We expose the Schrödinger quantum mechanics with traditional applications to Hydrogen atom. We discuss carefully the experimental and theoretical background for the introduction of the Schrödinger, Pauli and Dirac equations, as well as for the Maxwell equations. We explain in detail all basic theoretical concepts. We explain all details of the calculations and mathematical tools: Lagrangian and Hamiltonian formalism for the systems with finite degree of freedom and for fields, Geometric Optics, the Hamilton-Jacobi equation and WKB approximation, Noether theory of invariants including the theorem on currents, four conservation laws (energy, momentum, angular momentum and charge), Lie algebra of angular momentum and spherical functions, scattering theory (limiting amplitude principle and limiting absorption principle), the Lienard-Wiechert formulas, Lorentz group and Lorentz formulas, Pauli theorem and relativistic covariance of the Dirac equation, etc. We give a detailed oveview of the conceptual development of the quantum mechanics, and expose main achievements of the ``old quantum mechanics'' in the form of exercises.
One of our basic aim in writing this book, is an open and concrete discussion of the problem of a mathematical description of the following two fundamental quantum phenomena: i) Bohr's quantum transitions and ii) de Broglie's wave-particle duality. Both phenomena cannot be described by autonomous linear dynamical equations, and we give them a new mathematical treatment related with recent progress in the theory of global attractors of nonlinear hyperbolic PDEs.
The main goal of these lectures -- introduction to Quantum Mechanics for mathematically-minded readers. The second goal is to discuss the mathematical interpretation of the main quantum postulates: transitions between quantum stationary orbits, wave-particle duality and probabilistic interpretation. We suggest a dynamical interpretation of these phenomena based on the new conjectures on attractors of nonlinear Hamiltonian partial differential equations. This conjecture is confirmed for a list of {\it model Hamiltonian nonlinear} PDEs by the results obtained since 1990 (we survey sketchy these …
These notes have been prepared for a series of lectures given at the Sarajevo Stochastic Analysis Winter School, from January 28 to February 1, 2019. There already exist several excellent lecture notes and reviews on the subject, such as (Hairer 2009) on (non-singular) stochastic PDEs, and (Hairer 2015, Chandra and Weber 2017) on singular stochastic PDEs and regularity structures. The present notes have two main specificities. The first one is that they focus on a particular example, the Allen-Cahn equation, which allows to introduce several of the difficulties of the theory in a gradual way, …
爱因斯坦的布朗运动的动力学理论，基于轻的水分子不断轰击重的花粉，提供了一个从牛顿力学的扩散解释。自量子力学发现以来，验证薛定谔方程中扩散的出现一直是一个挑战。该程序的第一步是验证线性玻尔兹曼方程作为具有随机势的薛定谔方程的某一标度极限。在第二步中，我们考虑一个更长的时间尺度，对应于无限多个玻尔兹曼碰撞。直觉是，玻尔兹曼方程然后收敛到一个扩散方程类似的中心极限定理马尔可夫过程充分混合。在这些讲义（为2010年8月的Les Houches暑期学校准备的）中，我们提出了数学工具来严格证明这种直觉。新材料依赖于与H. T. Yau和M.萨姆霍夫
This script is based on the notes the author prepared to give a set of six lectures at the Les Houches School "Integrability in Atomic and Condensed Matter Physics" in the summer of 2018. The school had its focus on the application of integrability based methods to problems in non-equilibrium statistical mechanics. The lectures were meant to complement this subject with background material on the equilibrium statistical mechanics of quantum spin chains from a vertex model perspective. The author was asked to provide a minimal introduction to quantum spin systems including notions like the redu…
We examine the duality relating the equilibrium dynamics of the mean-field $p$-spin ferromagnets at finite size in the Guerra's interpolation scheme and the Burgers hierarchy. In particular, we prove that - for fixed $p$ - the expectation value of the order parameter on the first side w.r.t. the generalized partition function satisfies the $p-1$-th element in the aforementioned class of nonlinear equations. In the light of this duality, we interpret the phase transitions in the thermodynamic limit of the statistical mechanics model with the development of shock waves in the PDE side. We also o…
We review very briefly the main mathematical structures and results in some important areas of Quantum Mechanics involving PDEs and formulate open problems.
These are lecture notes for a short winter course at the Department of Mathematics, University of Coimbra, Portugal, December 6--8, 2018. The course was part of the 13th International Young Researchers Workshop on Geometry, Mechanics and Control. In three lectures I trace the work of three heroes of mathematics and mechanics: Euler, Poincaré, and Arnold. This leads up to the aim of the lectures: to explain Arnold's discovery from 1966 that solutions to Euler's equations for the motion of an incompressible fluid correspond to geodesics on the infinite-dimensional Riemannian manifold of volume p…
The classical Kubo-Martin-Schwinger (KMS) condition is a fundamental property of statistical mechanics characterizing the equilibrium of infinite classical mechanical systems. It was introduced in the seventies by G. Gallavotti and E. Verboven as an alternative to the Dobrushin-Lanford-Ruelle (DLR) equation. In this article, we consider this concept in the framework of nonlinear Hamiltonian PDEs and discuss its relevance. In particular, we prove that Gibbs measures are the unique KMS equilibrium states for such systems. Our proof is based on Malliavin calculus and Gross-Sobolev spaces. The mai…
These are lecture notes for a master-level course given at KTH, Stockholm, in the spring of 2017, with the primary aim of proving the stability of matter from first principles using modern mathematical methods in many-body quantum mechanics. General quantitative formulations of the uncertainty and the exclusion principles of quantum mechanics are introduced, such as the Hardy, Sobolev and Poincaré functional inequalities as well as the powerful Lieb-Thirring inequality that combines these two principles. The notes are aimed to be both fairly self-contained and at the same time complementary to…
Partial differential equations (PDEs) are at the heart of many mathematical and scientific advances. While great progress has been made on the theory of PDEs of standard types during the last eight decades, the analysis of nonlinear PDEs of mixed type is still in its infancy. The aim of this expository paper is to show, through several longstanding fundamental problems in fluid mechanics, differential geometry, and other areas, that many nonlinear PDEs arising in these areas are no longer of standard types, but lie at the boundaries of the classification of PDEs or, indeed, go beyond the class…
These lecture notes cover a brief introduction into some of the algebro-geometric techniques used in the construction of BPS algebras. The first section introduces the derived category of coherent sheaves as a useful model of branes in toric Calabi-Yau three-folds. This model allows a rather simple derivation of quiver quantum mechanics describing low-energy dynamics of various brane systems. Vacua of such quantum mechanics can be identified with the critical equivariant cohomology of the moduli space of quiver representations. These are often counted by various crystal configurations. Using c…
These are lecture notes for the course "Poisson geometry and deformation quantization" given by the author during the fall semester 2020 at the University of Zurich. The first chapter is an introduction to differential geometry, where we cover manifolds, tensor fields, integration on manifolds, Stokes' theorem, de Rham's theorem and Frobenius' theorem. The second chapter covers the most important notions of symplectic geometry such as Lagrangian submanifolds, Weinstein's tubular neighborhood theorem, Hamiltonian mechanics, moment maps and symplectic reduction. The third chapter gives an introd…
