CHAPTER 2 The State of a Physical System

Franck Laloë

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.

Andrzej Posiewnik

John M. Brown

This chapter discusses the physical properties of atoms and molecules, which are quantized. This means that they can only take certain discrete values rather than a full range of continuously varying values. The chapter examines the quantized energy levels of a molecule. These are called stationary states, which describe an energy that is independent of time. The chapter also outlines the procedure for constructing a quantum mechanical Hamiltonian, which includes writing down the complete and classical expression for the energy of the system. The chapter reviews the general approach to the determination of energy levels of a physical system. This involves two stages: First, the appropriate Hamiltonian is defined and, in a second step, the eigenfunctions and eigenvalues of the Hamiltonian are determined. The chapter also discusses the Schrödinger equation, which can only be solved for one-body systems.

F. S. Levin

Chapter 1 was concerned with the need for a quantal description of microscopic phenomena. The thrust of Chapter 3 is an analysis and comparison of a classical and a quantum system; some mathematical generalizations of the analysis are presented as well. Chapter 4 provides an introduction to Hilbert-space concepts that are the mathematical framework of quantum theory, while, in Chapter 5, the axioms of quantum mechanics for pure states are stated and illustrated. The present chapter is intended to be a bridge between those chapters and Chapter 1. Here we review many of the concepts of classical mechanics that have relevance in quantum mechanics. Equally important, if not more so, we stress the lack of equivalence between a number of classical concepts and their quantal counterparts. This inequivalence is emphasized because of the vocabulary common to classical and quantum mechanics. Classical physics is, of course, the source for much of the vocabulary in quantum physics. However, the ways in which the words are used in the two domains often differ substantially, while the concepts are almost necessarily not identical. Some classical concepts have no direct quantal counterparts. For example, velocity and force are not primary quantal entities. This fact means that the quantal definitions of linear and angular momentum cannot be the same as those in classical mechanics. Similarly, the concept of a “state” in classical physics – the collection of the generalized positions and generalized velocities or momenta – does not carry over to quantum physics.

Krystyna Bugajska, Slawomir Bugajski