Lawrence Krauss - The Greatest Story Ever Told--So Far

by Why Are We Here (pdf)

  when Heisenberg’s paper appeared. Heisenberg’s friend and

  contemporary the brilliant and irascible physicist Wolfgang Pauli

  (another future Nobel laureate assistant to Sommerfeld) thought the

  work to be essentially mathematical masturbation, leading

  Heisenberg to respond in jocular form:

  You have to allow that, in any case, we are not seeking to ruin

  physics out of malicious intent. When you reproach us that we are

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  such big donkeys that we have never produced anything new in

  physics, it may well be true. But then, you are also an equally big

  jackass because you have not accomplished it either. . . . Do not

  think badly of me and many greetings.

  Physics doesn’t proceed in the linear fashion that textbooks

  recount. In real life, as in many good mystery stories, there are false

  leads, misperceptions, and wrong turns at every step. The story of

  the development of quantum mechanics is full of them. But I want to

  cut to the chase here, and so I will skip over Niels Bohr, whose ideas

  laid out the first fundamental atomic rules of the quantum world as

  well as the basis for much of modern chemistry. We’ll also skip

  Erwin Schrödinger, who was a remarkably colorful character,

  fathering at least three children with various mistresses, and whose

  wave equation is the most famous icon of quantum mechanics.

  Instead I will focus first on Heisenberg, or rather not Heisenberg

  himself, but instead the result that made his name famous: the

  Heisenberg uncertainty principle. This is often interpreted to mean

  that the observations of quantum systems affect their properties—

  which was manifest in our earlier discussions of electrons or photons

  passing through two slits and impinging on a screen behind them.

  Unfortunately this leads to the misimpression that somehow

  observers, in particular human observers, play a key role in quantum

  mechanics—a confusion that has been exploited by my Twitter

  combatant Deepak Chopra, who, in his various ramblings, somehow

  seems to think the universe wouldn’t exist if our consciousness

  weren’t here to measure and frame its properties. Happily the

  universe predates Chopra’s consciousness and was proceeding pretty

  nicely before the advent of all life on Earth.

  However, the Heisenberg uncertainty principle at its heart has

  nothing to do with observers at all, even though it does limit their

  ability to perform measurements. It is instead a fundamental

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  property of quantum systems, and it can be derived relatively

  straightforwardly and mathematically, based on the wave properties

  of these systems.

  Consider for example a simple wavelike disturbance with a single

  frequency (wavelength) oscillating as it moves along the x direction:

  As I have noted, in quantum mechanics particles have a wavelike

  character. Thanks to Max Born we recognize that the square of the

  amplitude of the wave associated with a particle at any point—what

  we now call the wave function of the particle, following Schrödinger

  —determines the probability of finding the particle at that point.

  Because the amplitude of the oscillating wave above is more or less

  constant at all the peaks, such a wave, if it corresponded to the

  probability amplitude of finding an electron, would imply a more or

  less uniform probability for finding the electron anywhere along the

  path.

  Now consider what a disturbance would look like if it was the

  sum of two waves of slightly different frequencies (wavelengths),

  moving along the x axis:

  When we combine the two waves, the resulting disturbance will

  look like:

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  Because of the slightly different wavelengths of the two waves, the

  peaks and troughs will tend to cancel out, or “negatively interfere”

  with each other everywhere except for the rare places where the two

  peaks occur at the same point (one of these locations is shown in the

  figure above). This is reminiscent of the wave interference

  phenomenon in the Young double-slit experiment I described

  earlier.

  If we add yet another wave of slightly different wavelength

  the resulting wave then looks like this:

  The interference washes out more of the oscillations aside from

  the position where the two waves line up, making the amplitude of

  the wave at the peak much higher there than elsewhere.

  You can imagine what would happen if I continue this process,

  continuing to add just the right amount of waves with slightly

  different frequencies to the original wave. Eventually the resulting

  wave amplitudes will cancel out more and more at all places except

  for some small region around the center of the figure, and at faraway

  places where all the peaks might again line up:

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  The greater the number of slightly different frequencies that I add

  together, the narrower will be the width of the largest central peak.

  Now, imagine that this represents the wave function of some

  particle. The larger the amplitude of the central peak, the greater the

  probability of finding the particle somewhere within the width of

  that peak. But the width of that central peak is still never quite zero,

  so the disturbance remains spread out over some small, if

  increasingly narrow, region.

  Now recall that Planck and Einstein told us that, for light waves,

  at least, the energy of each quantum of radiation, i.e., each photon, is

  directly related to its frequency. Not surprisingly, a similar relation

  holds for the probability waves associated with massive particles, but

  in this case it is the momentum of the particle that is related to the

  frequency of the probability wave associated with the particle.

  Hence, Heisenberg’s uncertainty relation: If we want to localize a

  particle over a small region, i.e., have the width of the highest peak

  in its wave function as narrow as possible, then we must consider

  that the wave function is made up by adding lots of different waves

  of slightly different frequencies together. But this means that the

  momentum of the particle, which is associated with the frequency of

  its wave function, must be spread out somewhat. The narrower the

  dominant peak in space in the particle’s wave function, the greater

  the number of different frequencies (i.e., momenta) that must be

  added together to make up the final wave function. Put in a more

  familiar way, the more accurately we wish to determine the specific

  position of a particle, the greater the uncertainty in its momentum.

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  As you can see, there is no restriction here related to actual

  observations, or consciousness, or the specific technology associated

  with any observation. It is an inherent property of the fact that, in

  the quantum world, a wave function is associated with each particle,

  and for particles of a fixed specific momentum, the wave function

  has one specific frequency.

  After discovering this relation, Heisenberg was the first to provide

  a heuristic pictur
e of why this might be the case, which he posed in

  terms of a thought experiment. To measure the position of a particle

  you have to bounce light off the particle, and to resolve the position

  with great precision requires light of a wavelength small enough to

  resolve this position. But the smaller the wavelength, the bigger the

  frequency and the higher the energy associated with the quanta of

  that radiation. But bouncing light with a higher and higher energy

  off the particle clearly changes the particle’s energy and momentum.

  Thus, after the measurement is made, you may know the position of

  the particle at the time of the measurement, but the range of possible

  energies and momenta you have imparted to the particle by

  scattering light off it is now large.

  For this reason, many people confuse the Heisenberg uncertainty

  relation with the “observer effect,” as it has become known, in

  quantum mechanics. But, as the example I have given should

  demonstrate, inherently the Heisenberg uncertainty principle has

  nothing to do with observation at all. To paraphrase a friend of

  mine, if consciousness had anything to do with determining the

  results of quantum physics experiments, then in reporting the results

  of physics experiments we would have to discuss what the

  experimenter was thinking about—for example, sex—when

  performing the experiment. But we don’t. The supernova explosions

  that produced the atoms that make up your body and mine occurred

  quite nicely long before our consciousness existed.

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  The Heisenberg uncertainty principle epitomizes in many ways

  the complete demise of our classical worldview of nature.

  Independent of any technology we might someday develop, nature

  puts an absolute limit on our ability to know, with any degree of

  certainty, both the momentum and position of any particle.

  But the issue is even more extreme than this statement implies.

  Knowing has nothing to do with it. As I described in the earlier

  double-slit experiment example, there is no sense in which the

  particle has at any time both a specific position and a specific

  momentum. It possesses a wide range of both, at the same time, until

  we measure it and thereby fix at least one of them within some small

  range determined by our measurement apparatus.

  • • •

  Following Heisenberg, the next step in unveiling the quantum

  craziness of reality was taken by an unlikely explorer, Paul Adrien

  Maurice Dirac. In one sense, Dirac was the perfect man for the job.

  As Einstein is reputed to have later said of him, “This balancing on

  the dizzying path between genius and madness is awful.”

  When I think of Dirac, an old joke comes to mind. A young child

  has never spoken and his parents go to see numerous doctors to seek

  help, to no avail. Finally, on his fourth birthday he comes down for

  breakfast and looks up at his parents and says, “This toast is cold!”

  His parents nearly burst with happiness, hug each other, and ask the

  child why he has never before spoken. He answers, “Up to now,

  everything was fine.”

  Dirac was notoriously laconic, and a host of stories exist about his

  unwillingness to engage in any sort of repartee, and also about how

  he seemed to take everything that was said to him literally. Once,

  while Dirac was writing on a blackboard during one of his lectures,

  someone in the audience was reputed to have raised his hand and

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  said, “I don’t understand that particular step you have just written

  down.” Dirac stood silent for the longest while until the audience

  member asked if Dirac was going to answer the question. To which

  Dirac said, “There was no question.”

  I actually spoke to Dirac, one day, on the phone—and I was

  terrified. I was still an undergraduate and wanted to invite him to a

  meeting I was organizing for undergraduates around the country. I

  made the mistake of calling him right after my quantum mechanics

  class, which made me even more terrified. After a rambling request

  that I blurted out, he was silent for a moment, then gave a simple

  one-line response: “No, I don’t think I have anything to say to

  undergraduates.”

  Personality aside, Dirac was anything but timid in his pursuit of a

  new Holy Grail: a mathematical formulation that might unify the

  two new revolutionary developments of the twentieth century,

  quantum mechanics and relativity. In spite of numerous efforts since

  Schrödinger (who derived his famous wave equation during a two-

  week tryst in the mountains with several of his girlfriends), and since

  Heisenberg had revealed the basic underpinning of quantum

  mechanics, no one had been successful at fully explaining the

  behavior of electrons bound deep inside atoms.

  These electrons have, on average, velocities that are a fair fraction

  of the speed of light, and to describe them, we must use Special

  Relativity. Schrödinger’s equation worked well to describe the energy

  levels of electrons in the outer parts of simple atoms such as

  hydrogen, where it provided a quantum extension of Newtonian

  physics. It was not the proper description when relativistic effects

  needed to be taken into account.

  Ultimately Dirac succeeded where all others had failed, and the

  equation he discovered, one of the most important in modern

  particle physics, is, not surprisingly, called the Dirac equation. (Some

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  years later, when Dirac first met the physicist Richard Feynman,

  whom we shall come to shortly, Dirac said after another awkward

  silence, “I have an equation. Do you?”)

  Dirac’s equation was beautiful, and as the first relativistic

  treatment of the electron, it allowed correct and precise predictions

  for the energy levels of all electrons in atoms, the frequencies of light

  they emit, and thus the nature of all atomic spectra. But the equation

  had a fundamental problem. It seemed to predict new particles that

  didn’t exist.

  To establish the mathematics necessary to describe an electron

  moving at relativistic speeds, Dirac had to introduce a totally new

  formalism that used four different quantities to describe electrons.

  As far as we physicists can discern, electrons are microscopic

  point particles of essentially zero radius. Yet in quantum mechanics

  they nevertheless behave like spinning tops and therefore have what

  physicists call angular momentum. Angular momentum reflects that

  once objects start spinning, they will not stop unless you apply some

  force as a brake. The faster they are spinning, or the more massive

  they are, the greater the angular momentum.

  There is, alas, no classical way of picturing a pointlike object such

  as an electron spinning around an axis. Spin is thus one of the areas

  where quantum mechanics simply has no intuitive classical

  analogue. In Dirac’s relativistic extension of Schrödinger’s equation,

  electrons can possess only two possible values for their angular

  momentum, which we simply c
all their spin. Think of electrons as

  either spinning around one direction, which we can call up, or

  spinning around the opposite direction, which we can call down.

  Because of this, two quantities are needed to describe the

  configurations of electrons, one for spin-up electrons and one for

  spin-down electrons.

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  After some initial confusion, it became clear that the other two

  quantities that Dirac needed to describe electrons in his relativistic

  formulation of quantum mechanics seemed to describe something

  crazy—another version of electrons with the same mass and spin but

  with the opposite electric charge. If, by convention, electrons have a

  negative charge, then these new particles would have a positive

  charge.

  Dirac was flummoxed. No such particle had ever been observed.

  In a moment of desperation, Dirac supposed that perhaps the

  positively charged particle described by his theory was actually the

  proton, which, however, has a mass two thousand times larger than

  that of the electron. He gave some hand-waving arguments for why

  the positively charged particle might get a heavier mass. The larger

  weight could be caused by different possible electromagnetic

  interactions it had with otherwise empty space, which he envisaged

  might be populated with a possibly infinite sea of unobservable

  particles. This is actually not as crazy as it sounds, but to describe

  why would force us toward one of those twists and turns that we

  want to avoid here. In any case, it was quickly shown that this idea

  didn’t hold water—first, because the mathematics didn’t support this

  argument, and the new particles would have to have the same mass

  as electrons. Second, if the proton and the electron were in some

  sense mirror images, then they could annihilate each other so that

  neutral matter could not be stable. Dirac had to admit that if his

  theory was true, some new positive version of the electron had to

  exist in nature.

  Fortunately for Dirac, within a year of his resigned capitulation,

  Carl Anderson found particles in cosmic rays that are identical to

  electrons but have the opposite charge. The positron was born, and

  Dirac was heard to say, in response to his unwillingness to accept the

  implications of his own mathematics, “My equation was smarter