Lectures on Quantum Mechanics (nonlinear PDE point of view)

TL;DR

This paper provides a comprehensive mathematical and conceptual overview of quantum mechanics, emphasizing nonlinear PDE approaches to fundamental phenomena like quantum transitions and wave-particle duality.

Contribution

It introduces a novel mathematical framework for quantum phenomena using nonlinear hyperbolic PDEs, extending traditional linear models.

Findings

Detailed analysis of Schrödinger, Pauli, and Dirac equations

Discussion of nonlinear PDE approach to quantum phenomena

Connection between quantum transitions, wave-particle duality, and nonlinear dynamics

Abstract

We expose the Schr\"odinger quantum mechanics with traditional applications to Hydrogen atom. We discuss carefully the experimental and theoretical background for the introduction of the Schr\"odinger, 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.