Physical Chemistry Third Edition

16.5 The Uncertainty Principle of Heisenberg 713

Other measures of the uncertainty give a different minimum value. For a Gaussian

probability distribution, the uncertainty in a prediction at the 95% probability level is

equal to 1.96 times the standard deviation. At this level of uncertainty the right-hand

side of Eq. (16.5-2) would be replaced by a larger value.

The value of the uncertainty product depends on the system and on the state

considered. The uncertainty product for then1 state of the particle in a box, 0.09038h,

is slightly larger thanh/( 4 π), which equals 0.079577h. The uncertainty product for the

v0 state of the harmonic oscillator is exactly equal toh/( 4 π)and seems to be

the smallest uncertainty product for any system and any state. (See Problem 16.36.)

The uncertainty product for higher-energy states is larger.

Coordinates and momenta are not the only variables that have nonzero uncertainty

products. There is a general relation

∆A∆B≥

∣

∣

∣

∣

1

2

∫

ψ∗

[

Â,̂B

]

ψdq

∣

∣

∣

∣ (16.5-5)

where

[

Â,B̂

]

is the commutator of the operatorŝAandB̂.^4 From the commutator of

two angular momentum components, we can see that two components of the angular

momentum obey an uncertainty relation. (See Problem 16.16.)

Exercise 16.12

Use Eqs. (16.5-5) and (16.3-16) to obtain the same uncertainty relation forxandpxthat was

given in Eq. (16.5-2).

The uncertainty principle is a rather subtle concept, and deserves more discussion

than we give it in this book. However, the main idea is that it requires that the statistical

case applies to at least one of a conjugate pair of variables, and if the predictable case

applies to one of the variables, the other variable has an infinite uncertainty.

EXAMPLE16.22

Find〈px〉andσpxfor a free particle in a state corresponding to the wave function of

Eq. (15.3-28) withFset equal to zero.

Solution

We cancel the time-dependent factors to obtain

〈px〉

̄h

i

D∗D

∫

e−iκx

d

dx

eiκxdx

D∗D

∫

e−iκxeiκxdx

hκ ̄ hκ ̄

D∗D

∫

e−iκxeiκxdx

D∗D

∫

e−iκxeiκxdx

hκ ̄

∫

dx

∫

dx

We specify that the limits of the integrals are−LandL, with the intention of taking the limit

thatL→∞since all values ofxare possible. We cancel the integrals in the last quotient of

integrals prior to taking the limit, and obtain

〈px〉hκ ̄ (16.5-6)

(^4) Levine (Quantum Chemistry, 5th ed., Prentice-Hall, Englewood Cliffs, NJ, 2000, pp. 96, 206) assigns
the proof as a homework problem. A lot of hints are included, but it is a fairly long proof.