Physical Chemistry Third Edition

16.4 Postulate 4 and Expectation Values 705

For motion of one particle in three dimensions and for sufficiently small∆x∆y∆z

(

Probability that the particle

is in∆x∆y∆zat timet′

)

≈|Ψ(x′,y′,z′,t′)|^2 ∆x∆y∆z (16.4-25)

For a region of appreciable size, the integration in Eq. (16.4-21) must be carried out.

For a stationary state, the time-dependent wave functionΨin Eq. (16.4-24) or

Eq. (16.4-25) can be replaced by the coordinate wave functionψ.

EXAMPLE16.15

For thenx1,ny1,nz1 state of a particle in a three-dimensional box, find the prob-

ability that the particle is in a small rectangular region at the center of the box such that the

length of the region in each direction is equal to 1.000% of the length of the box in that

direction. Compare your probability with the fraction of the volume of the box represented

by your region.

Solution

The dimensions of the box areabybbyc.

Probability≈ψ∗ψ∆x∆y∆z

(

8

abc

)

sin^2 (πx/a) sin^2 (πy/b) sin^2 (πz/c)∆x∆y∆z

≈

(

8

abc

)

sin^6 (π/2)∆x∆y∆z

≈

(

8

abc

)

(1.000)(0. 01000 a)(0. 01000 b)(0. 01000 c)

≈8(0.01000)^3  8. 000 × 10 −^6

This value is 8 times as large as it would be if the probability were uniform within the box.

Exercise 16.8

Repeat the calculation of the previous example for thenx2,ny2,nz2 state. Explain

your result.

Distinguishing the Predictable Case from

the Statistical Case

A common measure of the “spread” or “width” of a probability distribution is the

standard deviation, which is defined for the measurement of a variableAby

σA

(

〈A^2 〉−〈A〉^2

) 1 / 2

(

definition of the

standard deviation

)

(16.4-26)

The square of the standard deviation is called thevariance. Equation (16.4-26) can also

be written

σA〈(A−〈A〉)^2 〉^1 /^2 (16.4-27)

Exercise 16.9

Show that Eq. (16.4-26) and Eq. (16.4-27) are equivalent.