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Section 1: Introduction

The quantum theory that evolved at the beginning of the 20th century is a strange creature: It assigns particulate properties to light—long thought to consist of waves—and, astonishingly, wave properties to individual fundamental particles that have definite masses. This seeming confusion between waves and particles never seems to bother us in ordinary life, where particles (such as grains of rice or baseballs) are simply particles, and what comes out of a flashlight or laser pointer can be focused or even diffracted, thereby revealing the wave-like properties we associate with light. Of course, large numbers of particles acting in collusion can create waves as a possible collective motion—on the surface of the ocean, for example. And a slinky or violin string can display wave properties, to the surprise of no one. But these are classical, not quantum, waves, a distinction we will make clear in this unit.

Some of the first experimental evidence for a gaseous macroscopic quantum state.

Figure 1: Some of the first experimental evidence for a gaseous macroscopic quantum state.

Source: © Mike Matthews, JILA. More info

We typically view quantum mechanics as applying only to the fundamental particles or fields of Units 1 and 2, and not to the objects we find in a grocery store or in our homes. If large objects like baseballs and our dining tables don't behave like waves, why do we bother about their possible quantum nature? More practically, we can ask: If we are supposed to believe that ordinary objects consist of wave-like quantum particles, how does that quantum nature disappear as larger objects are assembled from smaller ones? Or, more interestingly: Are there macroscopic objects large enough to be visible to the naked eye that still retain their quantum wave natures? To answer these questions, we need the rules for building larger objects out of smaller ones, and the means of applying them. The rules are of a highly quantum nature; that is, they have no counterpart at all in classical physics, nor are they suggested by the behavior of the ordinary material objects around us. But in surprising ways, they lead to macroscopic quantum behavior, as well as the classical world familiar to all of us.

To interpret these rules, we need to understand two critical factors. First, particles of all types, including subatomic particles, atoms, molecules, and light quanta, fall into one of two categories: fermions and bosons. As we shall see, the two play by very different rules. Equally important, we will introduce the concept of pairing. Invoked more than a century ago to explain the creation of molecules from atoms, this idea plays a key role in converting fermions to bosons, the process that enables macroscopic quantum behavior. This unit will explore all these themes and Unit 8 will expand upon them.

Building up atoms and molecules

We start with the empirically determined rules for building atoms out of electrons, protons, and neutrons, and then move on to the build-up of atoms into molecules. The quantum concept of the Pauli exclusion principle plays a key role in this build-up (or, in the original German, aufbau). This principle prevents more than one fermion of the same fundamental type from occupying the same quantum state, whether that particle be in a nucleus, an atom, a molecule, or even one of the atom traps discussed in Unit 5. But the exclusion principle applies only to identical fermions. Electrons, neutrons, and protons are all fermions; but, although all electrons are identical, and thus obey an exclusion principle, they certainly differ from neutrons or protons, and these non-identical fermions are not at all restricted by the presence of one another. Chemists and physicists understood the exclusion principle's empirical role in constructing the periodic table well before the discovery of the quantum theory of Unit 5; but they did not understand its mathematical formulation or its theoretical meaning.

Particles of light, or photons, are of a different type altogether: They are bosons, and do not obey the exclusion principle. Thus, identical photons can all join one another in precisely the same quantum state; once started, in fact, they are actually attracted to do so. This is the antithesis of the exclusion principle. Large numbers of such photons in the same quantum state bouncing back and forth between two mirrors several inches or even many meters apart create an essential part of a laser that, in a sense, is a macroscopic quantum system with very special properties.

Diffraction of green laser light passing though a random medium.

Figure 2: Diffraction of green laser light passing though a random medium.

Source: © William P. Reinhardt. More info

Lasers are everywhere in our technology, from laser pointers to surgical devices and surveying equipment, to the inner workings of CD and DVD players. Laser light sent from telescopes and reflected from mirrors on the Moon allows measurement of the distance between the Earth and Moon to better than one millimeter, allowing tests of gravitational theory, measurement of the slowly increasing radius of the moon's orbit around the Earth, and even allows geophysicists to observe the day-to-day motion of the continents (actually the relative motion of plate tectonics) with respect to one another: A bright light indeed. In addition to being bright and intense, laser light is coherent. Not only do all the particulate photons wave with the same wavelength (or color), but they are also shoulder to shoulder in phase with one another. This phase coherence of very large numbers of photons is a quantum property that follows from their bosonic nature. Yet, it persists up into the macroscopic world.

Superfluids and superconductors

Are there other macroscopic quantum systems in which actual particles (with mass) cooperate in the same coherent manner as the photons in a laser? We might use the molecules we will build up to make a dining room table, which has no evident quantum wave properties; so perhaps quantum effects involving very large numbers of atoms simply don't persist at the macroscopic level or at room temperature. After all, light is quite special, as particles of light have no mass or weight. Of course, we fully expect light to act like a wave, it's the particulate nature of light that is the quantum surprise. Can the wave properties of massive particles appear on a macroscopic scale?

The surprising answer is yes; there are macroscopic quantum systems consisting of electrons, atoms, and even molecules. These are the superfluids and superconductors of condensed matter physics and more recently the Bose condensates of atomic and molecular physics. Their characteristics typically appear at low (about 1–20 K), or ultra-low (about 10-8 K and even lower) temperatures. But in some superconductors undamped currents persist at rather higher temperatures (about 80 K). These are discussed in Unit 8.

Superfluids are liquids or gases of uncharged particles that flow without friction; once started, their fluid motion will continue forever—not a familiar occurrence for anything at room temperature. Even super-balls eventually stop bouncing as their elasticity is not perfect; at each bounce, some of their energy is converted to heat, and eventually they lie still on the floor.

Superconducting magnets enable MRI machines to produce dramatic images.

Figure 3: Superconducting magnets enable MRI machines to produce dramatic images.

Source: © Left: Wikimedia Commons, Creative Commons Attribution 3.0 Unported License. Author: Jan Ainali, 12 February 2008. Right: NIH. More info

Superconductors have this same property of flowing forever once started, except that they are streams of charged particles and therefore form electric currents that flow without resistance. As a flowing current generates a magnetic field, a superconducting flow around a circle or a current loop generates a possibly very high magnetic field. This field never dies, as the super-currents never slow down. Hospitals everywhere possess such superconducting magnets. They create the high magnetic fields that allow the diagnostic procedure called MRI (Magnetic Resonance Imaging) to produce images of the brain or other soft tissues. This same type of powerful superconducting magnetic field is akin to those that might levitate trains moving along magnetic tracks without any friction apart from air resistance. And as we saw in Unit 1, physicists at CERN's Large Hadron Collider use superconducting magnets to guide beams of protons traveling at almost the speed of light.

Thus, there are large-scale systems with quantum behavior, and they have appropriately special uses in technology, engineering, biology and medicine, chemistry, and even in furthering basic physics itself. How does this come about? It arises when electrons, protons, and neutrons, all of which are fermions, manage to arrange themselves in such a way as to behave as bosons, and these composite bosons are then sucked into a single quantum state, like the photons in a laser.

Pairing and exclusion

What is required to create one of these composite quantum states, and what difference does it make whether the particles making up the state are fermions or bosons? Historically, the realization that particles of light behaved as bosons marked the empirical beginning of quantum mechanics, with the work of Planck, Einstein, and, of course, Bose, for whom the boson is named. In Unit 5, we encountered Planck's original (completely empirical) hypothesis that the energy at frequency in an equilibrium cavity filled with electromagnetic radiation at temperature T should be n, where h is Planck's constant, is the photon frequency, and n is a mathematical integer, say 0, 1, 2, 3.... Nowadays, we can restate the hypothesis by saying that n photons of energy are all in the same quantum mode, which is the special property of bosons. This could not happen if photons were fermions.

Another, at first equally empirical, set of models building up larger pieces of matter from smaller came from Dmitri Mendeleev's construction and understanding of the periodic table of the chemical elements. American chemist G. N. Lewis pioneered the subsequent building up of molecules from atoms of the elements. His concept that pairs of electrons form chemical bonds and pairs and octets of electrons make especially stable chemical elements played a key role in his approach. This same concept of pairing arises in converting fermions into the bosons needed to become superconductors and superfluids, which will be our macroscopic quantum systems.

Early periodic table.

Figure 4: Early periodic table.

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Austrian theorist Wolfgang Pauli took the next step. The theoretical development of modern quantum mechanics by Heisenberg and Schrödinger allowed him to make a clear statement of the exclusion principle needed to build up the periodic table of elements. That principle also made clear the distinction between fermions and bosons as will be discussed in Section 4. But first, we will address how models of atomic structure were built, based on the discovery of the electron and the atomic nucleus. Then we will follow the quantum modeling of the atom, explaining these same empirical ideas once the quantum theory of Unit 5 is combined with Pauli's exclusion principle.