Physical Chemistry Third Edition

702 16 The Principles of Quantum Mechanics. II. The Postulates of Quantum Mechanics

(^01) x/a
(a)
x/a
Probability
01
(b)
Probability
Figure 16.3 The Probability Density for Positions of a Particle in a One-Dimensional
Box according to Classical Mechanics.(a) The instantaneous probability. (b) The prob-
ability averaged over a long time.
speed, the classical probability density of a moving particle averaged over a long time
is spread over the entire box and is uniform, as shown in Figure 16.3b.
The quantum mechanical probability density for finding a particle in a box is
distributed over the entire box but is not uniform. There are points at which the prob-
ability density vanishes. However, if we consider a very large value ofn, these points
become closer and closer together, as schematically shown in Figure 16.4, which is
drawn forn10. Asnis larger and larger the width of the oscillations in the curve
becomes smaller and smaller until it is smaller than any experimental inaccuracy, and
the probability density resembles more nearly that of the long-time average classi-
cal probability distribution. This behavior conforms to thecorrespondence principle,
which states that for sufficiently large energies and masses, the behavior predicted by
quantum mechanics approaches the behavior predicted by classical mechanics. In this
case, the classical behavior that is approached is a time-average behavior.
To obtain the probability that a particle is to be found in a finite region, we integrate
|Ψ|^2 over that region. For motion parallel to thexaxis the probability thatxlies between
bandcis
(Probability thatb<x<c)
∫c
b
|Ψ(x,t)|^2 dx (16.4-21)
The total probability of all positions is equal to the integral of the square of the mag-
nitude of the wave function over all values of the coordinates. This probability equals
unity if the wave function is normalized.
0.0
0
1
Wave function squared
3
a
2
3
0.2 0.4 0.6
x/a
0.8 1. 0
Figure 16.4 The Probability Density
for Positions of a Particle in a One-
Dimensional Box forn= 10.

EXAMPLE16.13

For a particle in a one-dimensional box in the stationary state withn2, find the probability

that the particle will be found in each of the regions making up thirds of the box.

Solution

For 0<x<a/3,

(Probability)

∫a/ 3

0

ψ(x)^2 dx

2

a

∫a/ 3

0

sin^2

(

2 πx

a

)

dx



2

a

a

2 π

∫ 2 π/ 3

0

sin^2 (y)dy

1

π

[

y

2

−

1

4

sin(2y)

]∣

∣∣^2 π/^3

0



1

π

[

π

3

−

1

4

sin(4π/3)

]

 0. 402249