Absolutely Small: How Quantum Theory Explains Our Everyday World - Michael D. Fayer (2010)
Chapter 20. Think Quantum
WHEN A PARENT HOLDS A TODDLER and points to the moon saying “that’s the moon,” the toddler certainly has an awareness of the bright object in the sky. The toddler may learn that the light in the sky is called “moon,” but has no understanding of what and where the moon is. By the time a child is seven or eight, understanding of the moon has grown considerably. The child knows that it is very different from the street light at the end of the block, that it is very far away, and that you cannot touch the moon or even go there, although many years ago people actually did go to the moon. As an adult, a person has a good understanding of what the moon is even without knowing how to use Newton’s classical mechanics to calculate its orbit around the Earth. The adult knows that the moon’s apparent motion across the sky is caused by the rotation of the Earth, that the moon is far away, but not nearly as far as the planets in the solar system, and that a person feels much lighter and can jump higher on the moon than on Earth because the moon has less mass and, therefore, it exerts a weaker gravitational pull.
EXPERIENCE TEACHES US TO UNDERSTAND THE CLASSICAL WORLD
As we grow up, our increased understanding of the moon comes not just from education, but from the intuitive logic in the description of the moon in orbit around the Earth. This description is, in many respects, consistent with everyday experiences. If you throw a baseball, it arcs before falling back to Earth. If you throw the ball harder and aim higher, it goes higher and farther, making a bigger arc before hitting the ground. It is a simple and reasonable extension to accept that if you use a rocket and get an object moving really fast, aiming very high, you can make the arc extend halfway around the Earth, which is the basis for intercontinental missiles. After that, it is not a great leap to accept that if you use an even bigger rocket and get an object going even faster, the object will have an arc that is more or less a circle going around the Earth in orbit. Then the moon is just a very large object moving very fast so that it is in orbit around the Earth.
The fact that we can progress from a baseball to the moon orbiting the Earth is based on everyday classical mechanics experiences. However, it does require abstract reasoning to put the pieces together. The ancients reasonably thought that the moon circled the Earth. After all, you can see it move across the sky. We can do a simple experiment to see why the moon appears to circle the Earth. Standing in the middle of a room with a single light on a wall, if you slowly spin around, you will see the light come and go. With your back to the light, you do not see it. As you rotate, the light will come into the edge of your vision, move to the center of your view, and then disappear at the other edge of your view. It does not appear again until you have made a half circle of your rotation. Given this simple experience combined with what we know about baseballs and intercontinental missiles, it is easy to accept and, in fact, understand that the moon is in orbit around the Earth and that the Earth rotates on its axis causing the “rising” and “setting” of the moon.
Our experiences and the basic nature of systems that obey classical mechanics allow us to develop an intuition for the behavior of many things we see about us daily. Even a novice pool player quickly grasps that if you hit the cue ball so that it strikes the left side of another ball, that ball will take off to the right. The pool ball collision is classical, and the balls move according to the precepts of classical mechanics with well-defined trajectories. However, the world around us, which is governed by the rules of quantum mechanics, is generally beyond consideration and understanding. When it comes to phenomena that are controlled by the properties of absolutely small systems, most people are like the toddler looking at the moon. They see it, but lack understanding of what they are seeing.
UNDERSTANDING WHAT WE SEE AROUND US REQUIRES SOME KNOWLEDGE OF QUANTUM MECHANICS
Why should we care? It is possible to go through life and see the moon with no idea of what it really is. A person can get up in the morning, go to work, eat, sleep, have a family, not know what the moon is, and be perfectly happy. It is also possible to have no conception of what makes the everyday things around us have their properties. We live in a sea of physical phenomena that can toss us around like ocean waves. We may not be able to control the physical world around us, but is it reasonable to completely lack any understanding of it? Do we want to be like a toddler or, even worse, an adult with no understanding of the moon? Do we really want to have no conception of why the element in an electric stove gets hot? I believe that the world is a more interesting place when we have some appreciation for the nature of the matter that surrounds us. From biological molecules to electrical conductivity, the natural world is driven by quantum phenomena. If we are adrift in an ocean of quantum physics, then some knowledge of quantum theory increases our appreciation for the wonders of the natural world.
COLOR IS A QUANTUM PHENOMENON
After plowing through the previous chapters, you have grown from toddler to adult in your quantum thinking. Now you know what color is. Let’s go back to the first sentence in the book. Why are cherries red and blueberries blue? The question is what gives objects color and what makes one thing have a different color than another. The answer is that matter is made of atoms and molecules. In contrast to classical mechanics, where energy varies continuously, atoms and molecules have discreet energy levels. Light is also not continuous. Light comes in discreet packets called photons. A photon has a particular energy, which means it has a particular color. Because energy must be conserved, a photon can only be absorbed by the atoms and molecules that make up matter if the energy of the photon matches the energy difference between two atomic or molecular quantum energy levels. If the energy matches, the photon can be absorbed, and the system makes a transition from a lower energy level to a higher energy level. The photons that do not match energy level differences reflect from the object. So if the energy level spacing of the molecules is such that red light is absorbed, blue light will bounce off, and the object looks blue. If the energy level differences are such that blue light is absorbed, then red light bounces off, and the object looks red.
ENERGY LEVELS AND COLOR COME FROM THE WAVELIKE NATURE OF PARTICLES
To reprise the color of objects in a little more detail, we discussed the one-dimensional particle in a box problem in Chapter 8. We learned that absolutely small “particles” are not particles in the everyday classical sense. They are actually waves or wave packets that are more or less localized in space. In the particle in a box, only waves of certain shapes were allowed. In three-dimensional systems, such as the hydrogen atom discussed in Chapter 10, the shapes of the waves are more complicated, but again, only certain shapes, called orbitals, can exist. This is also true for larger atoms and molecules, where the molecular electron waves were described as molecular orbitals. The electron waves (wavefunctions) in an atom or molecule have associated with them well-defined energies, the energy levels. We say energy is quantized; the energy changes in discreet steps. The discreet quantum energy levels are one of the major departures from classical mechanics. In classical mechanics, energy changes continuously.
We solved the quantum particle in a box problem and found that the energy levels depend on the size of the box. A larger box (larger molecule) has energy levels that are more closely spaced than a smaller box. The result, which applies to real molecules as well as the particle in a box, is that large molecules tend to absorb light in the red part of the spectrum. Red light is low energy light, and for large molecules there are relatively small energy differences between the energy levels. Smaller molecules absorb light in the blue because the molecular energy level differences are larger, and blue light is higher energy than red light. Really small molecules, like benzene (see Chapter 18), absorb in the ultraviolet part of the spectrum. Thus, they do not absorb visible light. Crystals of small molecules like naphthalene (mothballs) look white because none of the visible wavelengths can be absorbed. The energy levels are too widely spaced to absorb visible light. All of the visible light bounces off of the crystals, and they look white. This is the reason that salt crystals in a salt shaker are white. It is also the reason sugar crystals are white. Both salt and sugar have widely spaced energy levels that absorb in the ultraviolet, and reflect all visible colors of light.
QUANTUM PHENOMENA HOLD ATOMS TOGETHER TO MAKE MOLECULES AND DETERMINE THEIR SHAPES
We now know what holds atoms together to make molecules, what gives molecules their shapes, and why details of the shapes are important. We have seen that the electron waves of atoms combine to make molecular orbitals. Sharing the electrons among atoms in molecular orbitals can result in chemical bonds that hold atoms together to form molecules. In Chapters 12 through 14, we looked at molecular orbitals in some detail. We learned that molecular orbitals come in two flavors, bonding and antibonding. Appropriately placing electrons in the simple molecular orbital energy level diagrams can provide a great deal of information. In the hydrogen molecule (Chapter 12), the two electrons from the two hydrogen atoms go into the lowest energy molecular orbital, the bonding MO. The result is a covalent bond in which the atoms share a pair of electrons. However, the same considerations enabled us to see why a diatomic helium molecule does not exist. Each helium atom contributes two electrons to the hypothetical helium diatomic molecule. The first two go into the bonding MO but because of the Pauli Exclusion Principle, the next two electrons must go into the antibonding MO. The result is no net bonding, and the He2 molecule does not exist. The covalent chemical bond is intrinsically a quantum phenomenon with no classical mechanics explanation.
For atoms larger than hydrogen, combining different s and p atomic orbitals produced hybrid orbitals with various shapes. The combination of different-shaped hybrid atomic orbitals to form molecular orbitals is responsible for the types of bonds that are formed (single, double, triple) and the shapes of molecules. We paid particular attention to organic molecules, which are molecules formed mainly from carbon, hydrogen, oxygen, and several other atoms. Organic molecules are important because they are the basis for life, as well as for materials such as plastics. We found that the types of bonds are very important. A molecule can readily rotate around a carbon-carbon single bond, changing the molecule’s shape, but a molecule cannot rotate around a carbon-carbon double bond. The inability of organic molecules to rotate around carbon-carbon double bonds is crucial in biology.
In Chapter 16, we discussed fats. Double bonds made all the difference. Fats with double bonds cannot change their shape around the double bond. Polyunsaturated fats have multiple double bonds. All naturally occurring fats, except those from ruminants, have cis double bonds. That means the fat molecule is bent at the double bond. However, chemical processing of polyunsaturated fats to produce monounsaturated fats generates trans double bonds. Fats with trans double bonds are called trans fats. The trans fat molecule is straight at a trans double bond rather than bent. This difference in shape, which is produced by the properties of the quantum mechanical covalent double bond, has a substantial influence on the biological activity of the molecules. Trans fats have been linked to a variety of deleterious effects on human health. The shapes of biomolecules, such as proteins, are central to biology. The shapes of molecules are controlled by the quantum mechanical interactions between atoms that give rise to different types of molecular orbitals and different types of bonds. Therefore, the processes of life are controlled by quantum mechanics.
CARBON DIOXIDE IS A GREENHOUSE GAS BECAUSE OF QUANTUM EFFECTS
We have seen that the greenhouse effect produced by carbon dioxide, which is driving global climate change, is fundamentally quantum mechanical in nature. Carbon dioxide is a perfect storm of quantum effects that produces a vicious greenhouse gas. Hot objects give off light that we call black body radiation. The colors given off by a hot object could not be explained with classical theory. In fact, the classical theory was so far off that it was called the Ultraviolet Catastrophe because the theory predicted that an infinite amount of energy would be given off in the ultraviolet portion of the spectrum by any hot object. Clearly a hot object does not and cannot give off an infinite amount of energy. So the failing of classical theory was monumental. In 1900 Planck made the first use of the idea of quantized energy levels for electrons in matter to explain black body radiation. He was able to derive a formula for the colors given off by a hot object that matched experiments virtually perfectly. The hotter an object is, the more high-energy photons it emits. However, Planck’s quantum theory showed that the amount of energy is not infinite and told us precisely how much light is emitted at each color. Stars are very hot, so they give off light in the visible and ultraviolet portions of the spectrum. An example of the black body spectrum for our Sun is shown in Figure 9.1. The Sun is a medium hot star. It appears slightly yellow. Very hot stars are blue and cooler stars than our Sun are red.
The Earth also emits black body radiation, but because it is very cool compared to a star, we cannot see the light the Earth emits with our eyes. The Earth’s black body spectrum is shown in Figure 17.1. The light emitted by the Earth is in the infrared, that is, the long wavelength (the low energy) part of the spectrum. Without the atmosphere, all of the black body radiation emitted by the Earth would go into space, and our world would be much colder, perhaps too cold for human life. However, the atmosphere absorbs some of the black body radiation, trapping the heat, and keeping the Earth warm. Most of the heat trapping is done by water, which has quantized rotational energy levels that can undergo transitions in the very far infrared (long wavelengths and low energy). We have not mentioned quantized rotations before. Here is where your quantum intuition comes into play. We have talked about quantized electronic energy levels and quantized vibrational energy levels. Classical objects can rotate, for example, a top. The classical energy associated with rotation is continuous. Spin the top a little bit faster, and the energy is increased a little bit. It should come as no surprise that molecules in the gas phase, such as water vapor in the air, can rotate, and because a water molecule is absolutely small, the rotational energy is quantized. Rotational energy can only change in discreet steps. A water molecule can rotate at one speed and then have a step to another speed, but it cannot rotate at speeds in between. Think of what this would mean if it applied to big classical systems. You are riding you bike. You can pedal at one speed, but you can’t pedal just a little bit faster. You would have to take a discreet step to the next quantized rotational energy level. Of course this doesn’t happen for absolutely large objects for which energy is continuous.
Water does not absorb the Earth’s black body radiation at the peak of the black body spectrum where a lot of energy is emitted. However, carbon dioxide does. As we discussed in Chapter 17, molecules have quantized vibrational energy levels. Carbon dioxide, CO2, is made up of three atoms, with the carbon in the center. It is a linear molecule that can undergo bending vibrations. The vibrational motion has quantized energy levels. By happenstance, the energy difference between two CO2 bending vibrational energy levels falls at the energy of the peak of the Earth’s black body light emission. Therefore, the CO2 molecules in the air absorb a significant part of the Earth’s radiated black body energy that would otherwise go into space. The more CO2 in the air, the less radiated energy escapes the Earth’s atmosphere. The result is that as the amount of CO2 in the air increases, more and more of the Earth’s heat is trapped in the atmosphere, and the planet warms. CO2 is a greenhouse gas because of the two quantum phenomena, black body radiation and quantized vibrational energy levels.
VERY HOT OBJECTS GIVE OFF VISIBLE BLACK BODY RADIATION
While we are on black body radiation, now you know that whenever you see something glowing red, like molten lava coming out of a volcano or the hot element in an electric stove, you are seeing black body radiation. When you turn an electric stove on low, the temperature is low enough that all of the black body radiation is emitted in the infrared, and you can’t see it with your eyes. If you used a spectrometer and an infrared detector, you could measure the IR colors emitted. The spectrum of the IR black body radiation from the stove element would tell you the temperature. When you turn the stove to high, the element turns red because it is much hotter. Most of the black body radiation is still in the IR, but the high-energy portion of the black body spectrum is in the low-energy portion of the visible spectrum, that is, it is red.
ELECTRICAL HEATING IS A QUANTUM PHENOMENON
But why does the stove element get hot at all when electricity is passed through it? In spite of the fact that the stove element itself is a large object, we saw in Chapter 19 that electrical conductivity and electrical resistance, which produces the heat, are consequences of fundamental quantum effects. Metal crystals, such as sodium or copper, have electrons in atomic orbitals that interact with each other. The atomic orbitals from the atoms in the entire crystal combine to make molecular orbitals that spatially span the crystal. Like the aromatic molecule benzene that has six electrons in six delocalized molecular orbitals formed from interacting carbon p orbitals (Chapter 18), the electrons are not associated with a particular atom or pair of atoms. Rather, the MOs span the system, and the electrons are free to roam about the entire system, that is, a benzene molecule or a metal crystal. For benzene, six interacting atomic orbitals give rise to six molecular orbitals that are delocalized over the molecule. In benzene, with only six MOs, the energy spacing between the MOs is large. In even a very small metal crystal, there are billions and billions of atoms that give rise to billions and billions of MOs. Because there are so many MOs, they are very closely spaced. In a metal, all of these MOs form a band of quantum energy states called the conduction band. Each of these MOs is spread over the entire crystal. However, we know that these energy quantum states, the energy eigenstates, can be superimposed to form electron wave packets that are more or less localized in a manner consistent with the Heisenberg Uncertainty Principle. These electron wave packets are virtually free to move about the crystal.
Electrons are negatively charged. When a battery or other electrical source is connected to a piece of metal, for example a piece of copper wire, the electrons will be drawn to the positive battery electrode and move away from the negative battery electrode. The electrons are accelerated toward the positive side of the battery, which increases their kinetic energy. However, the electrons are not the only types of wave packets moving about in a metallic crystal. The mechanical vibrations of atoms in a crystal lattice have quantized energy levels. As with the electron bands in a macroscopic piece of metal, because there are so many atoms, there are a vast number of quantized vibrational levels that form bands of mechanical energy levels. The quantized delocalized mechanical motions of the coupled atoms in a lattice are called phonons. These delocalized phonon waves combine to form phonon wave packets. These phonon wave packets propagate through the lattice.
Electron wave packets and phonon wave packets collide. The collision is called an electron-phonon scattering event, illustrated in Figure 19.7. Some of the extra kinetic energy that the electron picked up because of its acceleration by the electric field is transferred to the phonon. The electron has less energy and the phonon has more energy after the scattering event. Many such electron-phonon scattering events cause the bath of phonons to have increased energy. Mechanical energy is heat. Temperature is a measure of the amount of kinetic energy in a substance. So the electron-phonon scattering events slow the electrons, which is what we call electrical resistance. The increase in phonon energy increases the temperature of the metal; it gets hot. The heating of a piece of wire as electricity (electrons) flows through it is caused by the collisions of electron and phonon wave packets. The scattering of these two types of wave packets is an intrinsically quantum mechanical phenomenon. The more electricity (current) that flows through the metal, the more collisions occur, and the hotter the metal gets. This is what happens when you turn up an electric stove. You are increasing the current (number of electrons flowing), and therefore, you are increasing the number of electron-phonon scattering events. Increasing the number of electron-phonon scattering events increases the energy transformed into heat, which causes the temperature to rise. When the metal stove element gets hot enough, it glows read because the black body emission has moved into the visible part of the spectrum. The net result is that turning on an electric stove or electric space heater, and seeing the heating element emit red light, involves a number of quantum phenomena. So every time you see a red hot stove element, rather than it being a complete mystery like the moon to a toddler, think quantized electron states, electron wave packets, phonon wave packets, electron-phonon scattering generating heat, and finally black body radiation. Every day observations are filled with quantum phenomena.
All of the quantum physical phenomena that surround us come ultimately from the fact that size is absolute, and absolutely small particles just don’t behave the way classical objects, that is, absolutely big objects, behave. Baseballs are classical particles. Sound waves are classical waves. Baseballs and sound waves are big. In classical mechanics, the theory of big things, we have waves and particles. We discussed that light comes in discreet packets called photons. The description of photons and electrons as wave packets is profoundly different from anything in classical mechanics. Absolutely small particles, such as photons and electrons, are neither waves nor particles, as we saw in Chapters 4 through 7. They are wave packets. Sometimes they behave as waves (light diffraction from a grating and electron diffraction from a crystal surface) and sometimes they behave as particles (photons in the photoelectric effect and electrons in the cathode ray tube of old-style televisions). In fact, the essence of the nature of absolutely small particles is that they are really neither particles nor waves but a strange type of entity that has simultaneously particle and wave properties. This duality of matter is contained in the Heisenberg Uncertainty Principle. In contrast to a classical object like a baseball, we cannot know exactly simultaneously the position and the momentum (mass times velocity) of an electron or other absolutely small particles.
When is a particle absolutely small and subject to the new world of quantum physics? Dirac taught us that there is a minimum disturbance that accompanies a measurement, a disturbance that is inherent in the nature of things and can never be overcome by improved experimental technique. If the disturbance is negligible, then the object is large in an absolute sense, and it can be described by class physics. However, if the minimum disturbance accompanying a measurement is nonnegligible, then the object is absolutely small, and its properties fall in the realm of quantum mechanics. The quantum properties of absolutely small particles are not strange; they are just unfamiliar and not subject to our classical intuition. They are like the moon to the toddler. In this book the fundamental concepts of quantum theory have been explicated and applied to some important everyday phenomena. You are no longer a quantum toddler.