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Simultaneous observables

Efstratios Manousakis

Published Nov 1, 2015• 0 citations
Publisher

Abstract

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.
DOI
10.1093/acprof:oso/9780198749349.003.0006

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This chapter explains that a central feature of quantum theory is its representation of observables by ‘operators’, which act on the wavefunction and extract the information it contains. It shows how operators are constructed and used. Observables are represented by hermitian operators, which have real eigenvalues and orthogonal eigenfunctions. When the system is not described by a single eigenfunction of an operator, it may be expressed as a superposition of such eigenfunctions. One consequence of the use of operators is the ‘uncertainty principle’, one of the most profound departures of quantum mechanics from classical mechanics. The uncertainty principle restricts the precision with which complementary observables may be specified and measured simultaneously.
Abstract Heisenberg’s uncertainty principle states that: The product of the uncertainties in the simultaneous specifications of two °complementary observables can never be less than a small quantity of the order of h. The precise form of the principle is given in Box U.1. It implies that as one observable is constrained to have a more precise value, so its partner becomes more ill-defined. An experiment designed to measure the two observables simultaneously will be able to determine one precisely only at the expense of losing information about the other (see °duality). The principle was established by Werner Heisenberg in 1927, as a consequence of his formulation of °matrix mechanics.
In view of the Heisenberg uncertainty principle, introduced in Chapter 1 , the application of Postulate 3 to a dynamical system faces one problem never encountered in classical mechanics, namely to define the conditions which make possible the simultaneous measurement of different observables. Because each physical observable takes definite values in the eigenstates of its associated operator, the simultaneous measurements of different observable will only be possible if their associated operators have a common eigenbasis. If such a complete orthonormal set of vectors ψ i are simultaneous eigenfunctions of two operators  and B̂: 4.1  ψ i = λ i ψ i , B ^ ψ i = µ i ψ i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn4_1_B.tif"/> we have for each basis vector:  B ^ ψ i =  µ i ψ i = λ i µ i ψ i = µ i λ i ψ i = B ^  ψ i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn4_2_B.tif"/> It follows that for any state vector Ψ, which can be expanded in the form ( 3.55) , we have: 4.2 (  B ^ − B ^  ) Ψ = 0 or  B ^ − B ^  = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn4_3_B.tif"/> This means, in view of Eq.(2.80) , that simultaneous measurements on a dynamical system are restricted to those observables for which the associated operators commute.
Given two or more non-commuting observables, it is generally not possible to simultaneously assign precise values to each. This quantum mechanical uncertainty principle is widely understood to be encapsulated by some form of uncertainty relation, expressing a trade-off between the standard deviations or other measures of uncertainty of two (or more) observables, resulting from their non-commutativity. Typically, such relations are coarse, and miss important features. It was not until very recently that a broader perspective on quantum uncertainty was envisaged and explored, one that utilises the notion of an uncertainty region. Here we review this new approach, illustrating it with pairs or triples of observables in the case of qubit and qutrit systems. We recall some of the shortcomings of traditional uncertainty relations, and highlight their inability to identify the full uncertainty region. These shortcomings suggest a precautionary note that, we surmise, ought to accompany the presentation of the uncertainty principle in introductory quantum mechanics courses.
In this chapter we shall mainly present the basic formalism that was initially developed by Heisenberg, also known as matrix mechanics (see Subsec. 1.5.7). We will first introduce in Sec. 2.1 the concept of quantum observables. Then, the problem of discrete and continuous spectra will be discussed and the basic non-commutability of quantum-mechanical observables will be deduced. While in Sec. 2.1 we discuss observables on a general formal level, in Sec. 2.2 some basic quantum-mechanical observables will be defined, and then different representations discussed and commutation relations derived. In Sec. 2.3 a basic uncertainty relation is derived. In the same section the relationship between uncertainty, superposition, and complementarity will be discussed. Finally, in Sec. 2.4 complete subsets of commuting observables will be shown to be Boolean subalgebras pertaining to a quantum algebra which is not Boolean. BASIC FEATURES OF QUANTUM OBSERVABLES This section is devoted to a general and formal exposition of quantum observables. In Subsec. 2.1.1 we shall learn how one can mathematically represent quantum observables as Hermitian operators. In Subsec. 2.1.2 we shall see how to change a basis, while in Subsec. 2.1.3 we shall find the relationship between eigenvalues of the observables and probabilities and learn how to calculate mean values. In Subsec. 2.1.4 we shall deal with an operator diagonalization. Finally, in Subsec. 2.1.5, the basic non-commutability of quantum observables will be presented by means of an example.