The Amazing Story of Quantum Mechanics

by Kakalios, James

  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 = ħ photons, termed bosons, can be written as the sum of the two functions A and B, Ψ = A + B, rather than Ψ = 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 Ψ = A + B is unchanged if the positions of particles 1 and 2 are switched, in which case Ψ would be given by Ψ = B + A. But this is just the same as Ψ = A + B = B + A. When two particles for which the intrinsic angular momentum has values of spin = 0 or spin = ħ52 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 Ψ = A + B? Recall that for fermions such as electrons, the fact that the two-electron wave function is Ψ = A-Bmeant that the probability is exactly zero that both electrons would be in the same quantum state, for which A = B. For bosons, Ψ = 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 Ψ = A + A = 2A and the probability density Ψ2 = (2A) × (2A) = 4A2. For a single particle in state Ψ = A the probability density would be Ψ2 = A×A= 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 = + ħ/2 and the other with spin = -ħ/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 = ħ 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 entity, 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 Damned. 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 Nobel 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 clustering 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 noble, 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 mass 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 mass 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 molasses 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 pow
er of quantum mechanics!53

  Experimentalists in 1965 rotated a spherical container of liquid helium at 4 degrees above absolute zero about an axis passing through its center. The sphere was packed with glass 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 absolute zero, at which point some of the helium condensed into the superfluid state. When the container’s rotation was then stopped, the superfluid continued to move with no change in speed. When you stop stirring your coffee, the fluid comes to rest within a few seconds, but the superfluid helium maintained its circulating motion for hours, until the researchers eventually stopped the experiment.

  If this system were warmed higher than a temperature of 2.17 degrees above absolute zero, then the superfluid would transform into a normal fluid, and it would rapidly cease rotation. The low temperature is crucial. At low enough temperatures, the lower momentum of the helium atom corresponds to a long de Broglie wavelength. There is then sufficient overlap among the many helium atoms’ wave functions that all of the atoms can be described by a single macroscopic quantum mechanical wave function. To slow down even one helium atom in the condensate, it is necessary to decelerate the entire multi-atom wave function, and provided the rotation is not too fast, there isn’t sufficient energy to do this. So the superfluid keeps on rolling.

  There is an electrical analog to superfluidity that is found in many metals and even some nonmetallic materials, termed “superconductivity.” A wire that is a superconductor has no electrical resistance and is analogous to a garden hose through which a superfluid flows. Any nonviscous fluid pushed into one end of the hose would exit at the other end with the same velocity, no matter how long or clogged the hose, even if the tubing circled the equator. While we don’t know whether the electrical current circulating in a superconducting loop will flow forever, experiments have confirmed that even after a year, the supercurrent in a closed ring has decreased from its initial value by less than one part in one hundred trillion.

  In a normal electrical conductor, an externally applied voltage induces an electrical current. The smaller the resistance of the conductor, the greater the current for a given voltage. In the standard water-flow analogy for electrical circuits, water pressure plays the role of voltage. The greater the pressure, the more of a push exerted on the water. The flow of water out of the faucet through a garden hose is analogous to the electrical current. If the hose has imperfections and bumps along its length that make it difficult for the water to flow, this would be analogous to the electrical resistance of the wire. The water loses energy through collisions with these partial blockages, as well as with the walls of the hose, so a constant pressure is needed to maintain a steady water flow out of the end of the hose. Similarly, as the electrical current collides with imperfections in the wire, some of the current’s energy is lost. This is why a constant push (a voltage) produces a constant flow (electrical current) rather than an accelerating flow (Newton’s second law, that force = mass × acceleration, would suggest that if the force is constant, then the acceleration, that is, the rate of change of the electron’s speed, should also be constant). Superconductors have no electrical resistance, so that a current, once started, will continue unchanged without an applied voltage, just as the helium atoms in a superfluid are able to translate without viscosity.

  Currents in metals are carried by electrons, not helium atoms. Remember that electrons are fermions that have intrinsic angular momentum of ħ/2. In order to observe Bose-Einstein condensation, the electrons must form a composite particle consisting of two electrons, one having spin = +ħ/2 and the other with spin = -ħ/2. Thus, the two-electron composite would have a net intrinsic angular momentum of zero and would therefore be a boson. As such, at a low enough temperature, these paired electrons would condense into a ground state and be able to flow without resistance.

  Electrons are negatively charged, and as two negative charges repel each other, the question is, Why would two electrons bind together to form a composite particle that acts as a boson? The answer lies in the positively charged atoms, termed “ions,” that make up the metal. Recall from the preceding chapter that in a metal such as lead, the last unpaired electrons from each atom reside in quantized momentum states. The electrons are free to roam over the solid but can do so in well-defined energy states. As the metal atoms were initially electrically neutral, if an electron leaves the immediate vicinity of its atom, it leaves behind a positively charged ion (an “ion” is an atom with a net electrical charge due to the removal or addition of electrons). These metallic ions form ordered arrays and comprise the crystal. As a negatively charged electron moves through the metal, the positively charged ions are attracted to it. The positive ion is too large to leave its position in the crystal, but it strains toward the negatively charged electron, slave to the electrostatic attraction between them.

  As the electron speeds along, it leaves in its wake a trail of positively charged ions that are pulled along its trajectory, not unlike the way metallic objects bend towards Magneto (the mutant master of magnetism) from the X-Men comic books when he employs his mutant power. In time the ions would be repelled from each other and return to their normal crystalline locations. At temperatures less than 7 degrees above absolute zero, the lead ions move slowly, and this positively charged channel in the wake of the first electron can persist long enough for a second electron to be attracted into this positive valley. That is, the first electron polarizes the positive ions in the lattice, and a second electron is attracted to this positively charged channel and follows the same path. In this way the two negatively charged electrons are bound together and form what is known as a Cooper pair (after Leon Cooper, who first theoretically showed that such a binding mechanism could operate in metals at low temperatures). The lowest energy configuration corresponds to two electrons with spins of + ħ/2 and -ħ/2, respectively, so the Cooper pair formed from the two bound electrons has an intrinsic angular momentum of zero and acts as a boson.

  Once at least some of the electrons in a metal start acting like bosons and condense into a low energy state, superconductivity is observed. When the wave functions for the many Cooper pairs overlap, they form a single multiparticle wave function. In a normal metal, collisions with vibrating atoms or defects in the metal cause the current to lose energy, which is why a constant voltage is needed to maintain a uniform current. In order to slow down a supercurrent consisting of a condensate of overlapping Cooper-paired electrons, the collisions must break apart a Cooper pair, also changing the energies of all the overlapping pairs, and at low temperatures and moderate currents this is not energetically possible. The Cooper-paired electrons are able to carry electrical current (provided it isn’t too high) without any resistive loss, just as the helium atoms in a superfluid are able to flow (but not too fast) without viscosity.

  There are many free-range electrons in a metal, but not all of them have to form Cooper pairs for the metal to exhibit superconductivity. What about the other electrons that may not bind up in pairs to form boson composite particles? They still have a normal resistance, but their contribution is shorted out by the supercurrent. If I have two roads to a destination, one that is a bumpy, unpaved dirt road with a speed limit of 5 miles per hour, and another a sleek superhighway with no upper speed limit, I will take the second road.54 Any electrical current in a metal, which has been cooled below the temperature at which Cooper pairs form, will be carried by the superconducting paired electrons. Similarly, not all the helium atoms in a superfluid reside in the ground-state condensate. As long as some of the particles in the superconductor or superfluid are in a lower energy condensate, they will exhibit cooperative behavior.

  Superconductors do not just carry electrical current with no resistance whatsoever—they also are perfect diamagnets. This means th
at they resist any externally applied magnetic field. Some metals are attracted to magnets, while others are actually repelled. Gold and silver are examples of this latter type of metal. If you are able to pick up your “gold” jewelry with a refrigerator magnet, you should probably look into a refund (or at least check to see whether the jewelry is filled with chocolate). The internal magnetic fields of the gold atoms polarize in the opposite direction to an external magnetic field, such that they develop a north pole that faces the applied north pole. As north repels north, the gold ignores the magnet, or if the applied magnetic field is strong enough, the gold is pushed away from the outside magnet through its diamagnetism.

  Superconductors are perfect diamagnets, as they can set up electrical currents that generate magnetic fields that exactly cancel out inside the solid all of the externally applied magnetic field. As these materials have no electrical resistance, once the current is started, it can continue indefinitely as long as the outside magnetic field is applied—which would make superconductors ideal materials from which to construct rails for magnetically levitating trains. The drawback, at present, is the ultralow temperatures necessary to induce superconductivity in most metals. In Section 6 I discuss materials termed “high-temperature superconductors” that show superconductivity at much higher temperatures (though not yet at room temperature) and turn out to not even be metals.