Absolutely Small: How Quantum Theory Explains Our Everyday World - Michael D. Fayer (2010)
Chapter 7. Photons, Electrons, and Baseballs
PHOTONS, ELECTRONS, AND BASEBALLS are all described with quantum theory in the same way, but quantum theory isn’t necessary to describe baseballs. Baseballs act as classical particles and behave in a manner that is accurately described by classical mechanics. If photons, electrons, and baseballs all have the same quantum mechanical description, why do only baseballs act as classical particles? The answer is that baseballs are large in the absolute sense. Here, we will see why photons and electrons need a quantum theory description but baseballs do not. Real physical situations are discussed that bring out both the wavelike and particlelike nature of quantum particles, that is, absolutely small particles.
WAVES OR PARTICLES?
When a particle is in a superposition state, a wave packet, we have some knowledge of its position and some knowledge of its momentum. So are photons, electrons, and so on waves or particles? The answer is that they are wave packets. Whether they seem to be particles or seem to be waves depends on what experiment you do, that is, what question you ask. In the photoelectric effect, the photons act like particles. One photon hits one electron, and kicks it out of the metal (Figure 4.3). The photon is a wave packet formed from a spread of momentum eigenstates. The width of the spread, Δp, results in a relatively well-defined position, that is, a relatively small Δx. Thus, the photon wave packet is more or less well defined in position, so that it can act like a particle of light in the photoelectric effect. In the interference experiment (Figure 5.1), the photons act like waves. This may not be surprising because the wave packet is, in fact, a superposition of waves, but not waves in the normal classical sense, but rather probability amplitude waves. In the discussion of the interference phenomenon, the photon wave is discussed as if it is a single probability amplitude wave. Now it is clear that it is actually a wave packet, composed of a superposition of waves. When it hits the beam splitter, it becomes a superposition of the two translation states, T1 and T2. The probability amplitude waves in T1 interfere with their corresponding waves in T2, to produce the interference pattern as discussed above.
DIFFRACTION OF LIGHT
So a photon acts like a particle in the photoelectric effect, but it can also act like a wave. An experiment that clearly shows the wavelike properties of photons is diffraction of light from a diffraction grating. It is possible to see diffraction using a music compact disk (CD) and a bright light or sunlight. The CD has very fine grooves on its surface. These are the tracks on which the information is stored. As explained below, when white light from the sun or a light bulb falls on the CD, the grooves diffract the light, sending different colors in particular directions. Different parts of the CD make a range of angles relative to your eye, which causes the appearance of different colors emanating from different parts of the CD.
Diffraction from a grating is used in many optical instruments, called spectrometers. These instruments separate the different colors of incoming light so that the colors can be analyzed individually. A recording of the colors of light from a particular source is called a spectrum. For example, stars emit different colors of light depending on their temperature. Taking a spectrum of the light from a star can provide a great deal of information about the star. Star light traveling through space will encounter molecules in space. As discussed in Chapter 8 and subsequent chapters, different molecules absorb different colors of light. Star light traveling to Earth has some of its colors partially absorbed by the molecules in space. Astronomers mount spectrometers on telescopes and take spectra to determine what types of molecules are in space between a particular star and the Earth.
Figure 7.1 shows the geometry for light diffraction from a grating. The incoming light is at an angle α (Greek letter alpha) relative to the normal to the grating. The normal is the direction perpendicular to the surface of the grating. The grating is shown from the side. The surface of the grating, which looks like a flat mirror to the eye, has a dense set of parallel grooves in it. These grooves are called lines. The spacing between the lines is labeled d in Figure 7.2. The spacing is about the wavelength of light, approximately one ten millionth of a meter. The grooves are highly reflective. They are usually gold or silver. For incoming light that has a range of colors, the outgoing light will be separated by color such that each color goes in a unique direction. The separation into colors going in different directions is illustrated in Figure 7.1. The angle between the normal to the grating and a particular color is labeled β (Greek letter beta) in the figure. β is shown for the blue color. β is bigger for green and still bigger for red.
FIGURE 7.1. Geometry of light diffraction from a grating. The grating is composed of a reflective surface, usually silver or gold, with very fine parallel grooves in it. The grating is shown here from the side. The grooves run into the page. The grooves have a very uniform spacing, d; α is the angle of the incoming light. The outgoing angles β depend on the color. Therefore, the colors are separated by diffraction.
Diffraction of Light Shows Wave Character of Photons
The diffraction of light from a grating shows the wave nature of photon wave packets. To see that diffraction brings out the wave character of photons, we need to look at how diffraction works in terms of constructive and destructive interference of waves. Figure 7.2 shows incoming photon wave packets as a beam of light impinging on the diffraction grating. The light travels different distances before it strikes various parts of the grating. The light that reaches the upper left portion of the grating will travel a shorter distance than the light that hits the bottom right portion of the grating. The wave packet is composed of many colors, that is, many waves with wavelengths, λ. The different colors of light will come off of the grating in all directions. Here is the tricky part. The wave packets are more or less localized, but they are composed of different colors, each of which is a delocalized probability amplitude wave (see Figures 6.1, 6.2, 6.4, and 6.7). The more or less localized wave packet is formed by the interference of many different color waves (different λs, which correspond to different momenta, p). Consider one particular color, red, that comprises part of the wave packet. If the wave hits only one line in the grating, it would reflect off in many directions because of the shape of the groove. It would leave the single groove as a superposition state composed of probability amplitude waves propagating in many directions. In the interferometer (Figure 5.1), the incoming wave packet became a superposition state that had probability amplitude propagating in two directions. Here, after hitting a single line, the superposition would be heading in many directions.
The important feature of a grating is that the incoming wave hits many lines in the grating. For a particular color, red shown in Figure 7.2, there is a single direction in which the waves will add up constructively. In the figure, for the direction in which the red waves are propagating, all of the peaks and troughs of the waves add in phase even though they reflect from different places. (The wavelength has been exaggerated relative to the line spacing, d, to make it easy to see the alignment of the waves.) The in phase addition of many waves leaving the grating makes a very large outgoing wave. In all other directions, the red waves will add destructively because the peaks and null do not line up.
Diffraction from a grating causes the waves of a given wavelength (a particular color) to add constructively in one direction. The intensity associated with a probability amplitude light wave is proportional the square of the wave amplitude. Therefore, in the direction of constructive interference for a particular color, red in the example, the intensity of light is large. In other directions, red light will experience destructive interference, because the wavelength is such that the differences in the distances from each groove don’t equal an integral number of wavelengths. For another color, say blue, there is a different direction in which light coming off of all of the grooves will add constructively (see Figure 7.1). Therefore, the blue light component of the incoming photon wave packets will leave the grating as a large amplitude wave in its own direction, and in this direction the intensity of the blue component of the incoming light will appear bright.
FIGURE 7.2. Incoming photon wave packets are diffracted from a grating. The different colors reflect off of the grooves. For a particular color, there is a direction in which the waves corresponding to that color constructively interfere. They add to make a large amplitude wave, so the color looks very bright in that particular direction.
Electrons Act Like Bullets in a Cathode Ray Tube
Diffraction of light from a grating brings out the wave property of the photon wave packets, while the photoelectric effect demonstrates the more localized particle like properties of a photon wave packet. In the discussion of the de Broglie wavelength, which relates the momentum to a wavelength through the relation p = h/λ, it was mentioned that the descriptions of electrons and other types of “particles” are the same as the description of photons. Both photons and electrons are described in terms of probability amplitude waves. Both are more or less localized wave packets (see Figure 6.7). For an electron that is a free particle (no forces are acting on it) the wave packet is a superposition of the free particle momentum eigenstates. The uncertainty in the electron’s position, Δx, is determined by the uncertainty (spread) in the momentum, Δp, through the Heisenberg’s uncertainty relationship, ΔxΔp ≥ h/4π. The equality sign holds for Gaussian wave packets, which are the shapes shown in Figure 6.7.
To illustrate both the particle nature and wave nature of electrons, two examples are discussed: how a cathode ray tube (CRT) works and low energy electron diffraction from a crystal surface. Cathode ray tubes used to be ubiquitous. They are the devices that produce the pictures in the original televisions and computer monitors. CRTs are the large, boxy TVs and monitors that are rapidly being replaced by other devices, such as liquid crystal displays (LCDs). (There are actually several technologies used to make large-screen thin TVs, but thin computer monitors are all LCDs.)
Figure 7.3 is a schematic diagram of a CRT. Inside the CRT is a vacuum in which electrons can move without colliding with air molecules. The process that produces the picture begins with the filament, a piece of wire (left side of figure). An electrical current is passed through the filament, which causes it to get very hot like the filament in a conventional light bulb, the element in an electric stove, or the element in an electric space heater. The heat from the filament heats the cathode until it is also very hot. The cathode is a piece of metal that is connected to a negative voltage, like the negative end of a battery, but at a much higher voltage. The cathode becomes so hot that electrons boil off. Heat is a form of energy. The electrons are held in the metal by a binding energy that depends on the type of metal. When the metal is hot enough, the thermal energy can overcome the electron binding energy and some electrons will leave the metal. In the photoelectric effect, a photon provided the energy to overcome the electron binding to the metal. In a CRT, heat provides the energy to eject the electrons from the metal. The electrons that leave are replaced by the connection to the negative power supply that puts electrons back into the cathode, so the process can go on continuously. Electrons are negatively charged, and because the cathode is at a negative voltage, the electrons are repelled from the cathode. Therefore, the electrons fly away from the cathode. The movement of the electrons away from the cathode is helped by the positively charged acceleration grid (see Figure 7.3). Because the acceleration grid is connected to a positive power supply, the negatively charged electrons are attracted to it. Like charges repel. Unlike charges attract. The acceleration grid is a mesh of wires that is mostly holes. When the electrons reach the grid, most of them fly right through it and keep going, moving very fast.
FIGURE 7.3. A schematic of a cathode ray tube (CRT). The hot filament heats the cathode, which “boils” off electrons. The positively charged acceleration grid accelerates the negatively charged electrons. Voltages applied to the control grids steer the electrons to particular points on the screen. The screen is covered with tiny adjacent red, green, and blue spots that glow with their particular color when hit with electrons. By rapidly scanning the electron beam to hit the appropriate colors in a given spot on the screen the image is made.
The electrons then pass between the control grids (see Figure 7.3), which control the direction the electrons go. There is one pair of control grids for the vertical direction (shown in the figure) and an equivalent set for the horizontal direction (not shown). Consider the vertical direction. If a positive voltage is applied to the top control grid and a negative voltage is applied to the bottom control grid, the electrons will be deflected up, as shown in Figure 7.3, because the negatively charged electrons will be attracted toward the positive top grid and repelled from the negative bottom grid. If the polarity of the voltages on the two grids is reversed, the electrons will be deflected downward. If large voltages are applied to the grids, the electrons will be deflected a lot. If small voltages are applied, the electrons will be deflected a small amount. If no voltages are applied, the electrons will go straight ahead. The same thing happens by applying voltages to the horizontal control grids. Once the electrons pass by the control grids, they continue in a straight line. In this manner, the electrons can be aimed just like bullets. This part of the CRT is referred to as an electron gun. Electron guns are used in many scientific devices such as electron microscopes and the device discussed below. So even when there are no more CRTs used as TVs and computer monitors, the basic device described here will still be important.
Because there is no air and gravity is a very weak force, the electrons travel basically as free particles until they hit the screen shown on the right side of Figure 7.3. On the screen are very small and very closely spaced patches of chromophores. Chromophores are chemical species that emit light when excited, that is, when sufficient energy is imparted to them. In this case, the chromophores are excited when the electrons hit them. In each very small region of the screen, there are three chromophores, one red, one green, and one blue. The electron beam can be aimed to hit a particular spot with great accuracy. If at a given location the red chromophore is hit, the screen lights up for an instant with a tiny red dot of light. If the green chromophore is hit, there is a green dot of light, and if the blue chromophore is hit, a blue dot of light is generated,
The electronics that produce the voltages on the control grids sweep the electron beam across the screen horizontally, then move the beam down, and sweep horizontally again. This is continued until the entire screen is swept. The beam returns to the top, and the sweep is repeated. As the beam sweeps, it is directed to hit red, green, or blue chromophores. The patches of three chromophores are so close together horizontally and vertically that your eye cannot distinguish them as individual dots. The beam can also be turned off so if no chromophore is hit you get a black spot. The combination of the three colors is sufficient to make any color. The image that we see on a CRT computer screen or a TV is formed by controlling which colors are hit and which spots are not hit at all. The electron beam is moved across the screen so fast that our eyes cannot tell that what we see is actually a very rapid succession of still pictures.
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 direction. The lattice at a crystal surface is two-dimensional. As can be seen in Figure 7.5, there are many directions in which there are parallel rows of atoms. The solid lines show rows of atoms running in different directions. The dashed line parallel to each solid line shows that there are parallel rows of atoms for each of the directions indicated by the solid lines. For the rows of atoms running in different directions, the spacing between the atoms (the diffraction groove spacing) is different. The difference in spacing can be seen in Figure 7.5. Look at the separation between a pair of solid and dashed lines. Each pair has a different separation, which is the groove spacing.
FIGURE 7.5. The lattice from Figure 7.4, with examples of different rows of atoms shown by the lines. For each line passing through the centers of atoms in a row, it is possible to draw more lines that are parallel to the initial line and that also pass through the centers of atoms. The spacings between these distinct parallel rows are different. Each set of rows causes diffraction in a different direction.
Because there are many different atom-to-atom spacings with the “grooves” running in different directions, the electrons’ waves will be diffracted in many different directions. Figure 7.6 is an example of low-energy electron diffraction from a crystal surface. The black circle in the center is a piece of metal called a beam stop. It is supported by another piece of metal that appears as the vertical bar below the beam stop in the picture. The beam stop prevents the portion of the electron beam that is reflected from the crystal from hitting the detector. The brighter and dimmer white spots are produced by the diffracted electrons hitting the detector. From the location of the spots, the spacing and arrangement of the atoms can be determined. Electron diffraction from crystal surfaces is an important tool in the science of understanding the nature of surfaces. The electron diffraction pattern demonstrates conclusively that electrons can behave as waves, just like photons.
FIGURE 7.6. Experimental data showing diffraction of electrons from the surface of a crystal. The various light spots are the electron diffraction spots. There are many spots because the diffraction occurs from the many different parallel rows of atoms (see Figure 7.5).
ELECTRONS AND PHOTONS ARE PARTICLES AND WAVES, BUT BASEBALLS ONLY PARTICLES
Electrons act as particles in a CRT just as photons act as particles in the photoelectric effect. Electrons act as waves in low-energy electron diffraction just as photons act as waves when they diffract from a diffraction grating. In reality, photons, electrons, and all particles are actually wave packets that are more or less localized. Wave packets can display their wavelike properties or their particlelike properties depending on the circumstances.
If photons and electrons can show both wavelike and particlelike properties, why don’t baseballs? To see the reason that baseballs act like particles in the classical mechanics sense, we need to look at the wavelengths associated with particles versus their size.
First consider an electron in an atom such as hydrogen. We are going to talk about the quantum description of the hydrogen atom and other atoms in Chapters 10 and 11, but for now, we will only use a very simple qualitative discussion of the wavelike characteristics of the hydrogen atom. The de Broglie relation tells us that the wavelength λ = h/p. The momentum is p = mV, the mass times the velocity. The mass of an electron is me = 9.1×10-31 kg. In an atom, the typical velocity of an electron is V = 5.0×106 m/s. Then the de Broglie wavelength is
Note that 1.5 Å is approximately the size of an atom. Therefore, the wavelength of an electron in an atom is about the size of the atom. The wave properties of electrons will be very important when electrons are in systems that are very small like atoms.
What about a baseball? According to the rules of Major League Baseball, a baseball must weigh between 142 g and 149 g. We will take the mass to be 145 g = 0.145 kg. A pretty fast pitch goes 90 mph. 90 mph = 40 m/s. The momentum of a fast ball is p = 0.145 kg × 40 m/s = 5.8 kg-m/s. The de Broglie wavelength of the fast ball is:
This is an unbelievably small number. The size of one atom is about 1 Å. The size of the nucleus of an atom is about 10-5 Å. Then the wavelength of a baseball is 0.0000000000000000001 of the size of one atomic nucleus. This wavelength is small beyond small. It is so small that it will never be manifested in any measurement. There can never be a diffraction grating with line spacing small enough to show diffraction for a wavelength that is a ten millionth of a trillionth of the size of an atomic nucleus. Because this wavelength is so small we never have to worry that a baseball will diffract from a baseball bat. It will always act like a classical particle. Objects that are large in the absolute sense have the property that the wavelengths associated with them are completely negligible compared to their size. Therefore, large particles only manifest their particle nature; they never manifest their wave nature. In contrast, particles that are small in the absolute sense have de Broglie wavelengths that are similar to their size. Such absolutely small particles will act like waves or act like particles depending on the situation. They are wave packets. In the manner that we have discussed, they are both waves and particles.