From classical mechanics to quantum mechanics

These are the notes for five one hour lectures delivered at the 2016 CRM summer school in Spectral Theory at Laval University. They were intended to provide some general background on classical and quantum mechanics to advanced undergraduate and beginning graduate students in mathematics. The first four lectures contain an outline of these two theories from a mathematical point view and a comparison of the classical and quantum descriptions of some simple systems. The last lecture is about hidden variables and is meant to illustrate the essential strangeness of the quantum description of nature. I have tried to make these notes more accessible by including an outline of some basic ideas in measure theory and operator theory. A similar outline of the basics of manifolds, tangent and cotangent spaces and differential forms would have been useful for understanding the sections on Lagrangian submanifolds and Hamilton-Jacobi equations. Unfortunately I did not have time to include this. I am grateful to Cyrus Bhiladvala and the referee for making suggestions for improvement.