bei48482_FM

3.8 UNCERTAINTY PRINCIPLE II

A particle approach gives the same result

The uncertainty principle can be arrived at from the point of view of the particle prop-

erties of waves as well as from the point of view of the wave properties of particles.

We might want to measure the position and momentum of an object at a certain mo-

ment. To do so, we must touch it with something that will carry the required information

back to us. That is, we must poke it with a stick, shine light on it, or perform some sim-

ilar act. The measurement process itself thus requires that the object be interfered with in

some way. If we consider such interferences in detail, we are led to the same uncertainty

principle as before even without taking into account the wave nature of moving bodies.

Suppose we look at an electron using light of wavelength , as in Fig. 3.17. Each

photon of this light has the momentum h. When one of these photons bounces

off the electron (which must happen if we are to “see” the electron), the electron’s

Wave Properties of Particles 113

Figure 3.16The wave packet that corresponds to a moving packet is a composite of many individ-

ual waves, as in Fig. 3.13. The phase velocities of the individual waves vary with their wave lengths.

As a result, as the particle moves, the wave packet spreads out in space. The narrower the original

wavepacket—that is, the more precisely we know its position at that time—the more it spreads out

because it is made up of a greater span of waves with different phase velocities.

Wave packet

Classical particle

Ψ^2 t^1

t 2

t 3

x

x

x

Ψ^2

Ψ^2

Figure 3.17An electron cannot be observed without changing its momentum.

Original

momentum

of electron Final

momentum

of electron

Reflected

photon

Incident

photon

Viewer

bei48482_ch03_qxd 1/16/02 1:51 PM Page 113