Masterclass in Operator Methods in Quantum Mechanics

Speaker: Dr. Horst R. Beyer.

Date and Time

October 3 – December 8, 2017.

Place

Sala de Usos Múltiples de la Facultad de Ciencias en Física y Matemáticas (UNACH),
Ciudad Universitaria Carretera Emiliano Zapata Km. 8, Real del Bosque (Terán).
Tuxtla Gutiérrez, Chiapas, México.

Sponsors

MCTP, CONACYT, UNACH y FCFM-UNACH.

Organizers

Horst R Beyer.
Sendic Estrada Jiménez.
Florencio Corona

Summary

In addition to abundant experimental evidence, quantum theory has a firm mathematical foundation, provided by the spectral theorems for (densely-defined, linear and) self-adjoint operators in Hilbert Spaces, called “observables” in quantum theory. In particular, these theorems provide the existence of spectral measures associated to self-adjoint operators and elements of the Hilbert space, the latter called “states” in quantum theory. Such spectral measures are the primary observables in quantum theory. One well-known example in one space dimension is the absolute square of the wave-function in quantum mechanics, which generates the spectral measure corresponding to the position operator and that wave function.

Unfortunately, these mathematical facts are not widely known neither inside the mathematics nor the physics community and deserve a lot more distribution, since providing an additional firm foundation of quantum theory, turning it into a “mathematical theory.”

This short course introduces into basic aspects of applications of operator theory to quantum mechanics. It follows to a large extent Martin Schechter's book “Operator Methods in Quantum Mechanics," Dover 2003. “This advanced undergraduate and graduate-level text introduces the power of operator theory as a tool in the study of quantum mechanics, assuming only a working knowledge of advanced calculus."

Contents
  1. A brief Sketch of the Mathematical Structure of Quantum Theory
  2. Introduction to Banach and Hilbert Spaces
  3. Representation of Formal Differential Operators
  4. Linear Operators in Banach Spaces
  5. Definition and Properties of the Fourier Transform
  6. Spectra of Operators
  7. The Essential Spectrum
  8. One-Dimensional Motion
  9. The Negative Eigenvalues
  10. Estimating the Spectrum
  11. Scattering Theory
  12. Long-Range Potentials
  13. Time-Independent Theory
  14. Completeness
  15. Strong Completeness
  16. Oscillating Potentials
  17. Eigenfunction Expansions
  18. Restricted Particles
  19. Hard-Core Potentials
  20. The Invariance Principle
  21. Trace Class Operators
UNACH/MCTP, Ciudad Universitaria
Carretera Emiliano Zapata Km. 4,
Real del Bosque (Terán).
Tuxtla Gutiérrez, Chiapas, México.
C. P. 29050
Teléfono 52 (961) 617-80-00
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mctp@unach.mx
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