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Butterfly in the Quantum World

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Butterfly in the Quantum World 㔀 One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, and that we get more out of them than was originally put into them. —Heinrich Hertz FIG. 1: Starting with Max Planck at the top, this figure displays some of the pioneers who created the quantum revolution. In counterclockwise (but chronological!) order, they are: Albert Einstein, Niels Bohr, Louis de Broglie, Werner Heisenberg, and Erwin Schrodinger.¨ 1 The Quantum World The quantum revolution is usually considered to have started in 1900, with the proposal, made by German physicist Max Planck, that the vibration energies of atoms forming the walls of a black body are restricted to a set of discrete energy values — a very bold and wild-seeming yet very carefully worked-out hypothesis in which Planck himself did not fully believe, but which he nonetheless published since it was the first theory anyone ever proposed that fully agreed with the experimentally observed spectra of black-body radiation. It took another 25 years for others to develop this first very tentative foray into a mature theory — the theory of quantum mechanics that was then, and that still remains today, a profound challenge to our everyday experiences and ingrained intuitions. Perhaps the most inspired period ever of scientific creativity, the quarter-century of the quantum revolution was jointly created by many thinkers of extraordinary brilliance and imagination, some of whom are shown in Figure 1. This chapter cannot claim to be a genuine introduction to quantum science, but only a quick peek into the quantum world [1]. In it we highlight and partially explain some of the key quantum phenomena and quantum concepts that are indispensable to an understanding of the butterfly landscape. The aim of this chapter is thus twofold: • Beginning with the wave nature of matter, we present a simple picture of the quantization of energies in the physical world. It will be useful for readers to internalize this first image of quantization before learning about the quantization of Hall conductivity, which is an entirely different ballgame. The integer quanta that make up the butterfly fractal, related to the quantization of Hall conductivity, have their roots in phenomena that are quite distinct from those that give rise to the quantization of energies in an atom. Keeping this distinction in mind will help one to appreciate the exotic nature of the quantum Hall effect that will be presented in chapters to come. • This chapter also serves as a bridge between Parts I and III of the book, as it explains the crucial concepts that are necessary to make a transition from the quantum world of a solitary atom to that of a solid body — a pristine crystalline structure made up of millions of atoms. 2 FIG. 2: “WAVE/particle” ambigram by Douglas Hofstadter, reproduced here with his permission. I. WAVE OR PARTICLE — WHAT IS IT ? Try telling someone that an electron can sometimes act like a particle and sometimes like a wave. This news is bound to cause confusion and skepticism. When we imagine a particle, we tend to think of a tennis ball or a pebble, perhaps even an infinitesimal ball bearing, while the mention of waves makes us dream of huge long swells of water drifting in, every ten or twenty seconds, and breaking on beaches. Waves are gigantic and spread-out, while particles are tiny and local. No two things could be more different than a particle and a wave! Perhaps the only familiar connection between particles and waves is the fact that tossing a pebble in a pond generates ripples that gracefully spread out in perfect circles. Richard Feynman once summed up the situation by saying that although we do not know what an electron is, there is nonetheless something simple about it: It is like a photon. The quantum world is very counterintuitive, for it is inhabited by species that have dual personalities: they can behave both as particles and as waves. These species are, however, not necessarily alien or extraterrestrial beings. Some of them are very familiar entities. Indeed, they can be baseballs or even baseball players themselves, as well as more exotic entities, such as electrons, quarks, or Higgs particles. Interestingly, they can also be light waves or X-rays or waves vibrating on the string of a violin. The revolutionary idea of wave–particle duality was first proposed by Louis de Broglie in his doctoral thesis in 1924, for which he was awarded the Nobel Prize in physics in 1929. 3 A. Matter Waves FIG. 3: Matter-wave cartoon In the quantum world, any particle possesses a wavelength and a frequency (like waves on the ocean or ripples on a pond), in addition to having usual particle-like properties, such as mass, size, position, velocity, and electric charge. The quantum waves that de Broglie postulated, sometimes called “matter waves”, reflect the intrinsic wave–particle duality of matter. According to de Broglie’s hypothesis, the wavelength λ associated with any particle is inversely proportional to the particle’s momentum p, while the frequency ν associated with a particle is proportional to the particle’s total energy E. The constant of proportionality in both cases is the fundamental constant of nature h, called Planck’s constant, named after Max Planck, who, in his pioneering 1900 work on the blackbody spectrum, discovered this number and its central role in nature, and who, for those discoveries, was awarded the 1918 Nobel Prize in physics. The following two equations constitute the core of Louis de Broglie’s hypothesis: h E λ = ; ν = (1) p h In most situations, it turns out to be simpler and more natural to use the angular frequency ! = 2πν than to use the simple frequency ν. In the case of a rotating body, the quantity ν is the number of full rotations made by the body per second. Thus a body that rotates exactly once per second (360 degrees per second) has a simple frequency of 1 hertz, while its angular frequency is 2π hertz. The angular frequency thus equals the number of radians turned per second by the body (one radian equaling 360◦=2π ≈ 57◦). Since the angular frequency is generally more natural, whether we are talking about rotating bodies or wave phenomena, it is often useful to write quantum expressions not in terms of in terms of the simple frequency ν and Planck’s constant h, 4 but in terms of the angular frequency ! and the reduced Planck’s constant, which equals Planck’s constant divided by 2π, and is denoted by the symbol “~” (pronounced “h-bar”): h = = 1:054571726(47) · 10−34Joule − sec (2) ~ 2π Readers will notice the explicit presence of h or ~ in all equations involving quantum aspects of a particle. These constants, which have the units of angular momentum, lie at the heart of quantum science. Since they are extremely small compared to familiar amounts of angular momentum (such as the angular momentum of an ice skater doing a spin, or that of a frisbee flying through the air), quantum effects are usually completely unobservable in the macroscopic world. Three years after de Broglie announced his hypothesis about the wave nature of particles, the idea was confirmed in two independent experiments involving the observation of electron diffraction. At Bell Labs in New Jersey, Clinton Davisson and his assistant Lester Germer sent a beam of electrons through a crystalline grid and, to their amazement, observed interference patterns. They were utterly baffled at first, but when they heard about de Broglie’s wave–particle hypothesis, they realized that what they were seeing confirmed de Broglie’s predictions exactly. A few months later, at the University of Aberdeen in Scotland, George Paget Thomson passed a beam of electrons through a thin metal film and also observed the predicted interference patterns. For their independent contributions, Davisson and Thomson shared the 1937 Nobel Prize for physics. II. QUANTIZATION The idea that atoms might be like small solar systems, with negatively-charged particles in orbits around a positively-charged central particle, was first proposed by French physicist Jean Baptiste Perrin in 1901, and two years later in a far more detailed manner by Japanese physicist Hantaro Nagaoka. Unfortunately, almost no physicists took their ideas seriously. However, just a few years later, very careful scattering experiments performed by the New Zealander Ernest Rutherford in Manchester, England showed that some kind of planetary model was in fact correct. In 1910, Australian physicist Arthur Haas tried valiantly to incorporate Planck’s constant into a planetary-style atom but ran into roadblocks. Two years later, British physicist John William Nicholson went considerably further, but still, his theories did not match known data. Then in 1913, Danish physicist Niels Bohr entered the scene. Bohr knew well that a particle in a 5 FIG. 4: “Bohr-Atom” ambigram by Douglas Hofstadter, reproduced here with his permission. circular orbit is constantly undergoing accelerated motion (change of direction being a type of acceleration), and that, according to classical electromagnetic theory, an accelerating charged particle must radiate energy away, so it will quickly lose all of its energy and the system will collapse. In other words, according to classical physics, any planetary model of an atom is unstable. To explain the stability of atoms, Bohr introduced some counterintuitive and highly revolutionary concepts. For simplicity, we will deviate slightly from Bohr’s original way of presenting his model, and will instead present it using de Broglie’s wave–particle hypothesis described above.
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