum mechanics the wave function is introduced as an ordinary complex-valued function of space and time. In Dirac’s terminology it has the character of a “c number”. Quantum field theory goes one step further and treats the wave function itself as an object which has to undergo quantization. In this way the wave function ψ(x, t) is transmuted into a field operator (x, t), which is an operator-valued quantity (a “q number”) satisfying certain commutation relations. This process, often called “second quantization”, is quite analogous to the route that in ordinary quantum mechanics leads from a set of classical coordinates to a set of quantum operators . There is one important technical difference, though, since ψ(x, t) is a field, i.e., an object which depends on the coordinate x. The latter plays the role of a “continuous-valued index”, in contrast to the discrete index i, which labels the set q i . Field theory therefore is concerned with systems having an infinite number of degrees of freedom.

There are many examples in theoretical physics where a fundamental quantity can be considered a quadratic form ρ=∑iρi=∣Ψ∣2 and the corresponding linear form Ψ=∑iψi is highly relevant for the physical problem under study. This, in particular, is the case of the density and the wave function in quantum mechanics. In the study of N-identical-fermion systems we have the additional feature that Ψ is a function of the 3N configuration space coordinates and ρ is defined in three-dimensional real space. For many-electron systems in the ground state the wave function and the Hamiltonian are to be expressed in terms of the configuration space (CS), a replica of real space for each electron. Here we present a geometric formulation of the CS, of the wave function, of the density, and of the Hamiltonian to compute the electronic structure of the system. Then, using the new geometric notation and the indistinguishability and equivalence of the electrons, we obtain an alternative computational method for the ground state of the system. We present the method and discuss its usefulness and relation to other approaches.

Schrödinger propagation of initial discontinuities leads to divergence of moments

The general properties of the quasilinearization method (QLM), particularly its fast quadratic convergence, monotonicity, and numerical stability, are analyzed and illustrated on different physical problems. The method approaches the solution of a nonlinear differential equation by approximating the nonlinear terms by a sequence of linear ones and is not based on the existence of a small parameter. It is shown that QLM gives excellent results when applied to different nonlinear differential equations in physics, such as Blasius, Lane-Emden, and Thomas-Fermi equations, as well as in computation of ground and excited bound-state energies and wave functions in quantum mechanics (where it can be applied by casting the Schrodinger equation in the nonlinear Riccati form) for a variety of potentials most of which are not treatable with the help of perturbation theory. The convergence of the QLM expansion of both energies and wave functions for all states is very fast and the first few iterations already yield extremely precise results. The QLM approximations, unlike the asymptotic series in perturbation theory and 1/N expansions, are not divergent at higher orders. The method sums many orders of perturbation theory as well as of the WKB expansion. It provides final and accurate answers for large and infinite values of the coupling constants and is able to handle even supersingular potentials for which each term of the perturbation series is infinite and the perturbation expansion does not exist.

The problem of the dualism of properties of quantum objectsis still one of the constantly debated problems of physics. To solve this problem, Wheeler proposed a thought experiment with a change in diffraction conditions for a separately flying particle. Interpretation of real experiments on the diffraction scattering of microparticles suggests that their behavior is determined by both their past and their future state. The purpose of this paper is to explain the obtained paradoxical result by the methods of classical physics. The concept of the temporal state of an object is introduced, which is equal to the square root of the pulse, analogous to the wave function in quantum mechanics. The novelty of the adopted approach is to describe the interaction of the current temporary state, both with the past and with the future temporary states. In this case, the interaction forces are proportional to the product of neighboring temporary states and inversely proportional to the time interval between these temporary states. The range of applicability of the proposed model is limited by speeds that are significantly lower than the speed of light, as well as by the condition of relatively small (compared to speed) temporal velocity increments. On the basis of the adopted method, formulas of classical dynamics for uniformly accelerated motion and motion along a circle are derived. In the framework of new ideas, the principle of equality of action and reaction is justified for the force of inertia.

量子力学中的波函数与新概念物质波

Introduction When a crystal such as quartz or feldspar is bombarded by a stream of high-energy electrons in a scanning electron microscope or other suitable instrument, photons (“particles” of light) are emitted, a phenomenon called cathodoluminescence (CL). Much is known about the origin of CL in artificial crystals because of their economic importance in the manufacture of television screens, computer monitors, and the like. Although less is known about the origin of CL in naturally occurring minerals, the fundamental causes of CL emissions are moderately well understood. To visualize the factors responsible for CL emissions, it is useful to consider atoms in crystals in terms of the band theory of solids. Energy states of electrons in crystals depend upon whether the electrons are bound in particular atoms (inner-shell electrons) or are delocalized. Delocalized electrons are electrons that are not associated with individual atoms or identifiable chemical bonds, but are shared collectively by all the constituent atoms or ions of a substance. Delocalized electrons have wave functions (in quantum mechanics, a complex function of time and position) that in effect traverse the entire crystal. Individual atoms have discrete energy states that are associated with the orbits of shells of electrons in the atom. When atoms are spaced far apart, as in a gas, they have very little influence upon each other. By contrast, atoms within a solid, such as quartz or feldspar, have a marked effect upon other atoms in the crystal (atomic orbitals combine to form molecular orbitals).

The paper is concerned with a statistical problem on the detection of particles by macroinstruments in unitary quantum mechanics. Requirements of macroinstruments whose error is minimal are defined. The wave function in unitary quantum mechanics is interpreted. The probabilities of particle detection in unitary quantum mechanics and in conventional quantum mechanics are shown to be different by two orders of magnitude, which can be tested experimentally.

The article presents the possible role of consciousness in quantum-mechanical description of physical reality. The widely spread interpretations of quantum phenomena are considered as indicating the apparent connection between conscious processes (such as observation) and the properties of the microcosm. The reasons for discrepancies between the results of observations of the microcosm and macrocosm and the potential association of consciousness with these reasons are closely investigated. The mentioned connection is meant to be interpreted in the sense that the probable requirement for a complete understanding of quantum theory is the adequate description of consciousness within it and that the correct theory of consciousness should include quantum-mechanical theoretical apparatus. In this context, the question about the methods of scientific cognition is discussed, in particular, the problem of the place and the importance of intellectual intuition in science and philosophy of science. The author draws the conclusions about the current state of the “measuring” consciousness. problem in its relationship with

An extraordinary, if circumscribed, positive shift has occurred since the mid‐twentieth century in the perceived status of Hugh Everett III's 1956 theory of the universal wave function of quantum mechanics, now widely called the Many‐Worlds Interpretation (MWI). Everett's starkly new interpretation denied the existence of a separate classical realm, contending that the experimental data can be seen as presenting a state vector for the whole universe. Since there is no state vector collapse, reality as a whole is strictly deterministic. Explained jointly by the dynamical variables and the state vector, “this reality is not the reality we customarily think of, but is a reality composed of many worlds,” wrote Everett's colleague Bryce DeWitt. In this essay, I account briefly for the change of status in conventional scientific terms, yet chiefly in extended terms of three sets of ideas that I argue can be understood to affect believability in both scientific and religious contexts, illuminating helpfully the MWI phenomenon, and its engagement with theology: orthodoxy and heresy, language and reference, and faith and agnosticism. One's orientation relative to the variable content of these dynamic, socially oriented categories helps to make belief in ideas as metaphysically challenging as Everettian Quantum Mechanics, or particular ideas about God, either more or less believable. The categories will have the same function in a theology engaging Everett's theory, and in any theology at all written in a society deeply marked by what I further argue is a subtle, powerful, and pervasive mode of quasi‐scientific thinking we can call societal constructive agnosticism, of which anyone doing theology today must be aware.

Dynamic programming gives a new method for computing the wave function in quantum mechanics. The paper compares this method with Feynman's path integral method (1965).