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original momentum will be changed. The exact amount of the change pcannot be

predicted, but it will be of the same order of magnitude as the photon momentum

h. Hence

p (3.23)

The longer the wavelength of the observing photon, the smaller the uncertainty in the

electron’s momentum.

Because light is a wave phenomenon as well as a particle phenomenon, we cannot

expect to determine the electron’s location with perfect accuracy regardless of the in-

strument used. A reasonable estimate of the minimum uncertainty in the measurement

might be one photon wavelength, so that

x  (3.24)

The shorter the wavelength, the smaller the uncertainty in location. However, if we use

light of short wavelength to increase the accuracy of the position measurement, there will

be a corresponding decrease in the accuracy of the momentum measurement because

the higher photon momentum will disturb the electron’s motion to a greater extent. Light

of long wavelength will give a more accurate momentum but a less accurate position.

Combining Eqs. (3.23) and (3.24) gives

x p h (3.25)

This result is consistent with Eq. (3.22), x p 2.

Arguments like the preceding one, although superficially attractive, must be

approached with caution. The argument above implies that the electron can possess a

definite position and momentum at any instant and that it is the measurement process

that introduces the indeterminacy in x p. On the contrary, this indeterminacy is

inherent in the nature of a moving body. The justification for the many “derivations” of

this kind is first, they show it is impossible to imagine a way around the uncertainty

principle; and second, they present a view of the principle that can be appreciated in

a more familiar context than that of wave groups.

3.9 APPLYING THE UNCERTAINTY PRINCIPLE

A useful tool, not just a negative statement

Planck’s constant his so small that the limitations imposed by the uncertainty princi-

ple are significant only in the realm of the atom. On such a scale, however, this principle

is of great help in understanding many phenomena. It is worth keeping in mind that

the lower limit of 2 for x pis rarely attained. More usually x p , or even

(as we just saw) x ph.

Example 3.7

A typical atomic nucleus is about 5.0  10 ^15 m in radius. Use the uncertainty principle to

place a lower limit on the energy an electron must have if it is to be part of a nucleus.

h





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