Physical Chemistry Third Edition

16.6 Postulate 5. Measurements and the Determination of the State of a System 719

Exercise 16.15

Show that the linear combination in Eq. (16.6-1) is an eigenfunction of the operatorÂfor any

set of values ofc 1 ,c 2 ,c 3 ,...,cgi, so long as all of the functions in the linear combination have

the same eigenvalue of̂A.

If there are other variables whose operators commute withÂ, measurement of

enough of these variables can put the system into a known state. We say that such a set

of variables is acomplete set of commuting observables. For example, assume that the

operatorsÂandB̂commute and that they have a set of common eigenfunctionsf 11 ,

f 12 ,...,f 21 ,f 22 ,...such that

Af̂ ijaifij (16.6-2)

̂Bfijbjjij (16.6-3)

Assume that a measurement ofAgives the resultak. The wave function immediately

after this measurement is the same as in Eq. (16.6-1):

ψ′

∑gk

j 1

cj′fkj (16.6-4)

where the sum now includes only those eigenfunctions of̂Bthat are eigenfunctions of

̂Acorresponding to the eigenvalueak. Assume that a measurement ofBgives the result

bm. The state after this measurement corresponds to the wave function

ψ′′fkm (16.6-5)

If the variablesAandBconstitute a complete set of commuting observables the system

is now in the state corresponding to the wave functionfkm. If there are other commuting

observables in the complete set, additional measurements must be made. We will see

an important example of this principle in Chapter 17, when we will find that four

variables constitute a complete set of commuting observables for the electronic motion

of a hydrogen atom.

Information about the State Prior to a Measurement

A single measurement gives us information about the state after the measurement but

says very little about the state prior to the measurement. Some information about the

original wave function of a quantum mechanical system can be obtained by repeated

measurements if we have some way to put the system back into the original state before

each measurement or if we have a supply of identical systems, all of which are in the

same state.

Consider the nondegenerate case, that each eigenfunction of the operatorÂhas a

distinct eigenvalue. Since the set of eigenfunctions is assumed to be a complete set, the

wave function immediately prior to a measurement ofAcan be represented as a linear

combination of eigenfunctions ofÂ, as in Eq. (16.3-34):

ψ(prior)

∑∞

j 1

cj(prior)fj (16.6-6)