This chapter gives a summary of the mathematical formalism used in quantum mechanics, with a short bibliography put directly at its end. It should be seen as a complement, to be used by readers who wish to know more than what has been recalled about the mathematical tools in the other chapters. Some results are given without explicit proofs; they can be found for instance in Chapters II and IV of [1]. Many quantum mechanics textbooks introduce its general formalism in a more complete way, for instance Chapter VII of [2], Chapter 3 of [3], or Chapter 2 of [4].

We first summarize the general formalism of quantum mechanics for any physical system (§11.1), with the Dirac notation; we then study how this formalism treats the grouping of several physical systems into one single quantum system (§11.2); finally, we study a few simple special cases (§11.3), for instance a single particle in an external potential, with or without spin; the reader who prefers wave functions to more abstract reasonings in spaces of states may begin with this section.

General physical system

The general formalism of quantum mechanics applies to all physical systems, whether they contain a single particle, many particles of various sorts, one or several fields, etc.

Quantum space of states

The physical state of a system at each time is defined in quantum mechanics by a state vector which, in Dirac notation, is written ∣ψ〈 – or ∣ψ(t)〈 if one wishes to make the time dependence explicit.