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.

This school introduces operator-theoretic methods that are central to a rigorous treatment of quantum mechanics. It is designed for students who want a careful bridge from finite-dimensional linear algebra to the infinite-dimensional Hilbert space setting used in modern quantum theory.

Instructor:
Onirbam Islam, University of Potsdam

Organizers:
Prof. Nabil Iqban, University of Durham and University of Amsterdam
Sayeba Tashnuba Jahan, BRAC University
Onirbam Islam, University of Potsdam
Ahmed Rakin Kamal, Masaryk University and  BRAC University

The programme of the course can be found below.