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The electronic Schrödinger equation describes the motion of N-electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wave functions, depend on 3N variables, with three spatial dimensions for each electron. Approximating these solutions is thus inordinately challenging, and it is generally believed that a reduction to simplified models, such as those of the Hartree-Fock method or density functional theory, is the only tenable approach. This book seeks to show readers that this conventional wisdom need not be ironclad: the regularity of the solutions, which increases with the number of electrons, the decay behavior of their mixed derivatives, and the antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximated with an order of complexity which comes arbitrarily close to that for a system of one or two electrons. The text is accessible to a mathematical audience at the beginning graduate level as well as to physicists and theoretical chemists with a comparable mathematical background and requires no deeper knowledge of the theory of partial differential equations, functional analysis, or quantum theory.

Electron configuration --- Wave functions --- Schrèodinger equation --- Approximation theory --- Mathematics --- Physics --- Physical Sciences & Mathematics --- Atomic Physics --- Calculus --- Electron configuration. --- Wave functions. --- Wave function --- Configuration, Electron --- Electron correlation --- Mathematics. --- Approximation theory. --- Partial differential equations. --- Numerical analysis. --- Partial Differential Equations. --- Approximations and Expansions. --- Numerical Analysis. --- Mathematical analysis --- Partial differential equations --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Math --- Science --- Wave mechanics --- Configuration space --- Atomic orbitals --- Electrons --- Differential equations, partial.

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At extremely low temperatures, clouds of bosonic atoms form what is known as a Bose-Einstein condensate. Recently, it has become clear that many different types of condensates  -- so called fragmented condensates -- exist. In order to tell whether fragmentation occurs or not, it is necessary to solve the full many-body Schrödinger equation, a task that remained elusive for experimentally relevant conditions for many years. In this thesis the first numerically exact solutions of the time-dependent many-body Schrödinger equation for a bosonic Josephson junction are provided and compared to the approximate Gross-Pitaevskii and Bose-Hubbard theories. It is thereby shown that the dynamics of  Bose-Einstein condensates is far more intricate than one would anticipate based on these approximations. A special conceptual innovation in this thesis are optimal lattice models. It is shown how all quantum lattice models of condensed matter physics that are based on Wannier functions, e.g. the  Bose/Fermi Hubbard model, can be optimized variationally. This leads to exciting new physics.

Bose-Einstein condensation. --- Many-body problem. --- Schrödinger equation. --- Bose-Einstein condensation --- Many-body problem --- Schrèodinger equation --- Physics --- Physical Sciences & Mathematics --- Atomic Physics --- Bose condensed fluids --- Bose condensed liquids --- Bose fluids --- Bose liquids --- Einstein condensation --- Physics. --- Phase transformations (Statistical physics). --- Condensed materials. --- Condensed matter. --- Superconductivity. --- Superconductors. --- Quantum Gases and Condensates. --- Theoretical, Mathematical and Computational Physics. --- Strongly Correlated Systems, Superconductivity. --- Bosons --- Condensation --- Superfluidity --- Mathematical physics. --- Superconducting materials --- Superconductive devices --- Cryoelectronics --- Electronics --- Solid state electronics --- Electric conductivity --- Critical currents --- Physical mathematics --- Condensed materials --- Condensed media --- Condensed phase --- Materials, Condensed --- Media, Condensed --- Phase, Condensed --- Liquids --- Matter --- Solids --- Phase changes (Statistical physics) --- Phase transitions (Statistical physics) --- Phase rule and equilibrium --- Statistical physics --- Materials --- Mathematics

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Quantum mechanics, discovered by Werner Heisenberg and Erwin Schrödinger in 1925-1926, is famous for its radical implications for our conception of physics and for our view of human knowledge in general. While these implications have been seen as scientifically productive and intellectually liberating to some, Niels Bohr and Heisenberg, among them, they have been troublesome to many others, including Schrödinger and, most famously, Albert Einstein. The situation led to the intense debate that started in the wake of its discovery and has continued into our own time, with no end appearing to be in sight. Epistemology and Probability aims to contribute to our understanding of quantum mechanics and of the reasons for its extraordinary impact by reconsidering, under the rubric of "nonclassical epistemology," the nature of epistemology and probability, and their relationships in quantum theory. The book brings together the thought of the three figures most responsible for the rise of quantum mechanics—Heisenberg and Schrödinger, on the physical side, and Bohr, on the philosophical side—in order to develop a deeper sense of the physical, mathematical, and philosophical workings of quantum-theoretical thinking. Reciprocally, giving a special emphasis on probability and specifically to the Bayesian concept of probability allows the book to gain new insights into the thought of these figures. The book reconsiders, from this perspective, the Bohr-Einstein debate on the epistemology of quantum physics and, in particular, offers a new treatment of the famous experiment of Einstein, Podolsky, and Rosen (EPR), and of the Bohr-Einstein exchange concerning the subject. It also addresses the relevant aspects of quantum information theory and considers the implications of its epistemological argument for higher-level quantum theories, such as quantum field theory and string and brane theories. One of the main contributions of the book is its analysis of the role of mathematics in quantum theory and in the thinking of Bohr, Heisenberg, and Schrödinger, in particular an examination of the new (vis-à-vis classical physics and relativity) type of the relationships between mathematics and physics introduced by Heisenberg in the course of his discovery of quantum mechanics. Although Epistemology and Probability is aimed at physicists, philosophers and historians of science, and graduate and advanced undergraduate students in these fields, it is also written with a broader audience in mind and is accessible to readers unfamiliar with the higher-level mathematics used in quantum theory.

Schro ̈dinger equation. --- Schro ̈dinger, Erwin,. --- Quantum theory --- Physics --- Knowledge, Theory of --- Complementarity (Physics) --- Wave-particle duality --- Causality (Physics) --- Heisenberg uncertainty principle --- Schrèodinger equation --- Physical Sciences & Mathematics --- Atomic Physics --- History --- Mathematics --- Philosophy --- Knowledge, Theory of. --- Wave-particle duality. --- Heisenberg uncertainty principle. --- Schrd̲inger equation. --- History. --- Mathematics. --- Philosophy. --- Bohr, Niels, --- Schrödinger, Erwin, --- Heisenberg, Werner, --- Indeterminancy principle --- Uncertainty principle --- Causality --- Dualism, Wave-particle --- Duality principle (Physics) --- Wave-corpuscle duality --- Epistemology --- Theory of knowledge --- Geĭzenberg, V. --- Heisenberg, W. --- Heisenberg, Werner --- Schredinger, Ervin, --- Schrödinger, E. --- Boer, Niersi, --- Boerh, Niersi, --- Bohr, N. --- Bohr, Niels Henrik David, --- Bor, Nil's, --- Physics. --- Epistemology. --- Philosophy and science. --- Probabilities. --- Quantum physics. --- Elementary particles (Physics). --- Quantum field theory. --- Elementary Particles, Quantum Field Theory. --- Probability Theory and Stochastic Processes. --- Quantum Physics. --- Philosophy of Science. --- Nuclear physics --- Electromagnetic waves --- Matter --- Radiation --- Wave mechanics --- Psychology --- Constitution --- Schroedinger, Erwin, --- Schrdinger equation. --- Schrodinger, Erwin,