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Entanglement: the greatest mystery in physics
Entanglement: the greatest mystery in physics
Aczel, Amir D
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Since cyberspace  a word coined by a science fiction writer  became reality, the lines between "science" and "science fiction" have become increasingly blurred. Now, the young field of quantum mechanics holds out the promise that some of humanity's wildest dreams may be realized. Serious scientists, working off of theories first developed by Einstein and his colleagues seventy years ago, have been investigating the phenomenon known as "entanglement," one of the strangest aspects of the strange universe of quantum mechanics. According to Einstein, quantum mechanics required entanglement  the idea that subatomic particles could become inextricably linked, and that a change to one such particle would instantly be reflected in its counterpart, even if a universe separated them. Einstein felt that if the quantum theory could produce such incredibly bizarre effects, then it had to be invalid. But new experiments both in the United States and Europe show not only that it does happen, but that it may lead to unbreakable codes, and even teleportation ...Dust jacket. This is a book about science, the making of science, the philosophy that underlies science, the mathematical underpinnings of science, the experiments that verify and expose nature's inner secrets, and the lives of the scientists who pursue nature's most bizarre effect. These scientists [are] relentlessly in search of knowledge about a deep mystery of nature  entanglement ... This book tells the story of this search ... [This book] is about the search called modern science.Pref. Read more...
Abstract: Since cyberspace  a word coined by a science fiction writer  became reality, the lines between "science" and "science fiction" have become increasingly blurred. Now, the young field of quantum mechanics holds out the promise that some of humanity's wildest dreams may be realized. Serious scientists, working off of theories first developed by Einstein and his colleagues seventy years ago, have been investigating the phenomenon known as "entanglement," one of the strangest aspects of the strange universe of quantum mechanics. According to Einstein, quantum mechanics required entanglement  the idea that subatomic particles could become inextricably linked, and that a change to one such particle would instantly be reflected in its counterpart, even if a universe separated them. Einstein felt that if the quantum theory could produce such incredibly bizarre effects, then it had to be invalid. But new experiments both in the United States and Europe show not only that it does happen, but that it may lead to unbreakable codes, and even teleportation ...Dust jacket. This is a book about science, the making of science, the philosophy that underlies science, the mathematical underpinnings of science, the experiments that verify and expose nature's inner secrets, and the lives of the scientists who pursue nature's most bizarre effect. These scientists [are] relentlessly in search of knowledge about a deep mystery of nature  entanglement ... This book tells the story of this search ... [This book] is about the search called modern science.Pref
Abstract: Since cyberspace  a word coined by a science fiction writer  became reality, the lines between "science" and "science fiction" have become increasingly blurred. Now, the young field of quantum mechanics holds out the promise that some of humanity's wildest dreams may be realized. Serious scientists, working off of theories first developed by Einstein and his colleagues seventy years ago, have been investigating the phenomenon known as "entanglement," one of the strangest aspects of the strange universe of quantum mechanics. According to Einstein, quantum mechanics required entanglement  the idea that subatomic particles could become inextricably linked, and that a change to one such particle would instantly be reflected in its counterpart, even if a universe separated them. Einstein felt that if the quantum theory could produce such incredibly bizarre effects, then it had to be invalid. But new experiments both in the United States and Europe show not only that it does happen, but that it may lead to unbreakable codes, and even teleportation ...Dust jacket. This is a book about science, the making of science, the philosophy that underlies science, the mathematical underpinnings of science, the experiments that verify and expose nature's inner secrets, and the lives of the scientists who pursue nature's most bizarre effect. These scientists [are] relentlessly in search of knowledge about a deep mystery of nature  entanglement ... This book tells the story of this search ... [This book] is about the search called modern science.Pref
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Year:
2001
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Recording for the Blind & Dyslexic
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english
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303
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1568582323
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entanglement This Page Intentionally Left Blank E N TA N G L E M E N T The Greatest Mystery in Physics amir d. aczel F O U R WA L L S E I G H T W I N D OW S N E W YO R K © 2001 Amir D. Aczel Published in the United States by: Four Walls Eight Windows 39 West 14th Street, room 503 New York, N.Y., 10011 Visit our website at http://www.4w8w.com First printing September 2002. All rights reserved. No part of this book may be reproduced, stored in a data base or other retrieval system, or transmitted in any form, by any means, including mechanical, electronic, photocopying, recording, or otherwise, without the prior written permission of the publisher. Library of Congress CataloginginPublication Data: Entanglement: the greatest mystery in physics/ by Amir D. Aczel. p. cm. Includes bibliographical references and index. isbn 1568582323 1. Quantum theory. I. Title. qc174.12.A29 2002 530.12—dc21 2002069338 10 9 8 7 6 5 4 3 2 1 Printed in the United States Typeset and designed by Terry Bain Illustrations, unless otherwise noted, by Ortelius Design. for Ilana This Page Intentionally Left Blank Contents Preface / ix A Mysterious Force of Harmony / 1 Before the Beginning / 7 Thomas Young’s Experiment / 17 Planck’s Constant / 29 The Copenhagen School / 37 De Broglie’s Pilot Waves / 49 Schrödinger and His Equation / 55 Heisenberg’s Microscope / 73 Wheeler’s Cat / 83 The Hungarian Mathematician / 95 Enter Einstein / 103 Bohm and Aharanov / 123 John Bell’s Theorem / 137 The Dream of Clauser, Horne, and Shimony / 149 Alain Aspect / 177 Laser Guns / 191 Triple Entanglement / 203 The TenKilometer Experiment / 235 Teleportation: “Beam Me Up, Scotty” / 241 Quantum Magic: What Does It All Mean? / 249 Acknowledgements / 255 References / 266 Index / 269 vii This Page Intentionally Left Blank Preface “My own suspicion is that the universe is not only queerer than we suppose, but queerer than we can suppose.” —J.B.S. Haldane I n the fall of 1972, I was an undergraduate in mathematics and physics ; at the University of California at Berkeley. There I had the good fortune to attend a special lecture given on campus by Werner Heisenberg, one of the founders of the quantum theory. While today I have some reservations about the role Heisenberg played in history—at the time other scientists left in protest of Nazi policies, he stayed behind and was instrumental in Hitler’s attempts to develop the Bomb—nevertheless his talk had a profound, positive effect on my life, for it gave me a deep appreciation for the quantum theory and its place in our efforts to understand nature. Quantum mechanics is the strangest field in all of science. From our everyday perspective of life on Earth, nothing ix x 00 entanglement makes sense in quantum theory, the theory about the laws of nature that govern the realm of the very small (as well as some large systems, such as superconductors). The word itself, quantum, denotes a small packet of energy—a very small one. In quantum mechanics, as the quantum theory is called, we deal with the basic building blocks of matter, the constituent particles from which everything in the universe is made. These particles include atoms, molecules, neutrons, protons, electrons, quarks, as well as photons—the basic units of light. All these objects (if indeed they can be called objects) are much smaller than anything the human eye can see. At this level, suddenly, all the rules of behavior with which we are familiar no longer hold. Entering this strange new world of the very small is an experience as baffling and bizarre as Alice’s adventures in Wonderland. In this unreal quantum world, particles are waves, and waves are particles. A ray of light, therefore, is both an electromagnetic wave undulating through space, and a stream of tiny particles speeding toward the observer, in the sense that some quantum experiments or phenomena reveal the wave nature of light, while others the particle nature of the same light—but never both aspects at the same time. And yet, before we observe a ray of light, it is both a wave and a stream of particles. In the quantum realm everything is fuzzy—there is a hazy quality to all the entities we deal with, be they light or electrons or atoms or quarks. An uncertainty principle reigns in quantum mechanics, where most things cannot be seen or felt or known with precision, but only through a haze of probability and chance. Scientific predictions about outcomes preface 00 xi are statistical in nature and are given in terms of probabilities—we can only predict the most likely location of a particle, not its exact position. And we can never determine both a particle’s location and its momentum with good accuracy. Furthermore, this fog that permeates the quantum world can never go away. There are no “hidden variables,” which, if known, would increase our precision beyond the natural limit that rules the quantum world. The uncertainty, the fuzziness, the probabilities, the dispersion simply cannot go away—these mysterious, ambiguous, veiled elements are an integral part of this wonderland. Even more inexplicable is the mysterious superposition of states of quantum systems. An electron (a negativelycharged elementary particle) or photon (a quantum of light) can be in a superposition of two or more states. No longer do we speak about “here or there;” in the quantum world we speak about “here and there.” In a certain sense, a photon, part of a stream of light shone on a screen with two holes, can go through both holes at the same time, rather than the expected choice of one hole or the other. The electron in orbit around the nucleus is potentially at many locations at the same time. But the most perplexing phenomenon in the bizarre world of the quantum is the effect called entanglement. Two particles that may be very far apart, even millions or billions of miles, are mysteriously linked together. Whatever happens to one of them immediately causes a change in the other one.1 What I learned from Heisenberg’s lecture thirty years ago was that we must let go of all our preconceptions about the world derived from our experience and our senses, and instead let mathematics lead the way. The electron lives in a xii 00 entanglement different space from the one in which we live. It lives in what mathematicians call a Hilbert space, and so do the other tiny particles and photons. This Hilbert space, developed by mathematicians independently of physics, seems to describe well the mysterious rules of the quantum world—rules that make no sense when viewed with an eye trained by our everyday experiences. So the physicist working with quantum systems relies on the mathematics to produce predictions of the outcomes of experiments or phenomena, since this same physicist has no natural intuition about what goes on inside an atom or a ray of light or a stream of particles. Quantum theory taxes our very concept of what constitutes science— for we can never truly “understand” the bizarre behavior of the very small. And it taxes our very idea of what constitutes reality. What does “reality” mean in the context of the existence of entangled entities that act in concert even while vast distances apart? The beautiful mathematical theory of Hilbert space, abstract algebra, and probability theory—our mathematical tools for handling quantum phenomena—allow us to predict the results of experiments to a stunning level of accuracy; but they do not bring us an understanding of the underlying processes. Understanding what really happens inside the mysterious box constituting a quantum system may be beyond the powers of human beings. According to one interpretation of quantum mechanics, we can only use the box to predict outcomes. And these predictions are statistical in nature. There is a very strong temptation to say: “Well, if the preface 00 xiii theory cannot help us understand what truly goes on, then the theory is simply not complete. Something is missing— there must be some missing variables, which, once added to our equations, would complete our knowledge and bring us the understanding we seek.” And, in fact, the greatest scientist of the twentieth century, Albert Einstein, posed this very challenge to the nascent quantum theory. Einstein, whose theories of relativity revolutionized the way we view space and time, argued that quantum mechanics was excellent as a statistical theory, but did not constitute a complete description of physical reality. His wellknown statement that “God doesn’t play dice with the world” was a reflection of his belief that there was a deeper, nonprobabilistic layer to the quantum theory which had yet to be discovered. Together with his colleagues Podolsky and Rosen, he issued a challenge to quantum physics in 1935, claiming that the theory, was incomplete. The three scientists based their argument on the existence of the entanglement phenomenon, which in turn had been deduced to exist based on mathematical considerations of quantum systems. At his talk at Berkeley in 1972, Heisenberg told the story of his development of the approach to the quantum theory called matrix mechanics. This was one of his two major contributions to the quantum theory, the other being the uncertainty principle. Heisenberg recounted how, when aiming to develop his matrix approach in 1925, he did not even know how to multiply matrices (an elementary operation in mathematics). But he taught himself how to do so, and his theory followed. Mathematics thus gave scientists the rules of xiv 00 entanglement behavior in the quantum world. Mathematics also led Schrödinger to his alternative, and simpler, approach to quantum mechanics, the wave equation. Over the years, I’ve followed closely the developments in the quantum theory. My books have dealt with mysteries in mathematics and physics. Fermat’s Last Theorem told the story of the amazing proof of a problem posed long ago; God’s Equation was the tale of Einstein’s cosmological constant and the expansion of the universe; The Mystery of the Aleph was a description of humanity’s attempt to understand infinity. But I’ve always wanted to address the secrets of the quantum. A recent article in The New York Times provided me with the impetus I needed. The article dealt with the challenge Albert Einstein and his two colleagues issued to the quantum theory, claiming that a theory that allowed for the “unreal” phenomenon of entanglement had to be incomplete. Seven decades ago, Einstein and his scientific allies imagined ways to prove that quantum mechanics, the strange rules that describe the world of the very small, were just too spooky to be true. Among other things, Einstein showed that, according to quantum mechanics, measuring one particle could instantly change the properties of another particle, no matter how far apart they were. He considered this apparent actionatadistance, called entanglement, too absurd to be found in nature, and he wielded his thought experiments like a weapon to expose the strange implications that this process would have if it could happen. But experiments described in three forthcoming papers in the journal Physical Review Letters give a measure of just how badly Einstein has been routed. The preface 00 xv experiments show not only that entanglement does happen— which has been known for some time—but that it might be used to create unbreakable codes . . .2 As I knew from my study of the life and work of Albert Einstein, even when Einstein thought he was wrong (about the cosmological constant), he was right. And as for the quantum world—Einstein was one of the developers of the theory. I knew quite well that—far from being wrong—Einstein’s paper of 1935, obliquely alluded to in the Times article, was actually the seed for one of the most important discoveries in physics in the twentieth century: the actual discovery of entanglement through physical experiments. This book tells the story of the human quest for entanglement, the most bizarre of all the strange aspects of quantum theory. Entangled entities (particles or photons) are linked together because they were produced by some process that bound them together in a special way. For example, two photons emitted from the same atom as one of its electrons descends down two energy levels are entangled. (Energy levels are associated with the orbit of an electron in the atom.) While neither flies off in a definite direction, the pair will always be found on opposite sides of the atom. And such photons or particles, produced in a way that links them together, remain intertwined forever. Once one is changed, its twin—wherever it may be in the universe—will change instantaneously. In 1935, Einstein, together with his colleagues Rosen and Podolsky, considered a system of two distinct particles that was permissible under the rules of quantum mechanics. The state of this system was shown to be entangled. Einstein, xvi 00 entanglement Podolsky, and Rosen used this theoretical entanglement of separated particles to imply that if quantum mechanics allowed such bizarre effects to exist, then something must be wrong, or incomplete, as they put it, about the theory. In 1957, the physicists David Bohm and Yakir Aharonov analyzed the results of an experiment that had been performed by C.S. Wu and I. Shaknov almost a decade earlier, and their analysis provided the first hint that entanglement of separated systems may indeed take place in nature. Then in 1972, two American physicists, John Clauser and Stuart Freedman, produced evidence that entanglement actually exists. And a few years later, the French physicist Alain Aspect and his colleagues provided more convincing and complete evidence for the existence of the phenomenon. Both groups followed the seminal theoretical work in this area by John S. Bell, an Irish physicist working in Geneva, and set out to prove that the EinsteinPodolskyRosen thought experiment was not an absurd idea to be used to invalidate the completeness of the quantum theory, but rather the description of a real phenomenon. The existence of the phenomenon provides evidence in favor of quantum mechanics and against a limiting view of reality. A NOTE TO THE READER Quantum theory itself, and in particular the concept of entanglement, is very difficult for anyone to understand— even for accomplished physicists or mathematicians. I therefore structured the book in such a way that the ideas and preface 00 xvii concepts discussed are constantly being explained and reexplained in various forms. This approach makes sense when one considers the fact that some of the brightest scientists today have spent lifetimes working on entanglement; the truth is that even after decades of research, it is difficult to find someone who will admit to understanding the quantum theory perfectly well. These physicists know how to apply the rules of quantum mechanics in a variety of situations. They can perform calculations and make predictions to a very high degree of accuracy, which is rare in some other areas. But often these bright scientists will profess that they do not truly understand what goes on in the quantum world. It is exactly for this reason that in chapter after chapter in this book I repeat the concepts of quantum theory and entanglement, every time from a slightly different angle, or as explained by a different scientist. I have made an effort to incorporate the largest possible number of original figures, obtained from scientists, describing actual experiments and designs. My hope is that these figures and graphs will help the reader understand the mysterious and wonderful world of the quantum and the setting within which entanglement is produced and studied. In addition, where appropriate, I have incorporated a number of equations and symbols. I did so not to baffle the reader, but so that readers with an advanced preparation in science might gain more from the presentation. For example, in the chapter on Schrödinger’s work I include the simplest (and most restricted) form of Schrödinger’s famous equation for the benefit of those who might want to see what the equation looks like. It is perfectly fine for a reader, if she so chooses, to skip over the equations xviii 00 entanglement and read on, and anyone doing so will suffer no loss of information or continuity. This is a book about science, the making of science, the philosophy that underlies science, the mathematical underpinnings of science, the experiments that verify and expose nature’s inner secrets, and the lives of the scientists who pursue nature’s most bizarre effect. These scientists constitute a group of the greatest minds of the twentieth century, and their combined lifetimes span the entire century. These people, relentlessly in search of knowledge about a deep mystery of nature—entanglement—led and lead lives today that are, themselves, entangled with one another. This book tells the story of this search, one of the greatest scientific detective stories in history. And while the science of entanglement has also brought about the birth of new and very exciting technologies, the focus of this book is not on the technologies spawned by the research. Entanglement is about the search called modern science. 1 A Mysterious Force of Harmony “Alas, to wear the mantle of Galileo it is not enough that you be persecuted by an unkind establishment, you must also be right.” —Robert Park I s it possible that something that happens here will instantaneously make something happen at a far away location? If we measure something in a lab, is it possible that at the same moment, a similar event takes place ten miles away, on the other side of the world, or on the other side of the universe? Surprisingly, and against every intuition we may possess about the workings of the universe, the answer is yes. This book tells the story of entanglement, a phenomenon in which two entities are inexorably linked no matter how far away from each other they may be. It is the story of the people who have spent lifetimes seeking evidence that such a bizarre effect—predicted by the quantum theory and brought to wide scientific attention by Einstein—is indeed an integral part of nature. As these scientists studied such effects, and produced defin1 2 00 entanglement itive evidence that entanglement is a reality, they have also discovered other, equally perplexing, aspects of the phenomenon. Imagine Alice and Bob, two happily married people. While Alice is away on a business trip, Bob meets Carol, who is married to Dave. Dave is also away at that time, on the other side of the world and nowhere near any of the other three. Bob and Carol become entangled with each other; they forget their respective spouses and now strongly feel that they are meant to stay a couple forever. Mysteriously, Alice and Dave—who have never met—are now also entangled with each other. They suddenly share things that married people do, without ever having met. If you substitute for the people in this story particles labeled A, B, C, and D, then the bizarre outcome above actually occurs. If particles A and B are entangled, and so are C with D, then we can entangle the separated particles A and D by passing B and C through an apparatus that entangles them together. Using entanglement, the state of a particle can also be teleported to a faraway destination, as happens to Captain Kirk on the television series “Star Trek” when he asks to be beamed back up to the Enterprise. To be sure, no one has yet been able to teleport a person. But the state of a quantum system has been teleported in the laboratory. Furthermore, such incredible phenomena can now be used in cryptography and computing. In such futuristic applications of technology, the entanglement is often extended to more than two particles. It is possible to create triples of particles, for example, such that all three are 100% correlated with each other—whatever happens to one particle causes a similar instantaneous change in a mysterious force of harmony 00 3 the other two. The three entities are thus inexorably interlinked, wherever they may be. One day in 1968, physicist Abner Shimony was sitting in his office at Boston University. His attention was pulled, as if by a mysterious force, to a paper that had appeared two years earlier in a littleknown physics journal. Its author was John Bell, an Irish physicist working in Geneva. Shimony was one of very few people who had both the ability and the desire to truly understand Bell’s ideas. He knew that Bell’s theorem, as explained and proved in the paper, allowed for the possibility of testing whether two particles, located far apart from each other, could act in concert. Shimony had just been asked by a fellow professor at Boston University, Charles Willis, if he would be willing to direct a new doctoral student, Michael Horne, in a thesis on statistical mechanics. Shimony agreed to see the student, but was not eager to take on a Ph.D. student in his first year of teaching at Boston University. In any case, he said, he had no good problem to suggest in statistical mechanics. But, thinking that Horne might find a problem in the foundations of quantum mechanics interesting, he handed him Bell’s paper. As Shimony put it, “Horne was bright enough to see quickly that Bell’s problem was interesting.” Michael Horne took Bell’s paper home to study, and began work on the design of an experiment that would use Bell’s theorem. Unbeknownst to the two physicists in Boston, at Columbia University in New York, John F. Clauser was reading the same paper by Bell. He, too, was mysteriously drawn to the 4 00 entanglement problem suggested by Bell, and recognized the opportunity for an actual experiment. Clauser had read the paper by Einstein, Podolsky, and Rosen, and thought that their suggestion was very plausible. Bell’s theorem showed a discrepancy between quantum mechanics and the “local hidden variables” interpretation of quantum mechanics offered by Einstein and his colleagues as an alternative to the “incomplete” quantum theory, and Clauser was excited about the possibility of an experiment exploiting this discrepancy. Clauser was skeptical, but he couldn’t resist testing Bell’s predictions. He was a graduate student, and everyone he talked to told him to leave it alone, to get his Ph.D., and not to dabble in science fiction. But Clauser knew better. The key to quantum mechanics was hidden within Bell’s paper, and Clauser was determined to find it. Across the Atlantic, a few years later, Alain Aspect was feverishly working in his lab in the basement of the Center for Research on Optics of the University of Paris in Orsay. He was racing to construct an ingenious experiment: one that would prove that two photons, at two opposite sides of his lab, could instantaneously affect each other. Aspect was led to his ideas by the same abstruse paper by John Bell. In Geneva, Nicholas Gisin met John Bell, read his papers and was also thinking about Bell’s ideas. He, too, was in the race to find an answer to the same crucial question: a question that had deep implications about the very nature of reality. But we are getting ahead of ourselves. The story of Bell’s ideas, which goes back to a suggestion made thirtyfive years a mysterious force of harmony 00 5 earlier by Albert Einstein, has its origins in humanity’s quest for knowledge of the physical world. And in order to truly understand these deep ideas, we must return to the past. This Page Intentionally Left Blank 2 Before the Beginning “Out yonder there was this huge world, which exists independently of us human beings and which stands before us like a great, eternal riddle, at least partially accessible to our inspection.” —Albert Einstein “The mathematics of quantum mechanics is straightforward, but making the connection between the mathematics and an intuitive picture of the physical world is very hard” —Claude N. CohenTannoudji I n the book of Genesis we read: “God said: Let there be light.” God then created heaven and earth and all things that filled them. Humanity’s quest for an understanding of light and matter goes back to the dawn of civilization; they are the most basic elements of the human experience. And, as Einstein showed us, the two are one and the same: both light and matter are forms of energy. People have always striven to understand what these forms of energy mean. What is the nature of matter? And what is light? The ancient Egyptians and Babylonians and their successors the Phoenicians and the Greeks tried to understand the mysteries of matter, and of light and sight and color. The Greeks looked at the world with the first modern intellectual eyes. With their curiosity about numbers and geometry, cou7 8 00 entanglement pled with a deep desire to understand the inner workings of nature and their environment, they gave the world its first ideas about physics and logic. To Aristotle (300 B.C.), the sun was a perfect circle in the sky, with no blemishes or imperfections. Eratosthenes of Cyrene (c. 276 B.C.194 B.C.) estimated the circumference of our planet by measuring the angle sunlight was making at Syene (modern Aswan), in Upper Egypt, against the angle it made at the same time farther north, in Alexandria. He came stunningly close to the earth’s actual circumference of 25,000 miles. The Greek philosophers Aristotle and Pythagoras wrote about light and its perceived properties; they were fascinated by the phenomenon. But the Phoenicians were the first people in history to make glass lenses, which allowed them to magnify objects and to focus light rays. Archaeologists have found 3,000yearold magnifying glasses in the region of the eastern Mediterranean that was once Phoenicia. Interestingly, the principle that makes a lens work is the slowingdown of light as it travels through glass. The Romans learned glassmaking from the Phoenicians, and their own glassworks became one of the important industries of the ancient world. Roman glass was of high quality and was even used for making prisms. Seneca (5 B.C.A.D. 45) was the first to describe a prism and the breakingdown of white light into its component colors. This phenomenon, too, is based on the speed of light. We have no evidence of any experiments carried out in antiquity to determine the speed of light. It seems that ancient peoples thought that light moved instantly from place to place. Because light before the beginning 00 9 is so fast, they could not detect the infinitesimal delays as light traveled from source to destination. The first attempt to study the speed of light did not come for another 1,600 years. Galileo was the first person known to have attempted to estimate the speed of light. Once again, experimentation with light had a close connection with glassmaking. After the Roman Empire collapsed in the fifth century, many Romans of patrician and professional backgrounds escaped to the Venetian lagoons and established the republic of Venice. They brought with them the art of making glass, and thus the glassworks on the island of Murano were established. Galileo’s telescopes were of such high quality—in fact they were far better than the first telescopes made in Holland— because he used lenses made of Murano glass. It was with the help of these telescopes that he discovered the moons of Jupiter and the rings of Saturn and determined that the Milky Way is a large collection of stars. In 1607, Galileo conducted an experiment on two hilltops in Italy, in which a lantern on one hill was uncovered. When an assistant on the other hilltop saw the light, he opened his own lantern. The person on the first hill tried to estimate the time between opening the first lantern and seeing the light return from the second one. Galileo’s quaint experiment failed, however, because of the tiny length of time elapsed between the sending of the first lantern signal and the return of the light from the other hilltop. It should be noted, anyway, that much of this time interval was due to the human response time in uncovering the second lantern rather than to the actual time light took to travel this distance. 10 00 entanglement Almost seventy years later, in 1676, the Danish astronomer Olaf Römer became the first scientist to calculate the speed of light. He accomplished this task by using astronomical observations of the moons of Jupiter, discovered by Galileo. Römer devised an intricate and extremely clever scheme by which he recorded the times of the eclipses of the moons of Jupiter. He knew that the earth orbits the sun, and that therefore the earth would be at different locations in space visavis Jupiter and its moons. Römer noticed that the times of disappearance of the moons of Jupiter behind the planet were not evenly spaced. As Earth and Jupiter orbit the sun, their distance from each other varies. Thus the light that brings us information on an eclipse of a Jovian moon takes different lengths of time to arrive on Earth. From these differences, and using his understanding of the orbits of Earth and Jupiter, Römer was able to calculate the speed of light. His estimate, 140,000 miles per second, was not quite the actual value of 186,000 miles per second. However, considering the date of the discovery and the fact that time was not measurable to great accuracy using the clocks of the seventeenth century, his achievement—the first measurement of the speed of light and the first proof that light does not travel at infinite speed—is an immensely valuable landmark in the history of science. Descartes wrote about optics in 1638 in his book Dioptrics, stating laws of the propagation of light: the laws of reflection and refraction. His work contained the seed of the most controversial idea in the field of physics: the ether. Descartes put forward the hypothesis that light propagates through a medium, and he named this medium the ether. Sci before the beginning 00 11 ence would not be rid of the ether for another three hundred years, until Einstein’s theory of relativity would finally deal the ether its fatal blow. Christiaan Huygens (16291695) and Robert Hooke (16351703) proposed the theory that light is a wave. Huygens, who as a sixteenyearold boy had been tutored by Descartes during his stay in Holland, became one of the greatest thinkers of the day. He developed the first pendulum clock and did other work in mechanics. His most remarkable achievement, however, was a theory about the nature of light. Huygens interpreted Römer’s discovery of the finite speed of light as implying that light must be a wave propagating through some medium. On this hypothesis, Huygens constructed an entire theory. Huygens visualized the medium as the ether, composed of an immense number of tiny, elastic particles. When these particles were excited into vibration, they produced light waves. In 1692, Isaac Newton (16431727) finished his book Opticks about the nature and propagation of light. The book was lost in a fire in his house, so Newton rewrote it for publication in 1704. His book issued a scathing attack on Huygens’s theory, and argued that light was not a wave but instead was composed of tiny particles traveling at speeds that depend on the color of the light. According to Newton, there are seven colors in the rainbow: red, yellow, green, blue, violet, orange, and indigo. Each color has its own speed of propagation. Newton derived his seven colors by an analogy with the seven main intervals of the musical octave. Further editions of his book continued Newton’s attacks on Huygens’s theories and intensified the debate as to whether light 12 00 entanglement is a particle or a wave. Surprisingly, Newton—who codiscovered the calculus and was one of the greatest mathematicians of all time—never bothered to address Römer’s findings about the speed of light, and neither did he give the wave theory the attention it deserved. But Newton, building on the foundation laid by Descartes, Galileo, Kepler, and Copernicus, gave the world classical mechanics, and, through it, the concept of causality. Newton’s second law says that force is equal to mass times acceleration: F=ma. Acceleration is the second derivative of position (it is the rate of change of the speed; and speed, in turn, is the rate of change of position). Newton’s law is therefore an equation with a (second) derivative in it. It is called a (secondorder) differential equation. Differential equations are very important in physics, since they model change. Newton’s laws of motion are a statement about causality. They deal with cause and effect. If we know the initial position and velocity of a massive body, and we know the force acting on it and the force’s direction, then we should be able to determine a final outcome: where will the body be at a later point in time. Newton’s beautiful theory of mechanics can predict the motion of falling bodies as well as the orbits of planets. We can use these causeandeffect relationships to predict where an object will go. Newton’s theory is a tremendous edifice that explains how large bodies—things we know from everyday life—can move from place to place, as long as their speeds or masses are not too great. For velocities approaching the speed of light, or masses of the order of magnitude of stars, Einstein’s general relativity is the correct theory, and classical, before the beginning 00 13 Newtonian mechanics breaks down. It should be noted, however, that Einstein’s theories of special and general relativity hold, with improvements over Newton, even in situations in which Newtonian mechanics is a good approximation. Similarly, for objects that are very small—electrons, atoms, photons—Newton’s theory breaks down as well. With it, we also lose the concept of causality. The quantum universe does not possess the causeandeffect structure we know from everyday life. Incidentally, for small particles moving at speeds close to that of light, relativistic quantum mechanics is the right theory. One of the most important principles in classical physics— and one that has great relevance to our story—is the principle of conservation of momentum. Conservation principles for physical quantities have been known to physicists for over three centuries. In his book, the Principia, of 1687, Newton presented his laws for the conservation of mass and momentum. In 1840, the German physician Julius Robert Mayer (18121878) deduced that energy was conserved as well. Mayer was working as a ship’s surgeon on a voyage from Germany to Java. While treating members of the ship’s crew for various injuries in the tropics, Dr. Mayer noticed that the blood oozing from their wounds was redder than the blood he saw in Germany. Mayer had heard of Lavoisier’s theory that body heat came from the oxidation of sugar in body tissue using oxygen from the blood. He reasoned that in the warm tropics the human body needed to produce less heat than it would in colder northern Europe, and hence that more oxygen remained in the blood of people in the tropics, making the blood redder. Using arguments about how the 14 00 entanglement body interacts with the environment—giving and receiving heat—Mayer postulated that energy was conserved. This idea was derived experimentally by Joule, Kelvin, and Carnot. Earlier, Leibniz had discovered that kinetic energy can be transformed into potential energy and vice versa. Energy in any of its forms (including mass) is conserved— that is, it cannot be created out of nothing. The same holds true for momentum, angular momentum, and electric charge. The conservation of momentum is very important to our story. Suppose that a moving billiards ball hits a stationary one. The moving ball has a particular momentum associated with it—the product of its mass by its speed, p=mv. This product of mass times speed, the momentum of the billiards ball, must be conserved within the system. Once one ball hits another, its speed slows down, but the ball that was hit now moves as well. The speed times mass for the system of these two objects must be the same as that of the system before the collision (the stationary ball had momentum zero, so it’s the momentum of the moving one that now gets split in two). This is demonstrated by the figure below, where after the collision the two balls travel in different directions. before the beginning 00 15 11 11 In any physical process, total input momentum equals total output momentum. This principle, when applied within the world of the very small, will have consequences beyond this simple and intuitive idea of conservation. In quantum mechanics, two particles that interact with each other at some point—in a sense like the two billiards balls of this example—will remain intertwined with each other, but to a greater extent than billiards balls: whatever should happen to one of them, no matter how far it may be from its twin, will immediately affect the twin particle. This Page Intentionally Left Blank 3 Thomas Young’s Experiment “We choose to examine a phenomenon (the doubleslit experiment) that is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery.” —Richard Feynman T homas Young (17731829) was a British physician and physicist whose experiment changed the way we think about light. Young was a child prodigy who learned to read at age two, and by age six had read the Bible twice and learned Latin. Before the age of 19, he was fluent in thirteen languages, including Greek, French, Italian, Hebrew, Chaldean, Syriac, Samaritan, Persian, Ethiopic, Arabic, and Turkish. He studied Newton’s calculus and his works on mechanics and optics, as well as Lavoisier’s Elements of Chemistry. He also read plays, studied law, and learned politics. In the late 1700s Young studied medicine in London, Edinburgh, and Göttingen, where he received his M.D. In 1794, he was elected to the Royal Society. Three years later, he moved to Cambridge University, where he received a second 17 18 00 entanglement M.D. and joined the Royal College of Physicians. After a wealthy uncle left him a house in London and a large cash inheritance, Young moved to the capital and established a medical practice there. He was not a successful doctor, but instead devoted his energies to study and scientific experiments. Young studied vision and gave us the theory that the eye contains three types of receptors for light of the three basic colors, red, blue, and green. Young contributed to natural philosophy, physiological optics, and was one of the first to translate Egyptian hieroglyphics. His greatest contribution to physics was his effort to win acceptance of the wave theory of light. Young conducted the nowfamous doubleslit experiment on light, demonstrating the wavetheory effect of interference. In his experiment, Young had a light source and a barrier. He cut two slits in the barrier, through which the light from the source could pass. Then he placed a screen behind the barrier. When Young shone the light from the source on the barrier with the two slits, he obtained an interference pattern. x Source Wall Backstop thomas young’s experiment 00 19 An interference pattern is the hallmark of waves. Waves interfere with each other, while particles do not. Richard Feynman considered Young’s result of the doubleslit experiment— as it appears in the case of electrons and other quanta that can be localized—so important that he devoted much of the first chapter of the third volume of his renowned textbook, The Feynman Lectures on Physics, to this type of experiment.3 He believed that the result of the doubleslit experiment was the fundamental mystery of quantum mechanics. Richard Feynman demonstrated in his Lectures the idea of interference of waves versus the noninterference of particles using bullets. Suppose a gun shoots bullets randomly at a barrier with two slits. The pattern is as shown below. x Moveable Detector 1 Gun 2 Wall (a) P12 P1 x P2 P12 = P1+ P2 Backstop (b) (c) Water waves, if passed through a barrier with two slits, make the pattern below. Here we find interference, as in the Young experiment with light, because we have classical waves. The amplitudes of two waves may add to each other, producing a peak on the screen, or they may interfere destructively, producing a trough. 20 00 entanglement x Detector I12 1 Wave Source 2 Wall Absorber So the Young experiment demonstrates that light is a wave. But is light really a wave? The duality between light as wave and light as a stream of particles still remains an important facet of physics in the twentyfirst century. Quantum mechanics, developed in the 1920s and 1930s, in fact reinforces the view that light is both particle and wave. The French physicist Louis de Broglie argued in 1924 that even physical bodies such as electrons and other particles possess wave properties. Experiments proved him right. Albert Einstein, in deriving the photoelectric effect in 1905, put forward the theory that light was made of particles, just as Newton had argued. Einstein’s light particle eventually became known as a photon, a name derived from the Greek word for light. According to the quantum theory, light may be both a wave and a particle, and this duality—and apparent paradox—is a mainstay of modern physics. Mysteriously, light exhibits both phenomena that are characteristic of waves, interference and diffraction, and phenomena of particles, localized in their interaction thomas young’s experiment 00 21 with matter. Two light rays interfere with each other in a way that is very similar to sound waves emanating from two stereo speakers, for example. On the other hand, light interacts with matter in a way that only particles can, as happens in the photoelectric effect. Young’s experiment showed that light was a wave. But we also know that light is, in a way, a particle: a photon. In the twentieth century, the Young experiment was repeated with very weak light—light that was produced as one photon at a time. Thus, it was very unlikely that several photons would be found within the experimental apparatus at the same time. Stunningly, the same interference pattern appeared as enough time elapsed so that the photons, arriving one at a time, accumulated on the screen. What was each photon interfering with, if it was alone in the experimental apparatus? The answer seemed to be: with itself. In a sense, each photon went through both slits, not one slit, and as it appeared on the other side, it interfered with itself. The Young experiment has been carried out with many entities we consider to be particles: electrons, since the 1950s; neutrons, since the 1970s; and atoms, since the 1980s. In each case, the same interference pattern occurred. These findings demonstrated the de Broglie principle, according to which particles also exhibit wave phenomena. For example, in 1989, A. Tonomura and colleagues performed a doubleslit experiment with electrons. Their results are shown below, clearly demonstrating an interference pattern. 22 00 entanglement Anton Zeilinger and colleagues demonstrated the same pattern for neutrons, traveling at only 2 km/second, in 1991. Their results are shown below. thomas young’s experiment 00 23 INTENSITY (Neutrons/125min) 5000 4000 3000 2000 1000 100 m 0 SCANNING SLIT POSITION The same pattern was shown with atoms. This demonstrated that the duality between particles and waves manifests itself even for larger entities. INTENSITY (counts/5 min) 300 200 100 10 m 0 SCANNING GRATING POSITION Anton Zeilinger and his colleagues at the University of Vienna, where Schrödinger and Mach had worked, went one 24 00 entanglement step further. They extended our knowledge about quantum systems to entities that one would not necessarily associate any more with the world of the very small. (Although it should be pointed out that physicists know macroscopic systems, such as superconductors, that behave quantummechanically.) A bucky ball is a molecule of sixty or seventy atoms of carbon arranged in a structure resembling a geodesic dome. Buckminster Fuller made such domes famous, and the bucky ball is named after him. A molecule of sixty atoms is a relatively large entity, as compared with an atom. And yet, the same mysterious interference pattern appeared when Zeilinger and his colleagues ran their experiment. The arrangement is shown below. In each case, we see that the particles behave like waves. These experiments were also carried out one particle at a time, and still the interference pattern remained. What were these particles interfering with? The answer is that, in a sense, each particle went not through one slit, but rather through both slits—and then the particle “interfered with itself.” thomas young’s experiment 00 25 What we are witnessing here is a manifestation of the quantum principle of superposition of states. The superposition principle says that a new state of a system may be composed from two or more states, in such a way that the new state shares some of the properties of each of the combined states. If A and B ascribe two different properties to a particle, such as being at two different places, then the superposition of states, written as A + B, has something in common both with state A and with state B. In particular, the particle will have nonzero probabilities for being in each of the two states, but not elsewhere, if the position of the particle is to be observed. In the case of the doubleslit experiment, the experimental setup provides the particle with a particular kind of superposition: The particle is in state A when it passes through slit A and in state B when it passes through slit B. The superposition of states is a combination of “particle goes through slit A” with “particle goes through slit B.” This superposition of states is written as A + B. The two paths are combined, and there are therefore two nonzero probabilities, if the particle is observed. Given that the particle is to be observed as it goes through the experimental setup, it will have a 50% chance of being observed to go through slit A and a 50% chance that it will be observed to go through slit B. But if the particle is not observed as it goes through the experimental setup, only at the end as it collects on the screen, the superposition holds through to the end. In a sense, then, the particle has gone through both slits, and as it arrived at the end of the experimental setup, it interfered with itself. Superpo 26 00 entanglement sition of states is the greatest mystery in quantum mechanics. The superposition principle encompasses within itself the idea of entanglement. WHAT IS ENTANGLEMENT? Entanglement is an application of the superposition principle to a composite system consisting of two (or more) subsystems. A subsystem here is a single particle. Let’s see what it means when we say that the two particles are entangled. Suppose that particle 1 can be in one of two states, A or C, and that these states represent two contradictory properties, such as being at two different places. Particle 2, on the other hand, can be in one of two states, B or D. Again these states could represent contradictory properties such as being at two different places. The state AB is called a product state. When the entire system is in state AB, we know that particle 1 is in state A and particle 2 is in state B. Similarly, the state CD for the entire system means that particle 1 is in state C and particle 2 is in state D. Now consider the state AB + CD. We obtain this state by applying the superposition principle to the entire, twoparticle system. The superposition principle allows the system to be in such a combination of states, and the state AB + CD for the entire system is called an entangled state. While the product state AB (and similarly CD) ascribes definite properties to particles 1 and 2 (meaning, for example, that particle 1 is in location A and particle 2 is in location B), the entangled state—since it constitutes a superposition—does not. It only says that there are possibilities concerning particles 1 and 2 that are correlated, in the sense that if measurements are made, then if particle 1 is thomas young’s experiment 00 27 found in state A, particle 2 must be in state B; and similarly if particle 1 is in state C, then particle 2 will be in state D. Roughly speaking, when particles 1 and 2 are entangled, there is no way to characterize either one of them by itself without referring to the other as well. This is so even though we can refer to each particle alone when the two are in the product state AB or CD, but not when they are in the superposition AB + CD. It is the superposition of the two product states that produces the entanglement. This Page Intentionally Left Blank 4 Planck’s Constant “Planck had put forward a new, previously unimagined thought, the thought of the atomistic structure of energy.” —Albert Einstein T he quantum theory, with its bizarre consequences, was born in the year 1900, thirtyfive years before Einstein and his colleagues raised their question about entanglement. The birth of the quantum theory is attributed to the work of a unique individual, Max Planck. Max Planck was born in Kiel, Germany, in 1858. He came from a long line of pastors, jurists, and scholars. His grandfather and great grandfather were both theology professors at the University of Göttingen. Planck’s father, Wilhelm J. J. Planck, was a professor of Law in Kiel, and inspired in his son a deep sense of knowledge and learning. Max was his sixth child. Max’s mother came from a long line of pastors. The family was wealthy and vacationed every year on the shores of the Baltic Sea and traveled through Italy and Austria. The family was liberal in its views and, unlike many 29 30 00 entanglement Germans of the time, opposed Bismarck’s politics. Max Planck saw himself as even more liberal than his family. As a student, Max was good but not excellent—he was never at the top of his class although his grades were generally satisfactory. He exhibited a talent for languages, history, music, and mathematics, but never cared much for, nor excelled in, physics. He was a conscientious student and worked hard, but did not exhibit great genius. Planck was a slow, methodical thinker, not one with quick answers. Once he started working on something he found it hard to leave the subject and move on to something else. He was more a plodder than a naturally gifted intellect at the gymnasium. He often said that, unfortunately, he had not been given the gift of reacting quickly to intellectual stimulation. And he was always surprised that others could pursue several lines of intellectual work. He was shy, but was always wellliked by his teachers and fellow students. He saw himself as a moral person, one loyal to duties, perfectly honest, and pure of conscience. A teacher at the gymnasium encouraged him to pursue the harmonious interplay that he thought existed between mathematics and the laws of nature. This prompted Max Planck to study physics, which he did upon entering the University of Munich. In 1878, Planck chose thermodynamics as the topic for his dissertation, which he completed in 1879. The thesis dealt with two principles of classical thermodynamics: the conservation of energy, and the increase of entropy with time, which characterize all observable physical processes. Planck extracted some concrete results from the principles of thermodynamics and added an important premise: A stable equi planck’s constant 00 31 librium is obtained at a point of maximum entropy. He emphasized that thermodynamics can produce good results without any reliance whatever on the atomic hypothesis. Thus a system could be studied based on its macroscopic properties without the scientist having to worry about what happens or doesn’t happen to the system’s tiny components: atoms, molecules, electrons, and so on. Thermodynamic principles are still extremely important in physics as they deal with the energy of entire systems. These principles can be used to determine the output of an internal combustion engine, for example, and have wide applicability in engineering and other areas. Energy and entropy are key concepts in physics, so one would have thought that Planck’s work would have been wellreceived at the time. But it wasn’t. Professors at Munich, and Berlin—where Planck had studied for a year—were not impressed by his work. They did not think the work was important enough to merit praise or recognition. One professor evaded Planck so he could not even serve him with a copy of his doctoral work when preparing for its defense. Eventually Planck was awarded the degree and was fortunate enough to obtain the position of associate professor at the University of Kiel, where his father still had a number of friends who could help him. He took his position in 1885 and immediately attempted to vindicate both his own work and thermodynamics as a whole. He entered a competition organized by the University of Göttingen to define the nature of energy. Planck’s essay won second place—there was no first place. He soon realized that he would have had first place had not his article been critical of one of the professors at Göttingen. 32 00 entanglement Nonetheless, his award impressed the physics professors at the University of Berlin, and they offered him a post of associate professor in their faculty in 1889. In time, the world of theoretical physics came to appreciate the principles of thermodynamics with their treatment of the concepts of energy and entropy, and Planck’s work became more popular. His colleagues in Berlin, in fact, borrowed his dissertation so frequently that within a short time the manuscript started falling apart. In 1892 Planck was promoted to full professor in Berlin and in 1894 he became a full member of the Berlin Academy of Sciences. By the late 1800s, physics was considered a completed discipline, within which all explanations for phenomena and experimental outcomes had already been satisfactorily given. There was mechanics, the theory started by Galileo with his reputed experiment of dropping items from atop the Leaning Tower of Pisa, and perfected by the genius of Isaac Newton by the turn of the eighteenth century, almost two centuries before Planck’s time. Mechanics and the theory of gravitation that goes with it attempt to explain the motions of objects of the size we see in everyday life up to the size of planets and the moon. It explains how objects move; that force is the product of mass and acceleration; the idea that moving objects have inertia; and that the earth exerts a gravitational pull on all objects. Newton taught us that the moon’s orbit around the earth is in fact a constant “falling” of the moon down to earth, impelled by the gravitational pull both masses exert on each other. Physics also included the theory of electricity and electro planck’s constant 00 33 magnetism developed by Ampere, Faraday, and Maxwell. This theory incorporated the idea of a field—a magnetic or electric field that cannot be seen or heard or felt, but which exerts its influence on objects. Maxwell developed equations that accurately described the electromagnetic field. He concluded that light waves are waves of the electromagnetic field. In 1831 Faraday constructed the first dynamo, which produced electricity through the principle of electromagnetic inductance. By rotating a copper disc between two poles of an electromagnet, he was able to produce current. In 1887, during Planck’s formative years, Heinrich Rudolf Hertz (18571894) conducted his experiments that produced radio waves. By chance, he noticed that a piece of zinc illuminated by ultraviolet light became electrically charged. Without knowing it, he had discovered the photoelectric effect, which links light with matter. Around the same time, Ludwig Boltzmann (18441906) assumed that gases consist of molecules and treated their behavior using statistical methods. In 1897, one of the most important discoveries of science took place: the existence of the electron was deduced by J. J. Thomson. Energy was a crucial idea within all of these various parts of classical physics. In mechanics, half the mass times the velocity squared was defined as a measure of kinetic energy (from the Greek word kinesis, motion); there was another kind of energy, called potential energy. A rock on a high cliff possesses potential energy, which could then be instantly converted into kinetic energy once the rock is pushed slightly and falls from the cliff. Heat is energy, as we learn in high school physics. Entropy is a quality related to randomness 34 00 entanglement and since randomness always increases, we have the law of increasing entropy—as everyone who has tried to put away toys knows well. So there was every reason for the world of physics to accept Planck’s modest contributions to the theories of energy and entropy, and this was indeed what happened in Germany toward the end of the nineteenth century. Planck was recognized for his work in thermodynamics, and became a professor at the University of Berlin. During that time, he started to work on an interesting problem. It had to do with what is known as blackbody radiation. Logical reasoning along the lines of classical physics led to the conclusion that radiation from a hot object would be very bright at the blue or violet end of the spectrum. Thus a log in a fireplace, glowing red, would end up emitting ultraviolet rays as well as xrays and gamma rays. But this phenomenon, known as the ultraviolet catastrophe, doesn’t really take place in nature. No one knew how to explain this odd fact, since the theory did predict this buildup of energy levels of radiation. On December 14, 1900, Max Planck presented a paper at a meeting of the German Physical Society. Planck’s conclusions were so puzzling that he himself found it hard to believe them. But these conclusions were the only logical explanation to the fact that the ultraviolet catastrophe does not occur. Planck’s thesis was that energy levels are quantized. Energy does not grow or diminish continuously, but is rather always a multiple of a basic quantum, a quantity Planck denoted as hn, where n is a characteristic frequency of the system being considered, and h is a fundamental constant now known as Planck’s constant. (The value of Planck’s constant is 6.6262x1034 jouleseconds.) planck’s constant 00 35 The RayleighJeans law of classical physics implied that the brightness of the blackbody radiation would be unlimited at the extreme ultraviolet end of the spectrum, thus producing the ultraviolet catastrophe. But nature did not behave this way. According to nineteenth century physics (the work of Maxwell and Hertz), an oscillating charge produces radiation. The frequency (the inverse of the wavelength) of this oscillating charge is denoted by n, and its energy is E. Planck proposed a formula for the energy levels of a MaxwellHertz oscillator based on his constant h. The formula is: E=0, hn, 2hn, 3hn, 4hn . . . , or in general, nhn, where n is a nonnegative integer. Planck’s formula worked like magic. It managed to explain energy and radiation within a black body cavity in perfect agreement with the energy curves physicists were obtaining through their experiments. The reason for this was that the energy was now seen as coming in discrete packages, some large and some small, depending on the frequency of oscillation. But now, when the allotted energy for an oscillator (derived by other means) was smaller than the size of the package of energy available through Planck’s formula, the intensity of the radiation dropped, rather than increasing to the high levels of the ultraviolet catastrophe. Planck had invoked the quantum. From that moment on, physics was never the same. Over the following decades, many confirmations were obtained that the quantum is indeed a real concept, and that nature really works this way, at least in the microworld of atoms, molecules, electrons, neutrons, photons and the like. 36 00 entanglement Planck himself remained somewhat baffled by his own discovery. It is possible that he never quite understood it on a philosophical level. The trick worked, and the equations fit the data, but the question: “Why the quantum?” was one that not only he, but generations of future physicists and philosophers would ask and continue asking. Planck was a patriotic German who believed in German science. He was instrumental in bringing Albert Einstein to Berlin in 1914 and in promoting Einstein’s election to the Prussian Academy of Sciences. When Hitler came to power, Planck tried to persuade him to change his decision to terminate the positions of Jewish academics. But Planck never quit his own position in protest, as some nonJewish academics did. He remained in Germany, and throughout his life continued to believe in promoting science in his homeland. Planck died in 1947. By that time, the quantum theory had matured and undergone significant growth to become the accepted theory of physical law in the world of the very small. Planck himself, whose work and discovery of quanta had initiated the revolution in science, never quite accepted it completely in his own mind. He seemed to be puzzled by the discoveries he had made, and at heart always remained a classical physicist, in the sense that he did not participate much in the scientific revolution that he had started. But the world of science moved forward with tremendous impetus. 5 The Copenhagen School “The discovery of the quantum of action shows us not only the natural limitation of classical physics, but, by throwing a new light upon the old philosophical problem of the objective existence of phenomena independently of our observations, confronts us with a situation hitherto unknown in natural science.” —Niels Bohr iels Bohr was born in Copenhagen in 1885, in a sixteenthcentury palace situated across the street from the Danish Parliament. The impressive building was owned by a succession of wealthy and famous people, including, two decades after Bohr’s birth, King George I of Greece. The palace was bought by David Adler, Niels’s maternal grandfather, a banker and member of the Danish Parliament. Bohr’s mother, Ellen Adler, came from an AngloJewish family that had settled in Denmark. On his father’s side, Niels belonged to a family that had lived in Denmark for many generations, emigrating there in the late 1700s from the Grand Duchy of Mecklenburg in the Danishspeaking part of Germany. Niels’s father, Christian Bohr, was a physician and scientist who was nominated for the Nobel Prize for his research on respiration. N 37 38 00 entanglement David Adler also owned a country estate about ten miles from Copenhagen, and Niels was raised in very comfortable surroundings both in the city and in the country. Niels attended school in Copenhagen and was nicknamed “the fat one,” since he was a large boy who frequently wrestled with his friends. He was a good student, although not the first in his class. Bohr’s parents allowed their children to develop their gifts to the fullest. Bohr’s younger brother, Harald, always showed a propensity for mathematics, and, in time, became a prominent mathematician. Niels stood out as a curious investigator even as a very young child. While still a student, Niels Bohr undertook a project to investigate the surface tension of water by observing the vibrations of a spout. The project was planned and executed so well that it won him a gold medal from the Danish Academy of Sciences. At the university, Bohr was particularly influenced by Professor Christian Christiansen, who was the eminent Danish physicist of the time. The professor and the student had a relationship of mutual admiration. Bohr later wrote that he was especially fortunate to have come under the guidance of Christiansen, “a profoundly original and highly endowed physicist.” Christiansen, in turn, wrote Bohr in 1916: “I have never met anybody like you who went to the bottom of everything and also had the energy to pursue it to completion, and who in addition was so interested in life as a whole.”4 Bohr was also influenced by the work of the leading Danish philosopher, Harald Høffding. Bohr had known Høffding long before coming to the university, since he was a friend of the copenhagen school 00 39 Bohr’s father. Høffding and other Danish intellectuals regularly met at the Bohr mansion for discussions, and Christian Bohr allowed his two sons, Niels and Harald, to listen to the discussions. Høffding later became very interested in the philosophical implications of the quantum theory, developed by Niels Bohr. Some have suggested that, in turn, Bohr’s formulation of the quantum principle of complementarity (discussed later) was influenced by the philosophy of Høffding. Bohr continued on to his Ph.D. in physics at the university, and in 1911 wrote his thesis on the electron theory of metals. In his model, metals are viewed as a gas of electrons moving more or less freely within the potential created by the positive charges in the metal. These positive charges are the nuclei of the atoms of the metal, arranged in a lattice. The theory could not explain everything, and its limitations were due to the application of classical—rather than the nascent quantum—ideas to the behavior of these electrons in a metal. His model worked so well that his dissertation defense attracted much attention and the room was full to capacity. Professor Christiansen presided over the proceedings. He remarked that it was unfortunate that the thesis had not been translated into a foreign language as well, since few Danes could understand the physics. Bohr later sent copies of his thesis to a number of leading physicists whose works he had made reference to in the thesis, including Max Planck. Unfortunately few responded, since none could understand Danish. In 1920, Bohr made an effort to translate the thesis into English, but never completed the project. Having finished his work, Bohr went to England on a postdoctoral fellowship supported by the Danish Carlsberg 40 00 entanglement Foundation. He spent a year under the direction of J.J. Thomson at the Cavendish laboratory in Cambridge. The Cavendish laboratory was among the world’s leading centers for experimental physics, and its directors before Thomson were Maxwell and Rayleigh. The laboratory has produced some twentyodd Nobel laureates over the years. Thomson, who had won the Nobel Prize in 1906 for his discovery of the electron, was very ambitious. Often the film taken during experiments had to be hidden from him so he wouldn’t snatch it before it was dry to inspect it, leaving fingerprints that blurred the pictures. He was on a crusade to rewrite physics in terms of the electron, and to push beyond the impressive work of his predecessor, Maxwell. Bohr worked hard in the laboratory, but often had difficulties blowing glass to make special equipment. He broke tubes, and fumbled in the unfamiliar language. He tried to improve his English by reading Dickens, using his dictionary for every other word. Additionally, Thomson was not easy to work with. The project Thomson assigned to Bohr had to do with cathode ray tubes, and was a dead end that did not yield any results. Bohr found an error in Thomson’s calculations, but Thomson was not one who could accept criticism. He was uninterested in being corrected, and Bohr—with his poor English—did not make himself understood. In Cambridge, Bohr met Lord James Rutherford (18711937), who was recognized for his pioneering work on radiation, the discovery of the nucleus, and a model of the atom. Bohr was interested in moving to Manchester to work with Rutherford, whose theories had not yet received widespread acceptance. Rutherford welcomed him but suggested that he the copenhagen school 00 41 first obtain Thomson’s permission to leave. Thomson—who was not a believer in Rutherford’s theory of the nucleus— was more than happy to let Bohr go. In Manchester, Bohr began the studies that would eventually bring him fame. He started to analyze the properties of atoms in light of Rutherford’s theory. Rutherford set Bohr to work on the experimental problem of analyzing the absorption of alpha particles in aluminum. Bohr worked in the lab many hours a day, and Rutherford visited him and the rest of his students often, showing much interest in their work. After a while, however, Bohr approached Rutherford and said that he would rather do theoretical physics. Rutherford agreed and Bohr stayed home, doing research with pencil and paper and rarely coming into the lab. He was happy not to have to see anyone, he later said, as “no one there knew much.” Bohr worked with electrons and alpha particles in his research, and produced a model to describe the phenomena that he and the experimental physicists were observing. The classical theory did not work, so Bohr took a big step: He applied quantum constraints to his particles. Bohr used Planck’s constant in two ways in his famous theory of the hydrogen atom. First, he noted that the angular momentum of the orbiting electron in his model of the hydrogen atom had the same dimensions as Planck’s constant. This led him to postulate that the angular momentum of the orbiting electron must be a multiple of Planck’s constant divided by 2p, that is: mvr= h/2p, 2(h/2p), 3(h/2p), . . . Where the expression on the left is the classical definition of angular momentum (m is mass, v is speed, and r is the radius 42 00 entanglement of the orbit). This assumption of the quantizing of the angular momentum led Bohr directly to quantizing the energy of the atom. Second, Bohr postulated that when the hydrogen atom drops from one energy level to a lower one, the energy that is released comes out as a single Einstein photon. As we will see later, the smallest quantity of energy in a light beam, according to Einstein, was hn, where h was Planck’s constant and n the frequency, measured as the number of vibrations per second. With this development, and with his assumption of angular momentum, Bohr used Planck’s quantum theory to explain what happens in the interior of an atom. This was a major breakthrough for physics. Bohr finished his paper on alpha particles and the atom after he left Manchester and returned to Copenhagen. The paper was published in 1913, marking the transition of his work to the quantum theory and the question of atomic structure. Bohr never forgot he was led to formulate his quantum theory of the atom from Rutherford’s discovery of the nucleus. He later described Rutherford as a second father to him. Upon his return to Denmark, Bohr took up a position at the Danish Institute of Technology. He married Margrethe Nørlund in 1912. She remained by his side throughout his life, and was a power in organizing the physics group founded in Copenhagen by her husband. On March 6, 1913, Bohr sent Rutherford the first chapter of his treatise on the constitution of atoms. He asked his former mentor to forward the work to the Philosophical Magazine for publication. This manuscript was to catapult him the copenhagen school 00 43 from a young physicist who has made some important progress in physics to a world figure in science. Bohr’s breakthrough discovery was that it is impossible to describe the atom in classical terms, and that the answers to all questions about atomic phenomena had to come from the quantum theory. Bohr’s efforts were aimed at first understanding the simplest atom of all, that of hydrogen. By the time he addressed the problem, physics had already learned that there are specific series of frequencies at which the hydrogen atom radiates. These are the wellknown series of Rydberg, Balmer, Lyman, Paschen, and Brackett—each covering a different part of the spectrum of radiation from excited hydrogen atoms, from ultraviolet through visible light to infrared. Bohr sought to find a formula that would explain why hydrogen radiates in these particular frequencies and not others. Bohr deduced from the data available on all series of radiation of hydrogen that every emitted frequency was due to an electron descending from one energy level in the atom to another, lower level. When the electron came down from one level to another, the difference between its beginning and ending energies was emitted in the form of a quantum of energy. There is a formula linking these energy levels and quanta: EaEb=hnab Where Ea is the beginning energy level of the electron around the hydrogen nucleus; Eb is the ending energy level once the electron has descended from its prior state; h is Planck’s constant; and nab is the frequency of the light quantum emitted 44 00 entanglement during the electron’s jump down from the first to the second energy level. This is demonstrated by the figure below. 3rd ORB. R. ER SE BALM LYM AN SER . PAS SER CHEN IES 4th ORB. TT KE AC BR RIES SE 2nd ORB. 1st ORB. Rutherford’s simple model of the atom did not square well with reality. Rutherford’s atom was modeled according to classical physics, and if the atom was as simple as the model implied, it would not have existed for more than onehundredmillionth of a second. Bohr’s tremendous discovery of the use of Planck’s constant within the framework of the atom solved the problem beautifully. The quantum theory now explained all observed radiation phenomena about hydrogen, which had until then baffled physicists for decades. Bohr’s work has been partially extended to explain the orbits and energies of electrons in other elements and to bring us understanding of the periodic table of the elements, chemical bonds, and other fundamental phenomena. The quantum theory had just been put to exceptionally good use. It was becoming obvious that classical physics would not work the copenhagen school 00 45 well in the realm of atoms and molecules and electrons, and that the quantum theory was the correct path to take. Bohr’s brilliant solution to the question of the various series of spectral lines of radiation for the hydrogen atom left unanswered the question: Why? Why does an electron jump from one energy level to another, and how does the electron know that it should do so? This is a question of causality. Causality is not explained by the quantum theory, and in fact cause and effect are blurred in the quantum world and have no explanation or meaning. This question about Bohr’s work was raised by Rutherford as soon as he received Bohr’s manuscript. Also, the discoveries did not bring about a general formulation of quantum physics, applicable in principle to all situations and not just to special cases. This was the main question of the time, and the goal was not achieved until later, that is, until the birth of “the new quantum mechanics” with the work of de Broglie, Heisenberg, Schrödinger, and others. Bohr became very famous following his work on the quantum nature of the atom. He petitioned the Danish government to endow him with a chair of theoretical physics, and the government complied. Bohr was now Denmark’s favorite son and the whole country honored him. Over the next few years he continued to travel to Manchester to work with Rutherford, and traveled to other locations and met many physicists. These connections allowed him to found his own institute. In 1918, Bohr secured permission from the Danish gov 46 00 entanglement ernment to found his institute of theoretical physics. He received funding from the Royal Danish Academy of Science, which draws support from the Carlsberg brewery. Bohr and his family moved into the mansion owned by the Carlsberg family on the premises of his new institute. Many young physicists from around the world regularly came to spend a year or two working at the institute and drawing their inspiration from the great Danish physicist. Bohr became close with the Danish royal family, as well as with many members of the nobility and the international elite. In 1922, he received the Nobel Prize for his work on the quantum theory. Bohr organized regular scientific meetings at his institute in Copenhagen, to which many of the world’s greatest physicists came and discussed their ideas. Copenhagen thus became a world center for the study of quantum mechanics during the period the theory was growing: from its founding in the late first decade of the twentieth century until just before the Second World War. The scientists who worked at the institute (to be named the Niels Bohr Institute after its founder’s death), and many who came to attend its meetings, later developed what is called the Copenhagen Interpretation of the quantum theory, often called the orthodox interpretation. This was done after the birth of the “new quantum mechanics” in the middle 1920s. According to the Copenhagen interpretation of the rules of the quantum world, there is a clear distinction between what is observed and what is not observed. The quantum system is submicroscopic and does not include the measuring devices or the measuring process. In the years to come, the Copenhagen interpretation the copenhagen school 00 47 would be challenged by newer views of the world brought about by the maturing of the quantum theory. Starting in the 1920s, and culminating in 1935, a major debate would rage within the community of quantum physicists. The challenge would be issued by Einstein, and throughout the rest of his life, Bohr would regularly spar with Einstein on the meaning and completeness of the quantum theory. This Page Intentionally Left Blank 6 De Broglie’s Pilot Waves “After long reflection in solitude and meditation, I suddenly had the idea, during the year 1923, that the discovery made by Einstein in 1905 should be generalized by extending it to all material particles and notably to electrons.” —Louis de Broglie D uke Louis Victor de Broglie was born in Dieppe in 1892 to an aristocratic French family that had long provided France with diplomats, politicians, and military leaders. Louis was the youngest of five children. His family expected Louis’ adored older bother, Maurice, to enter the military service, and so Louis too decided to serve France. He chose the navy, since he thought it might allow him to study the natural sciences, which had fascinated him since childhood. He did indeed get to practice science by installing the first French wireless transmitter aboard a ship. After Maurice left the military and studied in Toulon and at the University of Marseilles, he moved to a mansion in Paris, where in one of the rooms he established a laboratory for the study of Xrays. To aid him in his experiments, the resourceful Maurice trained his valet in the rudiments of sci49 50 00 entanglement entific procedure, and eventually converted his personal servant into a professional lab assistant. His fascination with science was infectious. Soon, his younger brother Louis was also interested in the research and helped him with experiments. Louis attended the Sorbonne, studying medieval history. In 1911, Maurice served as the secretary of the famous Solvay Conference in Brussels, where Einstein and other leading physicists met to discuss the exciting new discoveries in physics. Upon his return, he regaled his younger brother with stories about the fascinating discoveries, and Louis became even more excited about physics. Soon, World War I erupted and Louis de Broglie enlisted in the French army. He served in a radio communication unit, a novelty at that time. During his service with the radiotelegraphy unit stationed at the top of the Eiffel Tower, he learned much about radio waves. And indeed he was to make his mark on the world through the study of waves. When the war ended, de Broglie returned to the university and studied under some of France’s best physicists and mathematicians, including Paul Langevin and Emile Borel. He designed experiments on waves and tested them out at his brother’s laboratory in the family’s mansion. De Broglie was also a lover of chamber music, and so he had an intimate knowledge of waves from a musictheory point of view. De Broglie immersed himself in the study of the proceedings of the Solvay Conference given to him by his brother. He was taken by the nascent quantum theory discussed in 1911 and repeatedly presented at later Solvay meetings throughout the following years. De Broglie studied ideal gases, which de broglie’s pilot waves 00 51 were discussed at the Solvay meeting, and came to a successful implementation of the theory of waves in analyzing the physics of such gases, using the quantum theory. In 1923, while working for a doctorate in physics in Paris, “all of a sudden,” as he later put it, “I saw that the crisis in optics was simply due to a failure to understand the true universal duality of wave and particle.” At that moment, in fact, de Broglie discovered this duality. He published three short notes on the topic, hypothesizing that particles were also waves and waves also particles, in the Proceedings of the Paris Academy in September and October 1923. He elaborated on this work and presented his entire discovery in his doctoral dissertation, which he defended on November 25, 1924. De Broglie took Bohr’s conception of an atom and viewed it as a musical instrument that can emit a basic tone and a sequence of overtones. He suggested that all particles have this kind of waveaspect to them. He later described his efforts: “I wished to represent to myself the union of waves and particles in a concrete fashion, the particle being a little localized object incorporated in the structure of a propagating wave.” Waves that he associated with particles, de Broglie named pilot waves. Every small particle in the universe is thus associated with a wave propagating through space. De Broglie derived some mathematical concepts for his pilot waves. Through a derivation using several formulas and Planck’s quantumtheory constant, h, de Broglie came up with the equation that is his legacy to science. His equation links the momentum of a particle, p, with the wavelength of 52 00 entanglement its associated pilot wave, l, through an equation using Planck’s constant. The relationship is very simply stated as: p=h/l De Broglie had a brilliant idea. Here, he was using the machinery of the quantum theory to state a very explicit relationship between particles and waves. A particle has momentum (classically, the product of its velocity and its mass). Now this momentum was directly linked with the wave associated with the particle. Thus a particle’s momentum in quantum mechanics is, by de Broglie’s formula, equal to the quotient of Planck’s constant and the wavelength of the wave when we view the particle as a wave. De Broglie did not provide an equation to describe the propagation of the wave associated with a particle. That task would be left to another great mind, Erwin Schrödinger. For his pioneering work, de Broglie received the Nobel Prize after many experiments verified the wave nature of particles over the following years. De Broglie remained active as a physicist and lived a long life, dying in 1987 at the age of 95. When de Broglie was already a worldfamous scientist, the physicist George Gamow (who wrote Thirty Years that Shook Physics) visited him in his mansion in Paris. Gamow rang the bell at the gate of the estate and was greeted by de Broglie’s butler. He said: “Je voudrais voir Professeur de Broglie.” The butler cringed. “Vous voulez dire, Monsieur le Duc de Broglie!” he insisted. “O.K., le Duc de Broglie,” Gamow said and was finally allowed to enter. de broglie’s pilot waves 00 53 *** Are particles also waves? Are waves also particles? The answer the quantum theory gives us is “Yes.” A key characteristic of a quantum system is that a particle is also a wave, and exhibits wave interference characteristics when passed through a doubleslit experimental setup. Similarly, waves can be particles, as Einstein has taught us when he developed his photoelectric effect Nobel Prizewinning paper, which will be described later. Light waves are also particles, called photons. Laser light is coherent light, in which all the light waves are in phase; hence the power of lasers. In 2001, the Nobel Prize in physics was shared by three scientists who showed that atoms, too, can behave like light rays in the sense that an ensemble of them can all be in a coherent state, just like laser light. This proved a conjecture put forth by Einstein and his colleague, the Indian physicist Saryendra Nath Bose, in the 1920s. Bose was an unknown professor of physics at the University of Dacca, and in 1924 he wrote Einstein a letter in which he described how Einstein’s light quanta, the photons, could form a kind of “gas,” similar to the one consisting of atoms or molecules. Einstein rewrote and improved Bose’s paper and submitted it for joint publication. This gas proposed by Bose and Einstein was a new form of matter, in which individual particles did not have any properties and were not distinguishable. The BoseEinstein new form of matter led Einstein to a “hypothesis about an interaction between molecules of an as yet quite mysterious nature.” The BoseEinstein statistics allowed Einstein to make groundbreaking predictions about the behavior of matter at 54 00 entanglement extremely low temperatures. At such low temperatures, viscosity of liquefied gases disappears, resulting in superfluidity. The process is called BoseEinstein condensation. Louis de Broglie had submitted his doctoral dissertation to Einstein’s friend in Paris, Paul Langevin, in 1924. Langevin was so impressed with de Broglie’s idea that matter can have a wave aspect, that he sent the thesis to Einstein, asking for his opinion. When Einstein read de Broglie’s thesis he called it “very remarkable,” and he later used the de Broglie wave idea to deduce the wave properties of the new form of matter he and Bose had discovered. But no one had seen a BoseEinatein condensate . . . until 1995. On June 5, 1995, Carl Weiman of the University of Colorado and Eric Cornell of the National Institute of Standards and Technology used highpowered lasers and a new technique for cooling matter to close to absolute zero to supercool about 2000 atoms of rubidium. These atoms were found to possess the qualities of a BoseEinstein condensate. They appeared as a tiny dark cloud, in which the atoms themselves had lost their individuality and entered a single energy state. For all purposes, these atoms were now one quantum entity, as characterized by their de Broglie wave. Shortly afterwards, Wolfgang Ketterle of M.I.T. reproduced the results and improved the experiment, producing what was the equivalent of a laser beam made of atoms. For their work, the three scientists shared the 2001 Nobel Prize in Physics, and de Broglie’s fascinating idea was reconfirmed in a new setting that pushed the limits of quantum mechanics up the scale toward macroscopic objects. 7 Schrödinger and His Equation “Entanglement is not one but rather the characteristic trait of quantum mechanics.” —Erwin Schrödinger E rwin Schrödinger was born in a house in the center of Vienna in 1887 to welltodo parents. An only child, he was doted on by several aunts, one of whom taught him to speak and read English before he even mastered his native German. As a young boy, Erwin started to keep a journal, a practice he maintained throughout his life. From an early age, he exhibited a healthy skepticism and tended to question things that people presented as facts. These two habits were very useful in the life of a scientist who would make one of the most important contributions to the new quantum theory. Questioning what from our everyday life we take as truth is essential in approaching the world of the very small. And Schrödinger’s notebooks would be crucial in his development of the wave equation. At age eleven, Erwin entered the gymnasium located a few 55 56 00 entanglement minutes’ walk from his house. In addition to mathematics and the sciences, the gymnasium taught its students Greek language and culture, Latin, and the classic works of antiquity, including Ovid, Livy, Cicero, and Homer. Erwin loved mathematics and physics, and excelled in them, solving problems with an ease and facility that stunned his peers. But he also enjoyed German poetry and the logic of grammar, both ancient and modern. This logic, in mathematics and in humanistic studies, shaped his thinking and prepared him for the rigors of the university. Erwin loved hiking, mountaineering, the theater, and pretty girls—amusements that would mark his behavior throughout his life. As a young boy, he worked hard at school, but also played hard. He spent many days walking in the mountains, reading mathematics, and courting his best friend’s sister, a darkhaired beauty named Lotte Rella.5 In 1906, Schrödinger enrolled at the University of Vienna—one of the oldest in Europe, established in 1365— to study physics. There was a long legacy of physics at the university. Some of the great minds that had worked there and left about the time Schrödinger enrolled were Ludwig Boltzmann, the proponent of the atomic theory, and Ernst Mach, the theoretician whose work inspired Einstein. There Schrödinger was a student of Franz Exner, and did work in experimental physics, some of it relating to radioactivity. The University of Vienna was an important center for the study of radioactivity, and Marie Curie in Paris received some of her specimens of radioactive material, with which she made her discoveries, from the physics department at Vienna. Schrödinger was admired by his fellow students for his shrödinger and his equation 00 57 brilliance in physics and mathematics. He was always sought out by his friends for help in mathematics. One of the subjects in mathematics that he took at the University of Vienna was differential equations, in which he excelled. As fate would have it, this special skill proved invaluable in his career: it helped him solve the biggest problem of his life and establish his name as a pioneer of quantum mechanics. But Schrödinger lived a multifacted life as a university student in Vienna at the height of its imperial glory. He retained his abilities as an athlete and was as highly social as he’d ever been: He found a number of good friends with whom he spent his free time climbing and hiking in the mountains. Once, in the Alps, he spent an entire night nursing a friend who had broken a leg while climbing. Once his friend was taken to the hospital, he spent the day skiing. In 1910, Schrödinger wrote his doctoral thesis in physics, entitled “On the conduction of electricity on the surface of insulators in moist air.” This was a problem that had some implications in the study of radioactivity, but the thesis was not a brilliant work of scholarship. Schrödinger had left out a number of factors about which he should have known, and his analysis was neither complete nor ingenious. Still, the work was enough to earn him his doctorate, and following his graduation he spent a year in the mountains as a volunteer in the fortress artillery. He then returned to the university to work as an assistant in a physics laboratory. Meanwhile, he labored on the required paper (called a Habilitationschrift) that would allow him to earn income as a private tutor at the university. His paper, “On the Kinetic Theory of Magnetism,” was a theoretical attempt to explain 58 00 entanglement the magnetic properties of various compounds, and was also not of exceptional quality, but it satisfied the requirements and allowed him to work at the university. His academic career had begun. Shortly afterwards, Schrödinger, who was now in his early twenties, met another teenage girl who caught his fancy. Her name was Felicie Krauss, and her family belonged to Austria’s lower nobility. The two developed a relationship and considered themselves engaged despite strong objections from the girl’s parents. Felicie’s mother, especially, was determined not to allow her daughter to marry a workingclass person; one who, she believed, would never be able to support her daughter in an appropriate style on his university income. In despair, Erwin contemplated leaving the university and working for his father, who owned a factory. But the father would hear nothing of it, and with the mounting pressure from Felicie’s mother, the two lovers called off their informal engagement. While she later married, Felicie always remained close to Erwin. This, too, was a pattern that continued throughout Schrödinger’s life: wherever he went— even after he was married—there were always young girlfriends never too far away. Schrödinger continued his study of radioactivity in the laboratory of the University of Vienna. In 1912, his colleague Victor Hess soared 16,000 feet in a balloon with instruments to measure radiation. He wanted to solve the problem of why radiation was detected not only close to the ground, where deposits of radium and uranium were its source, but also in the air. Up in his balloon, Hess discovered to his surprise that the radiation was actually three times as high as it was at shrödinger and his equation 00 59 ground level. Hess had thus discovered cosmic radiation, for which he later received the Nobel Prize. Schrödinger, taking part in related experiments on the background radiation at ground level, traveled throughout Austria with his own radiationdetecting instruments. This travel incidentally allowed him to enjoy his beloved outdoors—and make new friends. In 1913, he was taking radiation measurements in the open air in the area where a family he had known from Vienna was vacationing. With the family was a pretty teenage girl, Annemarie (“Anny”) Bertel. The twentysix year old scientist and the sixteenyearold girl were attracted to each other, and through meetings over the next several years, developed a romance that resulted in marriage. Anny remained devoted to Schrödinger throughout his life, even tolerating his perpetual relationships with other women. In 1914, Schrödinger reenlisted in the fortress artillery to fight on the Italian front of World War I. Even in the field, he continued to work on problems of physics, publishing papers in professional journals. None of his papers thus far had been exceptional, but the topics were interesting. Schrödinger spent much time doing research on color theory, and made contributions to our understanding of light of different wavelengths. During one of his experiments on color while still at the University of Vienna, Erwin discovered that his own color vision was deficient. In 1917, Schrödinger wrote his first paper on the quantum theory, on atomic and molecular heats. The research for this paper brought to his attention the work of Bohr, Planck, and Einstein. By the time the war was over, Schrödinger had addressed not only the quantum theory, but also Einstein’s 60 00 entanglement theory of relativity. He had now brought himself into the leading edge of theoretical physics. In the years following the war, Schrödinger taught at universities in Vienna, Jena, Breslau, Stuttgart, and Zurich. In 1920, in Vienna, Erwin married Anny Bertel. Her income was higher than his university salary, which made him upset and prompted him to seek employment at other universities throughout Europe. Through Anny, Erwin met Hansi Bauer, who later became one of the girlfriends he would maintain throughout his life. In Stuttgart, in 1921, Schrödinger began a very serious effort to understand and further develop the quantum theory. Bohr and Einstein, who were not much older than Schrödinger, had already made their contributions to the theory while in their twenties. Schrödinger was getting older, and he still had not had a major scientific achievement. He concentrated his efforts on modeling the spectral lines of alkali metals. In late 1921, Schrödinger was nominated for a coveted position of full Professor of Theoretical Physics at the University of Zurich. That year, he published his first important paper in the quantum area, about quantized orbits of a single electron, based on the earlier work of Bohr. Soon after his arrival in Zurich, however, he was diagnosed with pulmonary disease and his doctors ordered rest at high altitude. The Schrödingers decided on the village of Arosa in the Alps, not far from Davos, at an altitude of 6,000 feet. Upon his recovery, they returned to Zurich and there, in 1922, Schrödinger gave his inaugural lecture at the university. During 1923 and 1924, Schrödinger’s research was centered on shrödinger and his equation 00 61 spectral theory, light, atomic theory, and the periodic nature of the elements. In 1924, at the age of 37, he was invited to attend the Solvay Conference in Brussels, where the greatest minds in physics, including Einstein and Bohr, met. Schrödinger was there almost as an outside observer, since he had not produced any earthshattering papers. The quantum theory was nowhere near being complete, and Erwin Schrödinger was desperately seeking a topic in the quantum field with which he could make his mark. Time was running out on him, and if nothing happened soon, he would be condemned to obscurity, mediocrity, and to remain forever in the sidelines while others were making scientific history. In 1924, Peter Debye at the University of Zurich asked Schrödinger to report on de Broglie’s thesis on the wave theory of particles at a seminar held at the university. Schrödinger read the paper, started thinking about its ideas, and decided to pursue them further. He worked on de Broglie’s particlewave notion for a full year, but made no breakthrough. A few days before Christmas, 1925, Erwin left for the Alps, to stay in the Villa Herwig in Arosa, where he and Anny had spent several months while he was recuperating four years earlier. This time he came without his wife. From his correspondence, we know that he had one of his former girlfriends from Vienna join him at the villa, and stay with him there till early 1926. Schrödinger’s biographer Walter Moore makes much of the mystery as to who the girlfriend might have been.6 Could she have been Lotte, Felicie, Hansi, or one of his other liaisons? At any rate, according to the physicist Hermann Weyl, Schrödinger’s erotic encounters 62 00 entanglement with the mystery lady produced the burst of energy Schrödinger required to make his great breakthrough in the quantum theory. Over the Christmas vacation in the Alps with his secret lover, Schrödinger produced the nowfamous Schrödinger equation. The Schrödinger equation is the mathematical rule that describes the statistical behavior of particles in the microworld of quantum mechanics. The Schrödinger equation is a differential equation. Differential equations are mathematical equations that state a relationship between a quantity and its derivatives, that is, between a quantity and its rate of change. Velocity, for example, is the derivative (the rate of change) of location. If you are moving at sixty miles per hour, then your location on the road changes at a rate of sixty m.p.h. Acceleration is the rate of change of velocity (when you accelerate, you are increasing the speed of your car); thus acceleration is the second derivative of location, since it is the rate of change of the rate of change of location. An equation that states your location, as a variable, as well as your velocity, is a differential equation. An equation relating your location with your velocity and your acceleration is a secondorder differential equation. By the time Schrödinger started to address the problem of deriving the equation that governs the quantum behavior of a small particle such as the electron, a number of differential equations of classical physics were known. For example, the equation that governs the progression of heat in a metal was known. Equations governing classical waves, for example, waves on a vibrating string, and sound waves, were already well known. Having taken courses in differential equations, shrödinger and his equation 00 63 Schrödinger was well aware of these developments. Schrödinger’s task was to find an equation that would describe the progression of particle waves, the waves that de Broglie had associated with small particles. Schrödinger made some educated guesses about the form his equation must take, based on the known classical wave equation. What he had to determine, however, was whether to use the first or the second derivative of the wave with respect to location, and whether to use the first or the second derivative with respect to time. His breakthrough occurred when he discovered that the proper equation is firstorder with respect to time and secondorder with respect to location. HΨ=EΨ The above is the timeindependent Schrödinger equation, stated in its simplest symbolic way. The symbol Ψ represents the wave function of a particle. This is de Broglie’s “pilot wave” of a particle. But here this is no longer some hypothetical entity, but rather a function that we can actually study and analyze using the Schrödinger equation. The symbol H stands for an operator, which is represented by a formula of its own, telling it what to do to the wave function: take a derivative and also multiply the wave function by some numbers, including Planck’s constant, h. The operator H operates on the wave equation, and the result, on the other side of the equation, is an energy level, E, multiplied by the wave function. Schrödinger’s equation has been applied very successfully to a number of situations in quantum physics. What a physicist does is