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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
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entanglement

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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 Cataloging-in-Publication Data:
Entanglement: the greatest mystery in physics/ by Amir D. Aczel.
p. cm.
Includes bibliographical references and index.
isbn 1-56858-232-3
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

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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 Ten-Kilometer Experiment / 235
Teleportation: “Beam Me Up, Scotty” / 241
Quantum Magic: What Does It All Mean? / 249
Acknowledgements / 255
References / 266
Index / 269

vii

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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 negatively-charged
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 well-known statement that “God
doesn’t play dice with the world” was a reflection of his belief
that there was a deeper, non-probabilistic 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 action-at-a-distance, 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 Einstein-Podolsky-Rosen 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 little-known 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 thirty-five 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. Cohen-Tannoudji

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 3,000-year-old magnifying glasses in the region of the
eastern Mediterranean that was once Phoenicia. Interestingly,
the principle that makes a lens work is the slowing-down of
light as it travels through glass.
The Romans learned glass-making 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 vis-avis 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 (1629-1695) and Robert Hooke
(1635-1703) proposed the theory that light is a wave. Huygens, who as a sixteen-year-old 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 (1643-1727) 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 co-discovered 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 (second-order) 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 cause-and-effect 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 cause-and-effect 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
(1812-1878) 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 double-slit
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 (1773-1829) 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 now-famous doubleslit experiment on light, demonstrating the wave-theory 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 double-slit 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 double-slit experiment was the
fundamental mystery of quantum mechanics. Richard Feynman demonstrated in his Lectures the idea of interference of
waves versus the non-interference 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
twenty-first 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 non-zero 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 double-slit 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, two-particle 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.

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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, thirty-five 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 well-liked 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 well-received 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 (1857-1894) 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 (1844-1906) 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 black-body 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
6.6262x10-34 joule-seconds.)

planck’s constant 00 35

The Rayleigh-Jeans law of classical physics implied that the
brightness of the black-body 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 Maxwell-Hertz
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 micro-world 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 non-Jewish 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 sixteenth-century 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 Anglo-Jewish 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 Danish-speaking 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 twenty-odd 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 well-known 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:
Ea-Eb=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 one-hundred-millionth 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.

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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 X-rays. 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 radio-telegraphy 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 music-theory 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 wave-aspect 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 quantum-theory 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 world-famous 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 double-slit experimental setup. Similarly, waves can
be particles, as Einstein has taught us when he developed his
photoelectric effect Nobel Prize-winning 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 Bose-Einstein new form of
matter led Einstein to a “hypothesis about an interaction
between molecules of an as yet quite mysterious nature.”
The Bose-Einstein 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 Bose-Einstein 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 high-powered 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 Bose-Einstein 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 well-to-do 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
dark-haired 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 working-class
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 radiation-detecting 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 twenty-six year old scientist
and the sixteen-year-old 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

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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 earth-shattering 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 particle-wave 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 now-famous
Schrödinger equation. The Schrödinger equation is the mathematical rule that describes the statistical behavior of particles in the micro-world 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 second-order 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 first-order with respect
to time and second-order with respect to location.
HΨ=EΨ

The above is the time-independent 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