Quantum Mechanics

Abstract

For historical reasons, we present here the first quantum theories as they were understood by Heisenberg, Schrödinger, von Neumann and Dirac, among others. These quantization methods are essentially restricted to systems without interactions/symmetries. In this book, we favor the Lagrangian functional integral approach, because it allows a very general treatment of theories with symmetries. A hike through the Hamiltonian methods is however useful to better understand the physicists’ intuitions with quantum fields. The chapter starts with a description of the principles of quantum mechanics in von Neumann’s approach. We then describe the canonical quantization picture following Heisenberg-Schrödinger. We give a presentation of the algebraic setting underlying canonical quantization of the free classical and fermionic particles. We then explain the relation between Weyl quantization and pseudo-differential calculus. We describe the quantization of the harmonic operator, and finish by the canonical quantization of the free scalar field.