The Amazing Story of Quantum Mechanics

Chapter 9) broke. Dr. Solar, whose powers are normally energized by exposure to radiation, went underwater to fix the reactor but found himself weakened by the reactor"s radioactivity (through a process not clearly explained in the comic). Eventually he was rescued by a worker wearing a lead-lined safety suit, who would have done the job in the first place if Dr. Solar hadn"t attempted to "save the day." The additional radiation he absorbed from the reactor temporarily endowed Solar with a new superpower. As ill.u.s.trated in Figure 27, whenever Dr. Solar comes into physical contact with an object, he trans.m.u.tes it into the next element up the periodic table. In Figure 27, he transforms gold, with seventy-nine protons, into mercury, with eighty protons; earlier he grasps a copper rod (twenty-nine protons) and converts it into zinc (atomic number 30); and even when flying he begins to choke when the oxygen (atomic number 8) turns into fluorine gas (with nine protons). This newfound power of Dr. Solar"s appears to be the abil-ity to initiate beta decay of the neutrons in any object he touches, inducing elemental trans.m.u.tation via the weak nuclear force, an aspect of Watchmen"s "intrinsic field" that we have not discussed much yet.

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Before there was physicist Jon Osterman, there was physicist Philip Solar. In the 1986 DC comic book Watchmen, Osterman was disintegrated by the accidental removal of his intrinsic field at Gila Flats and reconstructed himself as the superpowered Dr. Manhattan. In the 1962 Gold Key comic book series Solar-Man of the Atom, Phillip Solar was exposed to a lethal dose of radiation in a sabotaged nuclear research experiment at Atom City, yet survived, though he acquired "quantum powers." In issue # 2 he is vaporized by an atomic bomb blast but manages through sheer force of will to reconst.i.tute himself as, well, Dr. Solar, which was his name after all. As a survivor of graduate school myself, I can empathize with Osterman and Solar"s inclination to retain the t.i.tle a.s.sociated with their Ph.D."s, in the lab or as a superpowered hero. Once you"ve pa.s.sed through the crucible of a graduate school candidacy exam, having to rea.s.semble yourself up from the subatomic level is not as challenging as you might think.

In writer Alan Moore"s initial outline of the DC comic book miniseries Watchmen, he intended to use comic book characters created by Charlton, another comic book publisher. Charlton had declared bankruptcy, and the company had been acquired by DC Comics, home of Superman and Batman. Moore"s initial outline for Watchmen made direct use of the Charlton characters, but the editors at DC Comics, seeing that some of these characters would not make it out of the miniseries unscathed, instructed Moore to 8 instead employ alternate versions of the Charlton heroes. Dr. Manhattan is the a.n.a.log of Captain Atom, an air force captain, who was disintegrated and (I"m sure you can see this coming at this stage) through force of will was able rea.s.semble himself into a quantum-powered superbeing.

Captain Atom"s powers were quantum based only in that he was able to manipulate energy, which he employed primarily for flight, superstrength, and energy blasts. Dr. Solar, though not a Charlton character, seems to be a closer antecedent for Dr. Manhattan, as Solar was also able to change size (Dr. Solar # 10 and # 11), split himself into multiple copies of himself (Dr. Solar # 12), and manipulate matter and energy, though unlike the blue Dr. Manhattan, Dr. Solar"s skin turned green when he used his powers. There are just enough differences among Captain Atom, Dr. Solar, and Dr. Manhattan that it is unlikely that they are all the same person, on three different versions of Hugh Everett"s many worlds, though further study appears warranted.

One of the more accurate manifestations of quantum mechanical powers was presented in "Solar"s Midas Touch," in 1965"s Dr. Solar, Man of the Atom # 14. In this tale an underwater nuclear reactor pile went critical when one of the control rods (whose role is to absorb neutrons, decreasing the rate of uranium fission, as described in Chapter 9) broke. Dr. Solar, whose powers are normally energized by exposure to radiation, went underwater to fix the reactor but found himself weakened by the reactor"s radioactivity (through a process not clearly explained in the comic). Eventually he was rescued by a worker wearing a lead-lined safety suit, who would have done the job in the first place if Dr. Solar hadn"t attempted to "save the day." The additional radiation he absorbed from the reactor temporarily endowed Solar with a new superpower. As ill.u.s.trated in Figure 27, whenever Dr. Solar comes into physical contact with an object, he trans.m.u.tes it into the next element up the periodic table. In Figure 27, he transforms gold, with seventy-nine protons, into mercury, with eighty protons; earlier he grasps a copper rod (twenty-nine protons) and converts it into zinc (atomic number 30); and even when flying he begins to choke when the oxygen (atomic number 8) turns into fluorine gas (with nine protons). This newfound power of Dr. Solar"s appears to be the abil-ity to initiate beta decay of the neutrons in any object he touches, inducing elemental trans.m.u.tation via the weak nuclear force, an aspect of Watchmen"s "intrinsic field" that we have not discussed much yet.

Figure 27: In Dr. Solar, Man of the Atom # 14, an additional nuclear accident endows Dr. Philip Solar (wearing the scuba suit and visor) with the temporary ability to induce beta decay via the weak nuclear force in any object he comes into direct contact with, thus trans.m.u.ting gold into mercury.

We saw in Chapter 9 that neutrons, through the strong force, hold the nucleus together by binding with protons and other neutrons and overwhelming the electrostatic repulsion that would, in their absence, cause the protons to fly out of the nucleus. Protons also exhibit the strong force, but without neutrons there is not sufficient binding energy to hold together a nucleus consisting only of protons. Neutrons themselves are not stable outside of a nucleus. A neutron sitting alone in the lab will decay into a proton and an electron with a half-life of about ten and one quarter minutes. The electron will be moving very near the speed of light, and when this process occurs within a nucleus, it is the source of the beta rays emitted from unstable isotopes.

As the total ma.s.s and energy of an isolated system must remain unchanged in any process, a "stationary" neutron43 can decay only to fundamental particles with less ma.s.s than the neutron"s. A neutron will thus decay into a proton, which has a slightly smaller ma.s.s, while a "stationary" proton could not decay into a heavier neutron. However, as the neutron is electrically neutral, and the proton is positively charged, the decay must also generate a negatively charged electron, in order for the total electrical charge to remain unchanged before and after the decay (we have not needed to invoke this principle before now, but another conservation principle in physics, comparable to conservation of energy or conservation of angular momentum, is conservation of charge, in that the net electrical charge can be neither created nor destroyed in any process). An electron is nearly two thousand times lighter than a proton, less than the ma.s.s difference between neutrons and protons, so adding an electron to the decay is still consistent with ma.s.s conservation. While a neutron decaying into a proton and an electron means that ma.s.s and electrical charge are balanced during the decay, examination of the kinetic energy of the proton and the high-speed electron (that is, the beta ray) and comparison to the rest-ma.s.s energy of the neutron indicates that some energy went missing in the process-not a lot, but enough to notice, and enough to cause trouble.

When physicists in the late 1920s discovered this phenomenon and realized that it appeared to violate the principle of conservation of energy, they were faced with two choices: (1) either abandon conservation of energy, at least for neutron decay processes, or (2) invent a miracle particle that was undetectable by instrumentation of the time but that carried off the missing energy. In 1930, Wolfgang Pauli (whose exclusion principle I address in the next section) suggested going with option 2. Knowing that this ghost particle had to be electrically neutral and had to have very little or no ma.s.s, Enrico Fermi called it the "little neutral one" in Italian, or "neutrino."44 Detectors were eventually constructed to observe these particles, and their existence was confirmed in 1956. These particles not only really exist, aside from photons they are the most common particle in the universe. Their interactions with matter are governed by the weak nuclear force, which is one hundred billion times weaker than electromagnetism (the force by which electrons interact with matter). Neutrinos consequently barely notice normal matter (it takes more than two light years of lead-that is, a length of more than ten trillion miles-to stop one). If you hold out your thumb and blink, during that time period more than a billion neutrinos will pa.s.s through your thumbnail.

Dr. Solar, after his radiation overdose, must have gained an uncontrolled ability to induce beta decay in any object with which he came into contact. If a gold atom, with seventy-nine protons, seventy-nine electrons, and 118 neutrons, has one of its neutrons spontaneously decay into a proton and an electron, then it will have eighty protons, eighty electrons, and 117 neutrons. The lightest, stable configuration of mercury has eighty protons, eighty electrons, and 118 neutrons, so Dr. Solar will have created an unstable isotope of mercury in Figure 27. The half-life of this isotope of mercury with 117 neutrons is roughly two and a half days, so there will be time for Solar to finish his adventure and try to restore the trans.m.u.ted mercury back to its original golden state. While transforming one element into its periodic-table neighbor via neutron beta decay is not quite the alchemist"s dream of trans.m.u.ting lead into gold (normal beta decay would convert platinum, with seventy-eight protons, into gold, with seventy-nine protons, so, depending on world exchange prices, you may wind up losing money on the deal), a process known as "reverse beta decay" would turn mercury into gold. While we cannot initiate such a conversion on Earth at will, fortunately this inverse process occurs constantly in the center of the sun, keeping the sun shining and providing the basis of all life.

The light from the sun-which is transformed by photosynthesis into chemical energy stored within plants, which in turn provides us with the energy we need to maintain our metabolisms-originates from nuclear transformations in the star"s core. Four protons, that is, hydrogen nuclei, subjected to the extreme pressures and temperatures at the center of the sun, are fused together to form helium nuclei. But a helium nucleus consists of two protons and two neutrons, not four protons. Recall that neutrons are necessary as mediators of the strong nuclear force that holds the nucleus together. Thus, to make helium out of hydrogen, you first have to combine two protons and then through reverse beta decay convert one of the protons into a neutron.

I argued above that a single proton cannot convert into a neutron, as the ma.s.s of the proton is less than that of the neutron, and lighter objects cannot decay into heavier products. If two protons collide, the weak force operates on the protons, turning one into a neutron through reverse beta decay, as ill.u.s.trated in Figure 28. The proton and neutron, subject to the strong force, become bound (now a deuterium nucleus-an isotope of hydrogen) and lower their en-ergy compared to an isolated proton and neutron. This lower energy is reflected in a smaller ma.s.s for the deuterium nucleus, relative to a free proton and neutron. While the ma.s.s difference is very small, through E = mc2 the energy difference of the bound deuterium is significant, and it emits a 2.225-million-electron-Volt gamma-ray photon during formation. In addition to the neutron generated by the weak force, the reaction creating a deuterium nucleus yields an antimatter electron (which has a positive electrical charge like a proton, but the ma.s.s of an electron) and a neutrino.

Figure 28: Sketch of the nuclear reactions in the center of the sun by which protons (hydrogen nuclei) combine to form alpha particles (helium nuclei). In step (a), two protons (represented by open circles) tunnel together, where the weak force converts one proton (open circle) into a neutron (dark circle). The proton and neutron then form a bound deuterium nucleus, with the release of a gamma ray photon (the positron and neutrino released are not shown for simplicity). The deuterium can then collide with another proton in step (b) and form a bound proton-proton-neutron nucleus, termed helium-3. In step (c) we indicate a possible reaction where two helium-3 nuclei collide and form a stable helium-4 nucleus (two protons and two neutrons), with the release of two protons and another gamma ray. Similar mechanisms result in the fusion of helium nuclei to synthesize heavier elements, such as carbon and oxygen, and up.

The weak force extends over a length scale roughly one thousand times smaller than that of the strong force, which itself acts only over distances less than the diameter of a nucleus. Two protons, both being positively charged, repel each other, and the closer they are, the greater the repulsive force. So one must force the two protons very close together, overcoming their electrical repulsion, in order for there to be an opportunity for the weak force to transform, through reverse beta decay, one of the protons into a neutron. The temperatures and pressures in the center of the sun are enormous, so that there are many opportunities for high-velocity collisions between two protons. However, even at the center of the sun the proton speeds are not sufficient to overcome the electrical repulsion when they draw too close. How do they manage to get past this electrical barrier? Through quantum mechanical tunneling!45 Just as the alpha particles in radioactive decay use tunneling to escape the strong-force barrier around the nucleus that keeps the protons and neutrons together, the two protons that join together, forming the simplest isotope of hydrogen, must tunnel to overcome the barrier of their mutual repulsion.

The deuterium nucleus created in the center of the sun is stable and continues to collide with other protons. Combining this deuterium with another proton forms a nucleus with two protons (that is, helium) but only one neutron (making it helium 3, a lighter isotope of helium). Here again quantum-mechanical tunneling is required to get the second proton close enough to the deuterium nucleus, overcoming the electrical proton-proton repulsion, for the strong force to hold the second proton in the now larger nucleus. The lower energy of this bound state results in the release of another gamma-ray photon. This reaction is much more likely than for two deuterium nuclei to combine to form normal helium (two protons and two neutrons).

There are then many different ways that the helium 3 or deuterium nuclei can interact to form a stable helium nucleus, all of which involve quantum mechanical tunneling to get the positively charged nuclei close enough for the strong force to operate, resulting in the release of a great deal of energy in the form of kinetic energy of the nuclei, gamma rays, and neutrinos. The neutrinos pa.s.s right through the sun and head off in all directions, while the gammas heat up the nuclei and electrons in the center, accelerating them and causing them to emit electromagnetic radiation at all wavelengths. The light created in the center of the sun is scattered many, many times before reaching the surface, where it then takes the brief, eight-and-a-half-minute journey to Earth. Before reaching the surface, the average photon spends forty thousand years colliding with the dense nuclear matter in the sun"s interior. The outward energy pressure counteracts the inward gravitational pull and keeps the diameter of the sun fairly stable.

In addition to providing us with energy, this fusion process is the mechanism by which elements heavier than helium are synthesized. Our sun is actually a second-generation star that formed after a much larger star pa.s.sed through its life cycle and "went supernova." Our sun converts a great deal of hydrogen as it generates energy-approximately six hundred million tons per second. But eventually stars exhaust their supply of hydrogen, and the star collapses until the temperature and pressure rise to the point where helium nuclei begin to fuse, forming carbon. The process continues, generating nitrogen, oxygen, silicon, and other heavy elements up the periodic table to iron and nickel. However, the larger the nucleus created, the less energy is released per reactant, and at the iron/nickel point, the outward flow of energy is insufficient to counteract the inward gravitational pull. At this stage the star collapses onto itself; in the process, all elements heavier than iron are created, and there is an explosive outpouring of energy as the star becomes a supernova, releasing as much energy in a period of several weeks as our sun does over its entire lifetime. It is from the elements synthesized in a much larger star that lived and underwent a violent demise that the planets and sun of our solar system formed.

The power of the atomic bomb results from the breaking apart of large nuclei, such as uranium or plutonium, in a fission process, described in Chapter 9. Current nuclear power plants, such as the one that went critical and injured Dr. Solar at the start of this chapter, are fission reactors. They require rare radioactive isotopes as fuel, and their by-products are unstable isotopes, which are themselves radioactive and harmful to people. After the atomic bomb, the hydrogen bomb was developed. This weapon utilizes a fission reaction to initiate a fusion reaction-the energy of an atomic bomb is employed to force heavy isotopes of hydrogen and helium to fuse and release even more energy. For more than fifty years, scientists have been attempting to construct a fusion reactor that could create energy for electricity production, harnessing the power of the hydrogen bomb and the sun for peaceful, controlled terrestrial needs. The required fuel for a fusion reactor involves isotopes of hydrogen (typically deuterium and tritium), which may be harvested from naturally occurring isotopes of seawater, and the reaction products are nonradioactive. The obstacle is to replicate, in a controlled manner, the temperatures and pressures at the center of the sun. While the engineering challenges have indeed been formidable, a consortium of nations including Europe, Russia, j.a.pan, and the United States are constructing a pilot fusion power plant (the International Thermonuclear Experimental Reactor, or ITER) to examine the feasibility of using nuclear fusion for electricity generation.

Back in the late 1980s there was a brief flurry of interest in reports that nuclear fusion had been achieved in a small tabletop experiment involving the electrolysis of heavy water using a palladium electrode. This so-called cold fusion process proposed that the deuterium nuclei, embedded within the metal electrode, were undergoing fusion and creating helium nuclei, with a concurrent release of excess heat. Whatever was going on in their device, it was not nuclear fusion, and it"s a good thing for the chemists involved in this project that they were not in fact generating fusion reactions. A by-product of this particular fusion reaction is high-energy neutrons that would have killed anyone unlucky enough to be in the lab at the time. Moreover, as discussed earlier, fusion reactions within the center of the sun, at temperatures of millions of degrees, require quantum mechanical tunneling for the protons to overcome their electrical repulsion. Fusion at room temperature in a palladium electrode is even more dependent on tunneling to proceed. A well-established feature of quantum mechanics is that the tunneling probability is very sensitive to the ma.s.s of the object involved. The smaller the ma.s.s, the lower the momentum and the longer the de Broglie wavelength, which can extend farther through the forbidden region, increasing the probability of finding the object on the other side of the barrier. Yet the initial investigators of "cold fusion" found no difference whether they used heavy water or ordinary tap water, whereas the difference in ma.s.s should have had a large effect on the fusion process.

For cold fusion to be a real phenomenon, it would require a suspension or violation of the principles of quantum mechanics, which underlies our understanding of solid-state physics, lasers, transistors, and all of the personal electronic devices they enable. Nevertheless, one might be tempted to give these up, if we could make cold fusion a physical reality. After all, a small cylinder capable of generating the power of the sun would make an awesome power supply for a jet pack!

SECTION 4.

WEIRD SCIENCE STORIES.

CHAPTER TWELVE.

Every Man for Himself.

The agreement between theoretical predictions of atomic properties using quantum mechanics, such as the wavelengths of light emitted when an excited hydrogen atom relaxes back to its ground state, and experimental measurements of these wavelengths is nothing short of amazing. But if that were all that quantum mechanics could do, it most certainly would not have "made the future." We would still be living in the "vacuum-tube age" and would not have laptop computers, cell phones, DVDs, or magnetic resonance imaging devices.

The quantum descriptions of Schrodinger and Heisenberg accurately account for the properties of a single atom, but very rarely does one encounter an isolated, single hydrogen atom, or any type of atom or molecule by itself. A typical cubic centimeter of a liquid or solid, about the size of a sugar cube, contains roughly a trillion trillion atoms. The power of quantum mechanics is that it also provides an understanding of the properties of these trillion trillion atoms and accounts for why some materials are metals, some are insulators, and others are semiconductors. Fortunately for us, it turns out that if one understands the behavior when two ent.i.ties are brought close enough to each other that their Schrodinger wave functions overlap, then this tells us nearly all we need to know to understand the results of a trillion trillion ent.i.ties in close quarters.

Up to now, we have made extensive use of the first two quantum principles listed in Section 1: that light consists of discrete packets of energy termed photons, and that there is a wave a.s.sociated with the motion of all matter. We have not needed to invoke the third principle: that all matter and light has an internal rotation that corresponds to discrete values of angular momentum. We would have needed this principle in order to understand details of how the electronic energy states in an atom are arranged, but for our purposes there was no call to head into this set of weeds. However, we cannot avoid a certain amount of weediness now, not if we wish to understand the basis for the semiconductor age and the foundations of the upcoming nanotechnology revolution.

In Chapter 4 we discussed the intrinsic angular momentum that all subatomic particles possess, termed spin. a.s.sociated with this internal spin is a magnetic field, so that every electron, proton, and neutron can also be considered a tiny bar magnet, with a north and a south pole (Figure 10b). While the concept of spin was introduced to account for experimental observations indicating that electrons possessed a built-in magnetic field, one cannot ascribe this magnetic field to a literal, physical rotation of the subatomic particles as if they were ballerinas. It is indeed confusing to imagine an intrinsic angular momentum, as integral to the properties of the electron and as real as its charge or ma.s.s, that is not a.s.sociated with a literal rotation. Nevertheless, spin is the term that has stuck, and we adhere to this nomenclature, as we are nothing if not slaves to convention.

As mentioned in Chapter 4, the intrinsic angular momentum of electrons is exactly /2 (recall that is defined as h/2).46 The "spinning" electron can have an intrinsic angular momentum of either + /2 or -/2, just as a real spinning ballerina can twirl either clockwise or counterclockwise. No other intrinsic angular momentum values are possible for electrons (or protons or neutrons). The collective behavior of quantum particles that have a spin of /2 was first worked out by Enrico Fermi and Paul Dirac in the 1920s. In honor of their contribution, physicists refer to all spin /2 particles as obeying "Fermi-Dirac statistics," or by the shorter nickname of fermions. (Fermi got the sweet part of this deal-the fact that electrons are spin /2 particles, and thus are fermions, has led to a host of quant.i.ties in solid-state physics as being labeled with his name-Fermi Energies, Fermi Surfaces, and so on-even though he made few direct contributions to this field of physics.) Consider two fermions, such as electrons. It really is true that all electrons look alike. This is not the prejudiced opinion of an anti-Fermite, but a reflection of the fact that all fundamental particles of a given type are identical. There is no way to distinguish or differentiate between electrons, for example. Similarly, all protons are identical, as are all neutrons. These three subatomic particles have different ma.s.ses and electrical charges, so they can be distinguished from one another. But if we bring two electrons so close to each other that their de Broglie waves overlap, then no observable property can possibly depend on which electron is which.

If I toss a rock into a pond, a series of concentric circular ripples forms (Figure 29a). When I toss two rocks into the water a small distance apart, each forms its own set of ripples, and the combined effect is a complicated interference pattern (Figure 29b). At some points the ripples from each rock add up coherently and create a larger disturbance on the water"s surface than generated by each rock separately. At other locations the two ripples are exactly out of phase, so that one ripple is at a peak while the other stone"s wave is at a trough, and the two exactly cancel each other out. Taken together, the resulting pattern is more than just a doubling of the result of one stone"s concentric ripples.

All objects have a quantum mechanical wave function. When two electrons are brought together such that their wave functions intersect, then they are described by a two-electron wave function. In the case of two rocks tossed into the pond, if the stones are identical and both are tossed into the water in the exact same way, then the interference pattern that is observed does not depend on which rock was tossed on the left and which on the right. Similarly, in atomic physics, nothing that we can measure, such as the wavelength of light emitted from transitions between quantized energy states, can depend on any artificial labeling of the electrons. In the case of the stones in the water, they are indeed distinguishable, for we can refer to the stone on the left and the stone on the right in a meaningful way. Heisenberg tells us that it is fruitless to try to specify the location of the electron more precisely than the extent of its de Broglie wave. When two de Broglie waves overlap, concepts such as "left" and "right" become irrelevant, and all we have is the composite two-electron wave function.

Figure 29: Cartoon of the wave patterns observed on the surface of a pond when one rock is tossed into the water (a), and when two rocks are simultaneously tossed, near but not touching each other (b).

Say I have two electrons, which I will creatively call electron 1 and electron 2. I bring them together so that their wave functions intersect. The electrons are indistinguishable, and no measurements can depend on which one is labeled "electron 1" and which one is "electron 2." Are there any differences at all between them at this point? Indeed yes! The two electrons, 1 and 2, have identical electrical charges and identical ma.s.ses, but they can have different intrinsic angular momentum. Both electron 1 and electron 2 can have spin values of +/2, or both could have a spin value of -/2, or one could have a spin of +/2 while the other has spin of -/2. These different values of spin will be crucial for understanding solid-state physics.

Think about a ribbon, one side of which is black and the other of which is white. The ribbon represents a single electron, and if I hold the ribbon so that the white side is facing you, it indicates that the electron"s spin is +/2, while if the black side is shown this means the spin is -/2. Now, if I hold two ribbons far away from each other, I can easily distinguish them-one is on the right and the other is on the left. Bring them so close that their waves overlap and I can no longer tell them apart. In this case I can describe them both with a single, longer ribbon. I can still represent the case where one electron has spin of +/2 and the other has spin of -/2, by having my right hand hold the ribbon with the white side facing out and my left hand hold the ribbon"s black side facing out. Figure 30 shows a ribbon where both ends have the white side facing out, indicating that both electrons have a spin of +/2. The arguments presented in this figure are a modification of those made by David Finkelstein, as described in Richard Feynman"s essay "The Reason for Antiparticles." I need hardly stress that the "ribbon" is simply a metaphor that will, I hope, a.s.sist in the visualization of a two-particle wave function, and is not intended as a literal representation.

The ribbon in Figure 30 represents a two-electron wavefunction with both electrons having a spin of +/2. The fact that the two electrons are so close that they are described by a single wave function is represented by the fact that I use one ribbon for both electrons. Any change to one electron is thus communicated to the other. What if I switch their positions, so that I move the right-hand side to the left and the left pa.s.ses to the right? If I do this-without letting go of either end of the ribbon-then by switching their locations, I will add a half twist to the ribbon (Figure 30b). This is not the same situation I started with-as the ribbon has a half twist that it did not have before switching their positions. One can tell from inspection of the ribbon that a swap from left to right has occurred.

And that"s it. That"s the heart of Fermi-Dirac statistics, which governs the way electrons interact with one another and is the basis of the periodic table of the elements, chemistry, and solid-state physics.

How do I mathematically combine the wave functions for two electrons so that switching their order changes the situation, but making another swap restores the original state? Easy: Let the two-electron wave function be described as the difference of two functions, A and B, that is, = A - B, where A and B each depend on the one-electron wave functions at positions 1 and 2.47 As in switching the two ends of our metaphoric ribbon, let"s move the electron that was at one position to the location of the other electron, and conversely. In this case the wave function would be written as = B - A. The process of switching the positions of the two electrons is the same as multiplying the original two-electron wave function by (-1). If I want to get back to the original configuration, I do another switch, which brings me to = A-Bagain.

Figure 30: Cartoon sketch of a ribbon with different colors on each side, where the ribbon is presented so that each end displays the same side (the white side in this case). Switching the two ends results in a half-twist in the ribbon. Only another rotation creates a full twist in the ribbon that can now be removed by flipping one side of the ribbon twice.

Nothing that I can measure should depend on which electron I label at position 1 and which one is at position 2. Now, there"s no problem with having a two-electron wave function written as A - B. The fact that switching the positions is the same as multiplying the wave function by -1 will not affect any measurement we make. Remember that while the wave function contains all the information about the quantum mechanical system, it is the wave function squared 2 that gives us the probability of finding the object at some point in s.p.a.ce and time.48 It is also the wave function squared 2 that is used in calculating the average position (we add up all possible positions when multiplied by the probability-2-of the electron being at that position), the average momentum, and so on. And since the square of negative one is (-1)2 = (-1) (-1) = +1, then = A-Bis a physically valid way to represent the two spin /2 electrons. In Chapter 8, Dr. Manhattan"s ability to change his size at will was ascribed to the fact that the Schrodinger equation is linear. This really becomes important here. Only for a linear equation will it be true that if A or B are separately solutions, then = A-B(or = A + B, discussed in the next chapter), will also be a valid solution of the Schrodinger equation.

Right off the bat there"s a big consequence of writing the two-electron wave function as = A - B. What happens if I try to make both electrons be at the same location, or have both electrons in the same quantum state (when they are close enough to overlap and are described by a single two-electron wave function), so that the function A is equal to the function B? Then the two-electron wave function would be = A-B= 0 when A = B. If = 0, then the square of the wave functionx= 2 = 0 as well. Physically, this means that the probability of finding two electrons at the same place in the exact same quantum state is zero-as in, this will never happen. Recall in Chapter 8 our discussion of quantum mechanical tunneling. In a tunneling situation an electron in one metal, separated by the vacuum of empty s.p.a.ce from another metal and not having sufficient energy to arc or jump from one metal to another, may nonetheless find itself in the second material. We pointed out that even though the probability for the electron to be outside of metal may be very small, as in one chance out of a trillion, there was still some chance of finding the electron in the second metal. The only time something will never be observed is if the probability of it happening is exactly zero. If something can never be observed, in physics we say that it is forbidden.

Right away, from the fact that electrons have an intrinsic angular momentum of /2, we can understand the structure of the periodic table of the elements. In Chapter 6 we discussed the solutions to Schrodinger"s equation when the potential V is that of the electrical attraction between the negatively charged electrons and the positively charged nucleus. Schrodinger found that there were a series of possible solutions corresponding to different energy states that we argued were not unlike a series of rows of seats in a cla.s.sroom, sketched in Figure 15. Some seats are close to the front of the cla.s.sroom, while there are other rows farther from the front of the room. The configuration of the rows of seats depends only on the attractive force between the positive nucleus and the negative electron. We now understand why all the electrons in an atom don"t just pile up in the chair in the front row, which is the lowest-energy quantum state available. For if they were to do that, then all of the electrons would be in the same location in the same quantum state, and as we have just shown, the probability of that happening is zero.

There"s a fancy term used to describe the fact that no two electrons can ever be in the same position in the same quantum state-the Pauli exclusion principle. Wolfgang Pauli, one of the founding fathers of quantum mechanics, postulated this principle in 1925 in order to account for the configuration of electrons in elements. Hydrogen with one electron has the lowest energy state occupied, shown in Figure 31a. As there is only one electron in this element, it is exempt from the exclusion principle. The next element in the periodic table is helium, with two electrons. We now extend this physical a.n.a.logy and propose that each "seat" in the auditorium is actually a "love seat" that can accommodate two electrons, provided that they face away from each other (that is, as long as one is spin "up" and the other is spin "down."49 As in Figure 31b, both of these electrons can reside in the lowest energy state, as long as one has a spin value of +/2 and the other has a spin of -/2, since each spin state counts as a different quantum state. As there are no other possible spin values, a third electron in lithium (the next element up the table, shown in Figure 31c) will have to reside in the next higher energy state. If all three electrons were to reside in the lowest energy state in lithium, then there would be at least two electrons both with spin = +/2 or spin = -/2, and the probability of this occurring is 2 = 0. Carbon, shown in Figure 31d, has six electrons-two sit in the ground state, and the remaining four sit in the next highest "row of seats"-and is able to form chemical bonds in a wide variety of ways. By forming these bonds, the carbon atom and the other atoms it chemically interacts with lower their energies, compared to their unbonded states. If all of carbon"s six electrons could drop down into the lowest energy state, there would be no energetic advantage to forming chemical bonds with other atoms. Consequently, there would be no methane, no diamond, no DNA, without the Pauli exclusion principle.

Figure 31: Representation of the allowed quantum state solutions to the Schroedinger equation for an electron in an atom as a set of seats in a cla.s.sroom. The Pauli principle indicates that each seat can accommodate two electrons provided they have opposite spins. Shown from left to right are the occupied quantum states for atoms containing one, two, three, six, and thirteen electrons, corresponding to hydrogen, helium, lithium, carbon, and aluminum, respectively.

Consider an atom such as aluminum, with thirteen electrons, shown in Figure 31e. All but one of these electrons are arranged in +/2 and -/2 pairs, and thus only this last electron can partic.i.p.ate in chemical bonds. The other twelve electrons are chemically inert and form the inner core of the aluminum atoms. Not that we can"t make use of these inner electrons. The Pauli principle forces the electrons to reside in higher and higher energy states, equivalent to having some students sit in rows far from the front of the cla.s.sroom, even when the atom is in its lowest energy configuration. If we could knock one of the electrons out of the ground state (a row close to the front of the lecture hall), then an empty position suddenly would open up, as if we had ejected a student sitting in a front-row seat. A student sitting in one of the upper rows could then jump down into the newly vacant seat. Just such a situation can arise when a high-energy beam of electrons strikes an atom. In that case, when one of the outer electrons falls down to occupy the lower energy state, it can emit an X-ray photon during the transition. This is in fact a very efficient way to generate X-rays, and most dental X-ray machines employ electron currents striking a copper target to create the penetrating radiation.

How do the last few electrons that are not residing in paired quantum states, and are thus available to partic.i.p.ate in chemical bonds, combine with those from neighboring aluminum atoms to hold all trillion trillion atoms together in a solid piece of aluminum? How do the last unpaired electrons between carbon atoms in diamond combine to bind this rigid insulator? In both cases, the electrons arrange themselves to satisfy the Pauli exclusion principle, though the resulting material properties in aluminum and diamond could not be more different.

One easy way to satisfy the Pauli principle is to never let the electrons be at the same place at the same time. If I have a line of atoms, and next to each atom is a barrier, then I can place an electron inside each theoretical box (we"ll see soon what this "box" really is), and all these negative charges can be in the same quantum state. This does not cause any problem, for by creating a series of containers for each electron, I have in principle made them distinguishable. I can tell which electron is in the box on the right and which on the left, just as I could tell apart the two stones that I tossed into the pond. The walls of the boxes prevent the de Broglie waves of each electron from overlapping with those of its neighbors, so the trillion trillion electrons can all be in the same quantum state, as the total wave function is just the one electron function repeated a trillion trillion times. Each electron is described by its own ribbon, as shown in Figure 30, and no ribbon is used for more than one electron. When I calculate the average energy of each electron in a box, a.s.suming the width of the box is the s.p.a.cing between atoms in my solid, I arrive at a number of about three electron Volts (the exact value obviously depends on all sorts of details of how the atoms in the solid are arranged-termed the crystalline configuration). This is the energy I would have to give to an electron to remove it from the box. Of course, as each electron has two possible values of spin, I can actually put two electrons in each box (each box contains a love seat), as long as they have intrinsic angular momentum +/2 and -/2.

Consider carbon, shown in Figure 31d. Carbon can easily "rearrange the seats in the rows" of the four upper electrons, mixing the quantum mechanical wave functions to form differing configurations of quantum states that allow for a variety of chemical interactions. Carbon can form strong bonds in a straight line, in proteins and DNA; it can form graphite, with three strong bonds in a plane and one weak bond above or below the plane, which is why graphite can be easily peeled apart when used in a pencil, for example; and when the "seats" are configured to form four equally strong bonds, we call this form of carbon "diamond." In each case, the carbon atom has four electrons that are capable of partic.i.p.ating in chemical bonds, represented by four boxes, each of which holds one electron. The Pauli principle tells us that each box can hold a second electron, provided it has an opposite spin from the first. When two carbon atoms come close enough to each other that the quantum states containing these unpaired electrons overlap, the two electrons can be represented by a two-electron wave function. It turns out that each of the unpaired electrons can lower its energy if the electrons fill up the love seats in each box (Figure 32a). That is, I must add energy to the atoms to remove the electrons from these boxes, and restore each one to its unpaired state. The overlapping electron wave functions form a chemical bond between the atoms, holding them together in the crystalline solid. The Pauli principle is satisfied by the localization of the electrons in s.p.a.ce. In a diamond crystal, each carbon is surrounded by four other carbon atoms in a tetrahedral arrangement, and their unpaired electrons can occupy the second seat in the first atom"s boxes (Figure 32b).

But there is another way to satisfy the Pauli exclusion principle. Let"s say that there are no boxes, and I let the electrons wander over the entire solid. In this case two electrons can be at the same place at the same time, so I have to ensure that they are each in different quantum states. How many different lowest-energy quantum levels are available for the last unpaired electron, such as the thirteenth electron for aluminum, shown in Figure 31e? As many as there are atoms in the solid. Since I have given up having any knowledge of where the electrons may be, I can compensate by having the electrons reside in states that have a well-defined momentum. I can have a matter wave with a very large wavelength, equal to the entire length of the solid. Since (momentum) (wavelength) = Planck"s constant, this large wavelength corresponds to a very small momentum, and hence energy. At the other extreme, the smallest wavelength that can be constructed corresponds to the distance between atoms in the solid. This is very short, so the momentum of this matter-wave is high. The highest energy of this shortest wavelength is also about three electron Volts, again, depending on the details of the atomic configurations in the solid.

Figure 32: Sketch of the lowering in energy when two unpaired electrons from adjacent carbon atoms overlap and form a carbon-carbon bond (a). Also shown is a sketch of the configuration of carbon when in the diamond configuration, allowing four chemical binds with its neighbors, in a tetrahedral orientation (b).

So, whether the electrons are put inside boxes in each atom, or allowed to roam over the solid, we still wind up with an energy of about three electron Volts. However, in the second situation, where the electrons can move around the solid in discrete momentum states, three electron Volts is approximately the energy of the most energetic electron, while when the electrons are placed in boxes, three electron Volts is roughly the energy of each electron. The average energy of an electron in the free-to-roam case is less than three electron Volts, and in fact will be closer to 1.5 electron Volts (recall the cla.s.s from our discussion of the Heisenberg uncertainty principle, where every student had a different exam score, from 0 to 100. In this case the average grade was 50 percent). For the electrons-in-a-box situation, every electron has the same energy, so the average energy is also three electron Volts (if every student scores a perfect 100 percent, then the cla.s.s average is also 100 percent). Consequently, depending on its chemical composition, the solid as a whole can lower its energy by letting the electrons wander around the crystal. This won"t be true for all solids. Some materials will be able to lower their total energy by keeping every electron localized in boxes around each atom. We call the free-to-roam cases "metals," and the electrons-in-a-box materials "insulators."

And that"s how quantum mechanics explains solid-state physics. At very low temperatures, all solids either conduct electricity or they don"t. We call the first case metals, and the second are insulators (the distinction between insulators and semiconductors is most relevant around room temperature, and I defer for now a discussion of the differences between the two).

Metals such as aluminum are good conductors of electricity because the outermost electrons satisfy the Pauli principle by residing in momentum states and are free to move around the entire solid, while insulators keep theirs in boxes (bonds) around each atom. To remove a metal atom from the solid, I must first grab one of the free-range electrons and localize it on a positively charged atom-in essence, put it in a box so that I can pull the neutral atom out of the solid. But this costs me a few electron Volts of energy, and this can be considered the binding energy holding the atoms together in the metal. There are no directional bonds between atoms, so it is easy to move atoms past each other, which is why metals are easy to pull into wires or pound into thin sheets, without losing their structural coherence. If light is absorbed by the solid, there is always a free electron that can absorb its energy and reemit it back again, which is why metals are reflective and shiny. The sea of free electrons makes metals good conductors of both electrical current and heat.

Insulators, on the other hand, such as diamond, have all the electrons tied into bonds between the atoms (the two electrons per box as discussed earlier). They are thus poor conductors of electricity and can conduct heat only by atomic vibrations (sound waves). The electrons in the box can a.s.sume only specific energy states, like electrons in atoms. Consequently, if you shine light on an insulator that does not correspond to an allowed transition, it will ignore the photons. This is why some insulators, such as diamond or window gla.s.s, are transparent to visible light. The details of the materials" properties are very sensitive to the configurations of the boxes in which the electrons reside, which is why changes in crystal structure-say, when carbon transforms from graphite to diamond-can yield big variations in optical and electrical properties. The boxes here are the directional, rigid chemical bonds between the atoms (Figure 32b), and changes in the type of these bonds and the chemical const.i.tuents lead to big variations in crystal structure and rigidity (diamond is very stiff, while graphite is so soft, we use it for pencil lead). All the differences between insulators and metals can be understood at the most basic level by whether the last few unpaired electrons of the atoms in the solid satisfy the Pauli exclusion principle by localizing themselves in real s.p.a.ce (insulators) or in momentum s.p.a.ce (metals).

Thus, from playing with a ribbon, with one side black and the other side white, we see why the world is the way it is. Note that not everything has an intrinsic angular momentum equal to /2. Some objects, such as helium atoms or photons, have spin values of either 0 or . This seemingly small difference leads to superconductors and lasers.

CHAPTER THIRTEEN.

All for One and One for All.

Bert Holldobler and Edward O. Wilson, in The Superorganism-The Beauty, Elegance, and Strangeness of Insect Societies, propose that colonies of wasps, ants, bees, or termites can be considered as a single animal. They argue that each insect is a "cell" in the "superorganism": foragers are the eyes and sense organs, the colony defenders act as the immune system, and the queen serves as the colony"s genitalia.50 An important difference between a superorganism and a regular animal is that the colony lacks a centralized brain or nervous system. Rather, each colony has its own rules for local interactions among the insects that govern its organization and size. In this way the colony can achieve levels of development that are far beyond the capabilities of the individual insects were they to act alone. As readers of science fiction pulps know, this is the mechanism (or at least one of them) by which humans defeat an alien invasion.

In Theodore Sturgeon"s 1958 science fiction novel The Cosmic Rape (an abridged version was simultaneously published in Galaxy magazine with the t.i.tle To Marry Medusa), an alien intelligence applies an unconventional approach to its efforts to conquer Earth. (Spoiler alert!) The alien, in fact a cosmic spore, is capable of controlling the intelligence of a single human. It is surprised to discover that the people of Earth are not in mental contact with one another, given that the planet is covered with complex structures such as buildings, bridges, and roads. In the spore"s experiences conquering other planets that contained advanced infrastructure, such architecture is possible only when the primitive intelligences of the individual agents interact in a cooperative manner, as in a colony of ants or a hive of honeybees. The invading spore had never encountered a species for which a single agent is capable of designing a bridge or building on is or her own, and it thus a.s.sumes that a previously existing collective connection has been severed. The spore sets upon a plan to reestablish this connection and force all humans to think and work together in unison. Unfortunately for the alien ent.i.ty, it succeeds in its plan. Once it finds itself dealing not with the intelligence of a single human, but with the collective consciousness of all several billion humans, the "hive mind" of humanity quickly devises an effective counterattack, destroying the alien spore. In this way Sturgeon has described the cooperative behavior of a Bose-Einstein condensate.

In Sturgeon"s novel, the alien spore creates from humanity a macroscopic quantum state, with a single wave function containing all the information about its const.i.tuent elements. Any change in one element, in Sturgeon"s case a human being, is instantly transmitted to every other element in the wave function, that is, the rest of humanity. Such situations occur frequently in the real world through quantum interactions between pairs of electrons in a superconductor or helium atoms in a superfluid. These collective states involve particles whose intrinsic angular momenta are multiples of , rather than /2. Particles with intrinsic angular momenta that are whole-number multiples of are called bosons, as they obey a form of quantum statistics elaborated by Satyendra Bose and Albert Einstein, termed Bose-Einstein statistics.

In the previous chapter we discussed fermions, for which the angular momentum could be either +/2 or -/2, but not any other value. This is intrinsically asymmetric, as we can distinguish a top twirling clockwise from one rotating counterclockwise. We represented this situation, when the two fermions are so close that their wave functions overlap, with a ribbon with one side black and the other white. The significance of the two colors was that we could readily distinguish the spin = +/2 electron from the spin = -/2 electron, as we could the black and white sides of the ribbon. But experiments have revealed situations where the intrinsic angular momentum can have values of 0 or or 2, and so on, but not any fractional values. Let"s consider the case of quantum objects with a spin of zero first, and then turn to spin = particles such as photons.

As the fundamental building blocks of atoms-electrons, protons, and neutrons-are all fermions, what sort of object would have spin of zero? One example is a helium atom. A helium nucleus has two protons and two neutrons, each having spin = +/2 or -/2. As the two protons are identical, in their lowest energy state in the nucleus they would pair up, +/2 and -/2, for a total spin of zero, as would the two identical neutrons. Similarly, the two electrons are spin paired, as indicated in the sketch in Figure 31b. Consequently, the total intrinsic angular momentum of a helium atom, when in its lowest energy configuration, has a spin value of zero.

Particles with zero value of spin are symmetric, in that we cannot describe the rotations as clockwise or counterclockwise. When two such particles are brought so close that their wave functions overlap, we will represent them by a ribbon whose sides are both white. I once again stress that the ribbon is employed as a metaphor for the resulting two-particle wave function, and as such, certain issues are being ignored here that would only distract from our discussion.

Let"s repeat the experiment with the ribbon from Chapter 12, only now using a ribbon with both sides the same color, for example, white (Figure 33). I can hold each end, and obviously a white side of the ribbon faces out (Figure 33a). Now, without letting go of either end, I will switch their positions, as before. The end of the ribbon on the left is now on the right, and vice versa. This procedure has introduced a half twist in the ribbon, as in Chapter 12 (Figure 33b). Of course, now both sides are still white. I can undo the half twist by flipping one end of the ribbon around, so that the back side turns outward (Figure 33c). When the ribbon had one side white and the other side black, this was a forbidden operation, as it changed the state of the ribbon (where before both ends had white facing out, one side would then have had a black side facing out). But if both sides of the ribbon are white, then this symmetry means I can flip one end of the ribbon and I have not changed anything except undoing the half twist. The important point is that a white ribbon can be restored to its original state following a single rotation, while the black/white ribbon requires two rotations to bring the original configuration back.

Figure 33: Cartoon sketch of a ribbon with the same color on each side (a). Switching the two ends results in a half-twist in the ribbon (b) that can be undone by rotating one side of the ribbon (c), restoring the original configuration.

This symmetry indicates that the two-particle wave function for spin = 0 particles, such as helium atoms, as well as spin = photons, termed bosons, can be written as the sum of the two functions A and B, = A + B, rather than = A - B, for fermions.51 As before, A and B depend on the product of the one-particle wave functions at positions 1 and 2. Now the two-particle wave function = A + B is unchanged if the positions of particles 1 and 2 are switched, in which case would be given by = B + A. But this is just the same as = A + B = B + A. When two particles for which the intrinsic angular momentum has values of spin = 0 or spin = 52 are brought close enough to each other that their de Broglie waves overlap, the resulting two-particle wave function is just the sum of the functions A and B, which are in turn functions of the one-particle wave functions.

What is the consequence of writing the two-electron wave function as = A + B? Recall that for fermions such as electrons, the fact that the two-electron wave function is = A-Bmeant that the probability is exactly zero that both electrons would be in the same quantum state, for which A = B. For bosons, = A + B indicates that the probability is large exactly when both particles are in the same quantum state, when A = B. Because when A = B, then = A + A = 2A and the probability density 2 = (2A) (2A) = 4A2. For a single particle in state = A the probability density would be 2 = AA= A2. For two single particles the probability would be A2 + A2 = 2A2. So just by bringing a second identical particle near the first, the probability that they would both be found in state A is double what it would be for the two particles separately. While the probability is not 100 percent that they will both be in the same state, it is enhanced compared to the single-particle situation. A larger probability of both particles being at the same location in the same quantum state indicates that it is more likely to occur than not.

As the temperature of a system is reduced, the particles will settle down into lower energy states. If we had particles that were somehow distinguishable, for example, if their wave functions did not overlap so we did not have to worry about Fermi-Dirac or Bose-Einstein statistics, then at low temperatures we would find many particles in the lowest energy state, some in the next available quantum level, a few more in the next higher level, and negligible occupation of very high-energy states. For fermions, such as electrons in a solid, only two electrons can occupy the lowest energy level (one with spin = + /2 and the other with spin = -/2), regardless of temperature. In contrast, bosons will have an enhanced probability of collecting into the lowest-energy ground state at low temperatures, relative to the distinguishable particle case. For these particles, the rule of one particle per spin orientation per seat (valid for fermions) is thrown out, and one can have many particles dog piling into a single state. These spin = 0 or spin = particles obey statistics described by Bose and Einstein, and this settling into the ground state is termed Bose-Einstein condensation.

Why do we need to go to low temperatures to see this condensation? If the particles are very far apart, then there will be little or no overlap of their wave functions, and the whole issue of indistinguishable particles is irrelevant. Temperature is just a bookkeeping device to keep track of the average energy per particle, so the lower the temperature, the less kinetic energy and the lower the momentum. From de Broglie"s relationship, a low momentum corresponds to a long matter-wavelength. If the particles involved have long de Broglie wavelengths, it will increase the opportunity for the waves of different identical particles to overlap. Similarly, confining the particles to a small volume also increases the possibility for interactions among wave functions. Consequently, low temperatures and small volumes (achieved by squeezing the system at high pressures) help induce Bose-Einstein condensation.

What are the special attributes of a Bose-Einstein condensate? We have considered the case of two identical bosons whose wave functions overlap such that they can be described by a single, two-particle wave function. As the temperature of a gas of bosons is lowered, millions of identical atoms" wave functions overlap, all in the same quantum state. We thus obtain one single wave function that describes the behavior of millions of atoms. In this way the individual indistinguishable bosons behave as a single ent.i.ty, and whatever happens to one atom is experienced by many. The Bose condensate is not unlike the demonically possessed children in the 1960 science fiction film Village of the d.a.m.ned. The fair-skinned, blond children play the role of indistinguishable particles, and the fact that knowledge gained by one child is instantly shared with all is a natural consequence of the multiparticle wave function that describes this collective phenomenon.

True condensation, confirming the theoretical predictions of Bose and Einstein from the 1920s, was experimentally observed by Eric Cornell and Carl Wieman in 1995, and independently by Wolfgang Ketterle the same year, a feat for which they shared the 2001 n.o.bel Prize in Physics. Their investigations involved thousands of particular isotopes of rubidium or sodium, cooled to temperatures below a millionth of a degree above absolute zero. While these Bose-Einstein condensates are ephemeral quantum objects, difficult to obtain and to probe, there are more robust systems that owe their striking properties to the cl.u.s.tering of bosons into a single low-energy quantum state.

As pointed out earlier, helium is an example of an atom that is characterized by total spin of zero, and is thus a boson. The two electrons in helium are spin paired in the ground state (Figure 31b), and helium thus does not have strong chemical interactions with other atoms-a feature it shares with other elements whose electrons are paired up in completely filled "rows," such as neon and argon, the inert, or n.o.ble, gases. These elements consequently remain gases until their temperature is so low that small fluctuations in their electrical charge distribution induce weak electrical attractions. Helium interacts so weakly with other helium atoms that it does not form a liquid until 4.2 degrees above absolute zero. If cooled even further at normal pressures, it does not form a solid but rather undergoes a quantum transition, where some of the atoms condense into the ground state.

Suppose the temperature of liquid helium is lowered all the way to absolute zero. We would expect that the helium would eventually become a solid, but it in fact remains a liquid, thanks to the uncertainty principle. At low temperatures, when the wave functions overlap, the uncertainty in the position of each atom is low. There is thus a large uncertainty in the momentum of each atom, which contributes to the ground-state energy of the helium atoms (called the "zero-point energy"). The lower the ma.s.s of the atom, the larger this zero-point energy, and for helium this contribution turns out to be just big enough to prevent the atoms from forming a crystalline solid, even at absolute zero. Hydrogen has an even lower ma.s.s than helium, but it forms a solid at 14 degrees above absolute zero due to strong electrical interactions between hydrogen molecules, while for heavier elements the uncertainty in the momentum of each atom is not enough to overwhelm the tendency to form a solid at low temperatures. While helium does not form a solid at normal pressures (if you squeeze the liquid, you can force it to form a crystal), it does undergo a "phase transition" at 2.17 degrees above absolute zero, as some of the helium forms a condensate in the ground state.

What would be the properties of a fluid for which some of the atoms have condensed into a single quantum state? One surprising feature would be that the fluid would have no viscosity! Viscosity describes the internal friction all normal fluids have; you can think of it as resistance to flow. Water has a pretty low viscosity, and mola.s.ses and motor oil have much larger viscosities. A fluid with no viscosity would, once it started moving, continue to flow at a constant speed through a hose without continued applied pressure. Such a state is termed a superfluid, for it does what a normal fluid does-but with the power of quantum mechanics!53 Experimentalists in 1965 rotated a spherical container of liquid helium at 4 degrees above absolute zero about an axis pa.s.sing through its center. The sphere was packed with gla.s.s particles, so the fluid would have to move through the small pores and gaps between the beads. The liquid helium, not yet a superfluid at this higher temperature, began to swirl along with the container. The temperature of the helium was then lowered to below 2.17 degrees above abso

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