Physical Chemistry Third Edition

700 16 The Principles of Quantum Mechanics. II. The Postulates of Quantum Mechanics

∫π

0

ysin^2 (y)dy

1

2

∫π

0

y[ 1 −cos(2y)]dy

1

2

∫π

0

ydy−

1

8

∫ 2 π

0

zcos(z)dz



π^2

4

−

1

8

∫ 2 π

0

zcos(z)dz

wherez 2 y. Integrating by parts,

∫ 2 π

0

zcos(z)dzzsin(z)|^20 π−

∫ 2 π

0

sin(z)dz 0 +cos(z)|^20 π 0

∫π

0

ysin^2 (y)dy

π^2

4

〈x〉

2

a

(

a

π

) 2

π^2

4



a

2

(16.4-10)

Exercise 16.6

Find〈x〉for a particle in a one-dimensional box of lengthafor then2 state. Look up the

integral if you want to do so. Comment on the comparison between the value of〈x〉forn 1

and the value forn2.

Probabilities and Probability Densities

Once we have an expectation value such as〈x〉in Example 16.12, we need to examine

whether this prediction of a mean corresponds to a set of equal results or whether

there can be a distribution of values. We need to discuss the probabilities of different

outcomes. Probabilities can be visualized by thinking of a set ofNdiscrete values of

some variable:w 1 ,w 2 ,w 3 ,...,wN. The mean value ofwis equal to

〈w〉

1

N

(w 1 +w 2 + ··· +wN)

1

N

∑N

i 1

wi (16.4-11)

where we introduce the notation for a sum. If some of the values are equal to other

values in the set, we place all of the distinct values at the beginning of the set and

number them from 1 toM. Let the number of the values in the entire set equal tow 1

beN 1 , the number equal tow 2 beN 2 , and so on up toNM. If all members of the set

are equally likely to be chosen the probability that a randomly chosen member of the

set is equal towiis

Probability ofwi

Ni

N

pi (16.4-12)

wherepiNi/N. The mean value can be simplified by grouping the terms together

that are equal to each other:

〈w〉

1

N

∑M

i 1

Niwi

∑M

w 1

piwi (16.4-13)