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Lecture Note
25/2005
Lectures on Quantum Mechanics (nonlinear PDEs point of view)
Alexander Komech
Abstract
We expose the Schrödinger quantum mechanics with traditional applications to Hydrogen atom: the calculation of the Hydrogen atom spectrum via Schrödinger, Pauli and Dirac equations, the Heisenberg representation, the selection rules, the calculation of quantum and classical scattering of light (Thomson cross section), photoeffect (Sommerfeld cross section), quantum and classical scattering of electrons (Rutherford cross section), normal and anomalous Zeemann effect (Landé factor), polarization and dispersion (Kramers-Kronig formula), diamagnetic susceptibility (Langevin formula).
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: the introduction of the quantum stationary states, charge density and electric current density, quantum magnetic moment, electron spin and spin-orbital coupling in "vector model" and in the Russel-Saunders approximation, differential cross section of scattering, the Lorentz theory of polarization and magnetization, the Einstein special relativity and covariance of the Maxwell Electrodynamics.
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.
Namely, we suggest that i) the quantum stationary states form a global attractor of the coupled Maxwell-Schrödinger or Maxwell-Dirac equations, in the presence of an external confining potential, and ii) the wave-particle duality corresponds to the soliton-like asymptotics for the solutions of the translation-invariant coupled equations without an external potential.
We emphasize, in the whole of our exposition, that the coupled equations are nonlinear, and just this nonlinearity lies behind all traditional perturbative calculations that is known as the Born approximation. We suggest that both fundamental quantum phenomena could be described by this nonlinear coupling. The suggestion is confirmed by recent results on the global attractors and soliton asymptotics for model nonlinear hyperbolic PDEs.
