Chapter 3 Observables and Uncertainty

Scott Corry

Efstratios Manousakis

Abstract This chapter discusses when two observables, which are represented by operators, –can be simultaneously observable. The chapter discusses the uncertainty principle in the general case and when two operators can share eigenstates. It also discuss the consequences of symmetry.

N. J. B. Green

This chapter discusses some of the fundamental principles in quantum mechanics. Towards the end of the nineteenth century, it had been thought that all the fundamental laws of nature were well-understood. However, the next 20 years saw the development of both quantum mechanics and relativity. The chapter then looks at the representation of a particle by a wavefunction. It also considers operators, observables, eigenfunctions, and eigenvalues, as well as commutation and uncertainty. Since it is not possible for a wavefunction to have characteristic values of two observables with non-commuting operators, it is not possible to specify exact simultaneous values for these observables. It should be noted that the uncertainty principle is rather deeper than the usual physical explanation might imply, it is actually inherent in the description of matter in terms of wavefunctions. Finally, the chapter explains expansion in eigenfunctions; the variational principle; and transition probabilities.

Günter Ludyk

Gary E. Bowman

Commutators and uncertainty relations have sparked much debate and confusion as to their physical meaning. As such, this chapter focuses partly on historical and interpretational aspects. Because it is central to the uncertainty relations, the commutator — including its manifestation in matrix mechanics — is introduced first. It is emphasized that the commutator of two operators is generally another operator. The general form of the uncertainty relations is then introduced. These relations reflect the fact that probability distributions for different observables are generally not independent. Moreover, application of the uncertainty relations generally requires knowledge of the quantum state — a fact that is obscured by the well known position-momentum uncertainty relation. Various proposed physical interpretations of the uncertainty relations are then discussed. The chapter concludes with a brief reflection on the significance of the uncertainty relations.