School on Mathematics for Physics 1: Operators on Hilbert Spaces

Instructor

Abstract

This short course is intended to offer a gentle introduction to unbounded operators on Hilbert spaces pertinent to quantum mechanics. Our starting point will be to point out the differences between operators on finite-dimensional and infinite-dimensional Hilbert spaces, and to understand why the latter is essential for formulating quantum mechanics in a mathematically rigorous fashion. We will begin with the basics of unbounded operators on Hilbert spaces and rigged Hilbert spaces in order to explain the standard Heisenberg matrix mechanics and Dirac bra-ket formalisms. Our aim is to comprehend the quantum mechanical simple harmonic oscillator and its spectrum from an operator-theoretic viewpoint. A rudimentary background in linear algebra is assumed, although a few topics will be recalled in due course in order to motivate the analysis in an infinite-dimensional Hilbert space setting.

Course Outline and Schedule