Feynman Simplified 3A: Quantum Mechanics Part One Everyone’s Guide to the Feynman Lectures on Physics by Robert L. Piccioni, Ph.D. Copyright © 2014 by Robert L. Piccioni Published by Real Science Publishing 3949 Freshwind Circle Westlake Village, CA 91361, USA Edited by Joan Piccioni All rights reserved, including the right of reproduction in whole or in part, in any form. Visit our web site www.guidetothecosmos.com Everyone’s Guide to the Feynman Lectures on Physics Feynman Simplified gives mere mortals access to the fabled Feynman Lectures on Physics. Quantum mechanics is rarely taught well in introductory physics courses, largely because this challenging subject was not well taught to many of today’s instructors. Few had the opportunity to learn quantum mechanics from some who understood it profoundly; almost none learned it from one of its creators. Here more than anywhere else, Feynman excels. Here more than anywhere else, Feynman Simplified can help you learn from the very best, but at a humane pace. This Book Feynman Simplified: 3A covers the first half of Volume 3 and chapters 37 and 38 of Volume 1 of The Feynman Lectures on Physics. The topics we explore include: Why the Micro-World is Different. Quantization and Particle-Wave Duality Indeterminism and the Uncertainty Principle Probabilities and Amplitudes Identical Particle Phenomena Bosons and Spin One Fermions and Spin One-Half Time Evolution and the Hamiltonian Operator The Two-State Ammonia Maser Readers will greatly benefit from a prior understanding of the material in Feynman Simplified 1A, 1B and 1C. A familiarity with elementary calculus is assumed. To find out about other eBooks in the Feynman Simplified series, and to receive corrections and updates, click HERE. Please help us make Feynman Simplified even better! Physics books are never completely error-free, and all become outdated when new discoveries are made. I welcome your comments and suggestions. Please contact me through my WEBSITE. If you enjoy this eBook please do me the great favor of rating it on Amazon.com or BN.com. Table of Contents Chapter 1: What Is Quantum Mechanics? Chapter 2: Particle-Wave Duality & Uncertainty Chapter 3: Particles, Waves & Particle-Waves Chapter 4: Probability Amplitudes Chapter 5: Identical Particles Chapter 6: Boson & Fermion Behaviors Chapter 7: Spin One Chapter 8: Rotations for Spin ½ Chapter 9: Time Dependence of Amplitudes Chapter 10: The Hamiltonian Chapter 11: Ammonia Maser Chapter 12: Review of Quantum Mechanics, Part One Chapter 1 What Is Quantum Mechanics? Much of the material in this chapter supplements the Feynman Lectures. Quantum mechanics is the physical theory of elementary particles, how they interact with one another, and how they form atoms and larger structures. It is often said that quantum mechanics is strange, “unnatural”, and bizarrely contrary to our innate sense of how things “really” are. For example, quantum mechanics says objects can be in different places at the same time, and can be simultaneously right-side-up and upside-down. It says particles are both everywhere and nowhere, until we look at them. To that last assertion, Einstein scoffed: “Would the Moon disappear if we did not look at it?” Moon no, but electrons yes. Other eminent physicists have also found quantum mechanics astonishing, including two Nobel Laureates honored for developing quantum mechanics: "If quantum mechanics hasn't profoundly shocked you, you haven't understood it yet" — Niels Bohr "I think that I can safely say that no one understands quantum mechanics" — Richard Feynman Quantum mechanics is strange, but true. Quantum mechanics correctly describes nature’s fundamental processes, the nuts-and-bolts of reality at its core. This strange theory is one of the most extensively tested and precisely confirmed creations of the human mind. The everyday world we perceive is a hazy, superficial, diluted version of the tempestuous reality at nature’s core. Perhaps, it is our perception of reality that is unnatural. How can Feynman say “no one understands quantum mechanics” when he and many others have filled library shelves with books explaining it? Let me address that with an analogy. Many people say they understand computers because they can email and surf the web. But few dive inside the box. And even fewer comprehend the internal structure of all those gray plastic centipedes that populate its guts. We know how to use quantum mechanics, but its gray centipedes are still bewildering. Quantum mechanics is truly how the world works: it is the heart of physics. Although many conclusions of quantum mechanics defy our intuition, physicists believe we now know all its rules. We can solve all its equations, even if we must laugh at some of the answers. In this sense, quantum mechanics is a mystery that we have solved but not fully digested. Key Principles of Quantum Mechanics The greatest physical theories blossom from just a few remarkable but simple-sounding ideas. Galileo’s Principle of Relativity has one idea: only relative velocities are physically meaningful. Einstein’s Special Theory of Relativity has one: the speed of light never changes. Einstein’s General Theory of Relativity has one: locally, gravity is equivalent to acceleration. I say quantum mechanics has two key principles. These two ideas are so interrelated that they could be combined into one idea. Perhaps, but it is easier to learn them separately, and put them together later. The two principles are: 1. Quantization 2. Particle-Wave Duality We will first describe what quantization means, then examine the development of particle-wave duality, and ultimately discover how duality leads to quantization. Quantization Quantization is the simple notion that some things in nature are countable — they come in integer quantities. Money is quantized. In the U.S., the amount of money in any transaction is an integer multiple of 1¢. In Japan it’s 1¥. Particles and people are also quantized: there is no such thing as 1.37 electrons or π people. Conversely, at least on a human scale, water, air, space, and time appear to be continuous. As our understanding has advanced, we have discovered that more and more entities that seem continuous are actually quantized. Perhaps we will ultimately discover that everything really is quantized. The staircase and ramp in Figure 1-1 illustrate the difference between quantized and continuous. On a ramp, every elevation is possible; just slide along to the right spot. Figure 1-1 Staircase and Ramp On a staircase, only a few discrete elevations are possible. One can be as high as the second step, or the third step, but never as high as the 2.7th step, because no such step exists. On a staircase, elevation changes abruptly and substantially. The micro-world of atoms and particles is replete with significant staircases — the steps are large and dramatically impact natural processes in this realm. We, however, live on a much larger scale, billions of times larger. As one’s perception zooms out from the atomic scale toward the human scale, the steps in nature’s staircases appear ever smaller and ever more numerous, as depicted in Figure 1- 2. Figure 1-2 Staircase Viewed On Ever-Larger Scales Eventually, the steps become too small to be seen individually, and we observe smooth ramps instead of staircases. Nature does not have one set of laws for the atomic scale and another set of laws for the human scale. Nature’s laws are universal; these steps exist always and everywhere. But at our level, the steps are so small that they almost never make a discernible difference. In our macro-world, planets can orbit stars at any distance, baseballs can have any speed, and nothing is ever in two different places at the same time. In the micro-world, electrons circle nuclei only in specific orbits, only with specific energies, and are everywhere simultaneously, until the macro-world intervenes. Quantum mechanics is all about understanding what happens when the staircase steps are important, when quantization dominates. The Beginning of Quantum Theory Feynman Simplified 1B Chapter 20 discusses how and why the first glimmers of quantum mechanics emerged in 1900, when Max Planck “solved” the Ultraviolet Catastrophe in the theory of thermal radiation. Thermal radiation is the light (often infrared light) that objects emit due to their heat energy, the random motion of “hot” atoms. Recall that classical physics predicts thermal radiation has the same wave amplitude at each frequency. Since frequency f has an unlimited range, when one integrates to infinite f, the total radiation becomes infinite. Lighting a match should cremate the entire universe — clearly that is ridiculous. For theorists it was a catastrophe at high frequency, ultraviolet and beyond. To fix this, Planck postulated that thermal radiation is quantized. He said energy is emitted only in integer multiples of hf: he said the emitted energy E must equal nhf, where n is an integer and h is a constant named in Planck’s honor. For a high enough frequency f, the available energy is less than 1•hf. Allowing only integer multiples of hf precludes any n•hf except 0•hf. This truncates high frequency emission and makes the integral finite. Planck offered no physical rationale for quantizing thermal emission; he viewed it as simply a mathematical formalism that worked. Truly, this was a solution without an explanation. But a profound explanation came five years later. Einstein & The Photoelectric Effect In 1887, Heinrich Hertz observed that when light strikes a metal, an electric current is produced sometimes. Careful experiments determined this photoelectric effect is due to light knocking electrons out of the atoms of the metal. In 1839, A. Edmond Becquerel discovered the closely related photovoltaic effect in which light sometimes elevates atomic electrons to higher energy states. Both originate from the same basic physics. But mysteriously, both effects only happen sometimes. Knocking an electron away from a positively charged nucleus requires energy. Since light carries energy, any beam of light of sufficient intensity should eject electrons. But here’s the mysterious part: blue light ejects electrons but red light doesn’t. Even extremely intense beams of red light fail to eject electrons. Conversely, even low intensity beams of blue light eject a few electrons. What’s wrong with red? Einstein solved this mystery in 1905 by proclaiming that light is both a particle and a wave. This was heresy ─ every other physicist was certain that waves and particles were entirely distinct and incompatible. Yet, Einstein claimed these two very different phenomena are actually two aspects of a more fundamental entity. Einstein said light beams are composed of vast numbers of individual particles, which we now call photons. He said each photon’s energy E is proportional to its frequency f, according to: E=hf. Blue light has twice the frequency of red light, hence a blue photon has twice the energy of a red photon. When a beam of photons strikes a metal surface, Einstein explained, the fundamental interaction is one photon hitting one electron ─ there’s no double-teaming. It takes one good whack to eject an electron; a thousand little nudges wouldn’t do the trick. An electron is ejected only if a single photon has sufficient energy. A blue photon does have enough energy, but a red photon doesn’t. That’s why blue works and red doesn’t. Einstein realized his concept of light being individual particles fit perfectly with Planck’s quantization of thermal radiation. Since particles always come in integer quantities, it is evident that the energy of radiation must be quantized, an integer multiple of the energy of one photon, hf. (Recall