Physical Chemistry Third Edition

704 16 The Principles of Quantum Mechanics. II. The Postulates of Quantum Mechanics

This integral cannot be carried out explicitly. A table of its numerical value is found in

Appendix C. We make the substitutiony

√

ax. From the table of values of the error function

in Appendix C,

(Probability)

2

√

π

∫ 1

0

e−y

2

dyerf(1) 0. 8427 (16.4-23)

The probability that the oscillating particle is farther away from its equilibrium position than

the classical turning point is 1. 0000 − 0. 8427  0 .1573, or 15.73% (7.86% probability of

being past either end of the classically permitted region). This probability is represented by

the two shaded areas in Figure 16.6, which shows the probability density for thev0 state

superimposed on a graph of the potential energy function.

Exercise 16.7

Calculate the value of the following ratio forv0 state of the harmonic oscillator

Ratio

|ψ 0 (xt)|^2

|ψ 0 (0)|^2

Explain in words what this ratio represents.

If a region is small enough, the probability is approximately equal to|Ψ|^2 multiplied

by the length or volume of the region. For motion of one particle in one dimension and

for a sufficiently small value of∆x,

(Probability thatx′<x<x′+∆x)≈|Ψ(x′,t′)|^2 ∆x (for a small value of∆x)

(16.4-24)

23

1

0

2

22

Classical

turning points

Area

representing

tunneling

Wave function squared

(different scale on graph)

Potential energy

Energy /

h^

or wave function squared

v 5 0 energy level

2101123

ŒWa z

Figure 16.6 The Probability Density of a Harmonic Oscillator in Its Ground State,

Showing Tunneling.