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
ͥ͡
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
ͥ͢
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:
ͥͣ
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:
ͥͤ
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.
ͥͥ
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.
͜͜͝
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
͜͝͝
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
͜͝͞
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.
͜͟͝
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
Lawrence Krauss - The Greatest Story Ever Told--So Far Page 10