Physical Chemistry Third Edition

16.4 Postulate 4 and Expectation Values 699

even if the system is in a known state. Information about this comes from part b of the

postulate, and we now explore this. There are two cases. For some systems, states, and

variables, it is possible to make a precise prediction of the outcome of a measurement

from knowledge of the wave function at the time of the measurement. We call this case

thepredictable case. For some states and some variables, the outcomes of individual

measurements will be distributed over various eigenvalues, even though the state is the

same before each measurement. We call this case thestatistical case. The existence of

the statistical case is one of the most striking differences between classical and quantum

mechanics. We first examine the statistical case, which is more interesting and more

complicated than the predictable case.

Position Measurements

A position measurement provides an important example of the statistical case. Consider

a particle that moves parallel to thexaxis. Assume that we make a set of position

measurements with the state of the system corresponding to the same wave function,

Ψ(x,t), at the time of each measurement. The outcome of any measurement must be

an eigenvalue of the position operator. The eigenfunction of the operatorxis theDirac

delta functionin Eq. (16.3-36). Any value ofxcan be an eigenvalue, denoted byain

Eq. (16.3-36). Any values ofxcan be an outcome of the position measurement.

The expectation value ofxis

〈x〉

∫

Ψ(x,t)∗xΨ(x,t)dx (16.4-7)

where we assume that the wave functionΨis normalized. Since the multiplication

operatorxcommutes with multiplication byΨ∗we can write

〈x〉

∫

xΨ(x,t)∗Ψ(x,t)dx

∫

x|Ψ(x,t)|^2 dx (16.4-8)

where we use the fact that any quantity times its complex conjugate is equal to the

square of the magnitude of the quantity (see Appendix B). If the wave function is a

product of an energy eigenfunction and a time factor, the time factor cancels against

its complex conjugate, as in Eq. (16.4-4):

〈x〉

∫

x|Ψ(x,t)|^2 dx

∫

x|ψ(x)|^2 dx (16.4-9)

EXAMPLE16.12

Find the expectation value for the position of a particle in a one-dimensional box of lengtha

forn1.

Solution

〈x〉

2

a

∫a

0

sin

(

πx

a

)

xsin

(

πx

a

)

dx

2

a

∫a

0

xsin^2

(

πx

a

)

dx

2

a

(

a

π

) 2 ∫π

0

ysin^2 (y)dy

whereyπx/a. We could look up the integral, but we will go through a little calculus for

practice. Using a trigonometric identity