Chapter 1
A Short Course on Quantum Mechanics
and Methods of Quantization
Elisa Ercolessi
Department of Physics and Astronomy, Universit`a di Bologna and
INFN-Sezione di Bologna, via Irnerio 46, 40127, Bologna, Italy.
email: [email protected]The first part of this paper aims at introducing a mathematical oriented reader to the realm
of Quantum Mechanics (QM) and then at presenting the geometric structures that underline the
mathematical formalism of QM which, contrary to what is usually done in Classical Mechanics
(CM), are usually not taught in introductory courses. The mathematics 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 theoretical physics. We will first discuss the
so-called “Coherent State Approach” which, mainly complemented by “Feynman Path Integral
Technique”, is the methods 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.
1.1 Introduction
The XIX century was the apex of Classical Mechanics (CM). The newly born tools
of differential and integral calculus and newtheoretical generalprinciples (such as
variational ones) allowed to put on a rigorous basis what we now call Analytical
Mechanics and provided the framework to study all mechanical problems, from the
simple case of a single point particle, to planetary motion, to rigid bodies. Also,
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