Simple Nature - Light and Matter

b/These two electron waves

are not distinguishable by any

measuring device.

voltage differences on the order of 1 V, so that a typical energy is

(e)(1 V), which is on the order of 10−^19 J. What is the wavelength

of an electron with this amount of kinetic energy?

.This energy is nonrelativistic, since it is much less thanmc^2.

Momentum and energy are therefore related by the nonrelativistic

equationK = p^2 / 2 m. Solving forpand substituting in to the

equation for the wavelength, we find

λ=

h

√

2 mK

= 1.6× 10 −^9 m.

This is on the same order of magnitude as the size of an atom,

which is no accident: as we will discuss in the next chapter in

more detail, an electron in an atom can be interpreted as a stand-

ing wave. The smallness of the wavelength of a typical electron

also helps to explain why the wave nature of electrons wasn’t dis-

covered until a hundred years after the wave nature of light. To

scale the usual wave-optics devices such as diffraction gratings

down to the size needed to work with electrons at ordinary ener-

gies, we need to make them so small that their parts are compa-

rable in size to individual atoms. This is essentially what Davisson

and Germer did with their nickel crystal.

self-check E

These remarks about the inconvenient smallness of electron wavelengths

apply only under the assumption that the electrons have typical ener-

gies. What kind of energy would an electron have to have in order to

have a longer wavelength that might be more convenient to work with?

.Answer, p. 1063

What kind of wave is it?

If a sound wave is a vibration of matter, and a photon is a
vibration of electric and magnetic fields, what kind of a wave is
an electron made of? The disconcerting answer is that there is
no experimental “observable,” i.e., directly measurable quantity, to
correspond to the electron wave itself. In other words, there are
devices like microphones that detect the oscillations of air pressure
in a sound wave, and devices such as radio receivers that measure
the oscillation of the electric and magnetic fields in a light wave,
but nobody has ever found any way to measure the electron wave
directly.

We can of course detect the energy (or momentum) possessed by
an electron just as we could detect the energy of a photon using a
digital camera. (In fact I’d imagine that an unmodified digital cam-
era chip placed in a vacuum chamber would detect electrons just as
handily as photons.) But this only allows us to determine where the

Section 13.3 Matter as a wave 893