Absolutely Small - Michael D. Fayer [40]
In the description of the CRT, the electrons acted very much like our general conception of particles. They could be aimed by the electron gun to hit very specific spots on the screen. This does not sound very much like a wave. However, it is certainly within our description of more or less localized wave packets. As long as Δx of the electron wave packets is small compared to the size of the chromophore spots (pixels), the fact that the wave packet is delocalized over a distance scale Δx doesn’t matter. The colored pixels are small, but not small on the “absolutely small” distance scale. They are smaller than the eye can see without a microscope, but that is still large compared to the length scales that are encountered in atomic and molecular systems. Therefore, wave packets with a reasonably small Δp can still have an uncertainty in position that is very small compared to the pixel size. For a particle, such as an electron, p = mV, where m is the mass of the electron and V is the velocity of the electron. The mass is well defined. The uncertainty in p comes from an uncertainty in the velocity. So what Δp means is that the velocity is not perfectly well defined. Measurements of the velocity on identically prepared electron wave packets will not give the same value from one measurement to the next. The uncertainty in the velocity yields an uncertainty in the momentum, Δp, which through the Uncertainty Principle, ΔxΔp ≥ h/4π, gives the uncertainty in x, Δx. The important point is that Δx can be significant on the distance scale of atoms and molecules but very small compared to the distance scale of the macroscopic colored pixels on the CRT screen. In such situations, the wave nature of a wave packet is not manifested, and the wave packet behaves like a classical particle.
Electrons Act Like Waves in Electron Diffraction
Electron wave packets also display their wave properties, as illustrated in Figure 7.4. In this experiment, a beam of electrons generated with an electron gun, like that described above, is aimed at a crystal surface rather than a TV screen. The electrons are not of sufficiently high energy to penetrate the crystal. The surface of the crystal is composed of rows of atoms called a lattice or crystal lattice. The rows of atoms are spaced a few angstroms apart; one angstrom is 1 × 10-10 m, or one ten billionth of a meter. On the atomic distance scale, the unit of angstroms comes up often. It is given a special symbol, Å. The spacing is determined by the size of the atoms. The rows of atoms act as the grooves in a diffraction grating, but they are much more closely spaced. The wavelength of the electrons is on the same distance scale as the lattice spacing (row spacing). The wavelength is given by the de Broglie relation λ = h/p. p = mV. The mass of the electron is 9.1 × 10-31 kg (kilograms). Then for a velocity of 7.3 × 105 m/s (730,000 meters per second, about 1.6 million miles per hour), λ = 10 A. This velocity is very easily obtained with a simple electron gun.
FIGURE 7.4. A schematic of low-energy electron diffraction (LEED) from the surface of a crystal. An incoming beam of electrons, with low enough energy not to penetrate the crystal, strikes the surface. The lines of atoms act like the grooves of the diffraction grating in Figure 7.1. They diffract the incoming electron waves.
The electron probability amplitude waves diffract from the crystal surface in a manner akin to the photons diffracting from the ruled grating discussed above. However, in the ruled grating, there is a single separation, d, because the grooves all run parallel to each other in a single