Lectures on Quantum Mechanics for mathematicians

Abstract

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 results). However, for the Maxwell--Schrödinger equations this conjecture is still an {\it open problem}. We calculate the diffraction amplitude for the scattering of electron beams and Aharonov--Bohm shift via the Kirchhoff approximation.

arXiv comment46 pages, 5 figures. Drawn substantially from arXiv:1907.04297

References in corpus (6)

• Global attractor for a nonlinear oscillator coupled to the Klein-Gordon field
• Solitary waves for Maxwell-Schrodinger equations
• Global Existence and Uniqueness of Solutions to the Maxwell-Schr{ö}dinger Equations
• High-Velocity Estimates for the Scattering Operator and Aharonov-Bohm Effect in Three Dimensions
• Global wellposedness in the energy space for the Maxwell-Schrödinger system
• Attractors of nonlinear Hamiltonian PDEs