Quantum Mechanics for Mathematicians (nonlinear PDEs point of view)
(May 2005)Abstract
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.
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