These notes collect the lectures given by the author to the “XXIII International Work- shop on Geometry and Physics” held in Granada (Spain) in September 2014. The first part of this paper aims at introducing a mathematical oriented reader to the realm of Quantum Mechanics (QM) and then to present the geometric structures that under- line the mathematical formalism of QM which, contrary to what is usually done in Classical Mechanics (CM), are usually not taught in introductory courses. The mathe- matics related to Hilbert spaces and Differential Geometry are assumed to be known by the reader. In the second part, we concentrate on some quantization procedures, that are founded on the geometric structures of QM — as we have described them in the first part — and represent the ones that are more operatively used in modern theoret- ical physics. We will discuss first the so-called Coherent State Approach which, mainly complemented by “Feynman Path Integral Technique”, is the method which is most widely used in quantum field theory. Finally, we will describe the “Weyl Quantization Approach” which is at the origin of modern tomographic techniques, originally used in optics and now in quantum information theory.
Ercolessi, E. (2015). A short course on quantum mechanics and methods of quantization. World Scientific Publishing Co. Pte Ltd [10.1142/S0219887815600087].
A short course on quantum mechanics and methods of quantization
ERCOLESSI, ELISA
2015
Abstract
These notes collect the lectures given by the author to the “XXIII International Work- shop on Geometry and Physics” held in Granada (Spain) in September 2014. The first part of this paper aims at introducing a mathematical oriented reader to the realm of Quantum Mechanics (QM) and then to present the geometric structures that under- line the mathematical formalism of QM which, contrary to what is usually done in Classical Mechanics (CM), are usually not taught in introductory courses. The mathe- matics related to Hilbert spaces and Differential Geometry are assumed to be known by the reader. In the second part, we concentrate on some quantization procedures, that are founded on the geometric structures of QM — as we have described them in the first part — and represent the ones that are more operatively used in modern theoret- ical physics. We will discuss first the so-called Coherent State Approach which, mainly complemented by “Feynman Path Integral Technique”, is the method which is most widely used in quantum field theory. Finally, we will describe the “Weyl Quantization Approach” which is at the origin of modern tomographic techniques, originally used in optics and now in quantum information theory.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
