Nodal domain distribution for a nonintegrable two-dimensional anharmonic oscillator

Courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of Laplacian-type operators. In particular, it holds for generic eigenfunctions of quantum graph. The theorem stipulates that, after ordering the eigenvalues as a non decreasing sequence, the number of nodal domains $\nu_n$ of the $n$-th eigenfunction satisfies $n\ge \nu_n$. Here, we provide a new interpretation for the Courant nodal deficiency $d_n = n-\nu_n$ in the case of quantum graphs. It equals the Morse index --- at a critical point --- of an energy functional on a suitably defined space of graph partitions. Thus, the nodal deficiency assumes a previously unknown and profound meaning --- it is the number of unstable directions in the vicinity of the critical point corresponding to the $n$-th eigenfunction. To demonstrate this connection, the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail.

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.

In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false. One is thus led to conclude that the description of reality as given by a wave function is not complete.

We consider the space of probabilities {P(x)}, where the x are coordinates of a configuration space. Under the action of the translation group, P(x) →P(x + θ), there is a natural metric over the parameters θ given by the Fisher-Rao metric. This metric induces a metric over the space of probabilities. Our next step is to set the probabilities in motion. To do this, we introduce a canonically conjugate field S and a symplectic structure; this gives us Hamiltonian equations of motion. We show that it is possible to extend the metric structure to the full space of the (P,S), and this leads in a natural way to introducing a Kähler structure; i.e., a geometry that includes compatible symplectic, metric and complex structures. The simplest geometry that describes these spaces of evolving probabilities has remarkable properties: the natural, canonical variables are precisely the wave functions of quantum mechanics; the Hamiltonian for the quantum free particle can be derived from a representation of the Galilean group using purely geometrical arguments; and it is straightforward to associate with this geometry a Hilbert space which turns out to be the Hilbert space of quantum mechanics. We are led in this way to a reconstruction of quantum theory based solely on the geometry of probabilities in motion.

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.

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).

Schrödinger propagation of initial discontinuities leads to divergence of moments

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.

Time-decay of scattering solutions and resolvent estimates for semiclassical Schrödinger operators

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

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.

The evolution of the wave function in quantum mechanics is deterministic like that of classical waves. Only when we bring in observers the fundamentally different quantum reality emerges. Similarly the introduction of observers changes the nature of spacetime by causing a split between past and future, concepts that are not well defined in the observer-free world. The induced temporal boundary leads to a resonance condition for the oscillatory vacuum solutions of the metric in Euclidean time. It corresponds to an exponential de Sitter evolution in real time, which can be represented by a cosmological constant , where r u is the radius of the particle horizon at the epoch when the observer exists. For the present epoch we get a value of Λ that agrees with the observed value within 2σ of the observational errors. This explanation resolves the cosmic coincidence problem. Our epoch in cosmic history does not herald the onset of an inflationary phase driven by some dark energy. We show that the observed accelerated expansion that is deduced from the redshifts is an ‘edge effect’ due to the observer-induced boundary and not representative of the intrinsic evolution. The new theory satisfies the BBN (Big Bang nucleosynthesis) and CMB (cosmic microwave background) observational constraints equally well as the concordance model of standard cosmology. There is no link between the dark energy and dark matter problems. Previous conclusions that dark matter is mainly non-baryonic are not affected.

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.

We can learn more about how our reality is made and what non-ordinary states of consciousness are by studying the Quantum Hologram Theory of Physics and Consciousness (QHTC). The QHTC says that consciousness is not local and that altered states of consciousness can help us understand how our minds work in more than one way. That's what Schrödinger thought. He thought that the quantum mechanical wave function was a field of consciousness. QHTC is based on holographic theories for human consciousness. These theories say that the brain works like a hologram and that it processes images into interference patterns that are then turned into virtual images, just like a laser hologram. These quantum waves can store a lot of information, which our brains use to make our three-dimensional world. This article says that this last theory should be the main framework for research on altered states of consciousness, and it talks about how to get data for analysis and how to get into an altered state for possible experiments.

Chapter 3 focused on Schrodinger’s argument that quantum mechanical wave functions (evolving always in accordance with Schrodinger’s own equation) cannot be understood as providing complete descriptions of the physical states of individual systems. In this chapter we focus on a second argument, due largely to Einstein, for the same conclusion – namely, the incompleteness of the quantum mechanical description of physical reality.