The Quantum Limit

One of the greater intellectual shocks of the Scientific Age to jolt Humanity was the discovery that the world present to our senses rests upon a foundation that seems to defy all reason. It comes to us as if someone had shown us that the typical suburban home sits on a foundation of cold fire that pulsates to some bizarre music. As physicists probed Reality on ever smaller scales, they discovered that the intuitively clear laws of Newtonian physics increasingly gave way to the fever-dream rules of the quantum theory.

How strange does it get? If I face a wall that has two six-inch-wide windows placed twenty feet apart and if I throw one thousand baseballs at the center of that wall, a point midway between the windows, we know that none of those balls will go through either of those windows. Now replace the wall with a razor blade and cut into it a pair of slits, each as wide as a human hair, spaced one micron apart. If I project electrons at the center of that barrier (and someone has actually done this), some of them will pass through the barrier and they will do so by passing through both slits simultaneously.

Two similar situations, apparently differing from each other only in scale, produce different phenomena. Yet they occur in the same Reality. At what scale, then, and how does Newtonian mechanics segue into quantum mechanics?

That’s actually a poorly framed question. Quantum
mechanics applies to everything, from electrons to baseballs, and Newtonian
mechanics applies to everything, from baseballs to electrons. There seems to be
a fundamental difference between the two kinds of mechanics only because in
Newtonian mechanics we have tacitly made an unfortunate assumption about how
precisely we can measure physical quantities. Newtonian mechanics contains the
tacit assumption that we can measure any physical quantity with infinite
precision. But the quantum theory denies that assumption, Heisenberg’s
indeterminacy principle putting finite limits on the precision of measurement.
We have the illusion of infinite precision in Newtonian mechanics because the
imprecisions in measurement are coordinated through Planck’s constant, which has
a value of h=6.6256x10^{-34} joule-second.

A standard baseball ponders five ounces (142 grams). If I
can throw it at 50 miles per hour (22.25 meters per second), then it will have a
forward momentum that will enact 3.16 joule-seconds of action for every meter
the ball crosses. If my laser-based surveying apparatus can measure the position
of the ball to within one-tenth of a millimeter (about the thickness of a sheet
of paper), then the imprecision in its momentum must be at least equal to
6.6256x10^{-30} joule second per meter (kilogram-meter per second).
Expressed in our baseball, that imprecision corresponds to a discrepancy in
velocity of 4.666x10^{-29} meter per second. An object moving at that
speed would take 6.79x10^{16} years to cross the one-tenth of a
millimeter imprecision in our position measurement: we can’t even measure such a
speed, much less notice it.

In the lateral direction we gain the possibility of a
baseball going through the windows if there exists an indeterminacy of at least
three meters in the ball’s location. That requires an indeterminacy in the
ball’s lateral velocity of less than 1.555x10^{-33} meter per second.
The thermal motion of the atoms that constitute the ball is vastly greater than
that, so the indeterminacy in the ball’s lateral location is less than the
diameter of an atom. We can treat a baseball as a quantum-mechanical object and
yet be incapable of detecting or noticing its quantum nature.

An electron leaves no doubt about its quantum nature. To conceive an idea of an electron imagine a minuscule cloud of completely frictionless, perfectly elastic vapor whose center of mass has an indeterminate location within the cloud. Describing the motion of that particle toward our razor-blade barrier obliges us to use a probability wave that we imagine the electron riding like a surfer, albeit a surfer who is smeared out across the wave because the electron’s position on the wave is indeterminate until the electron interacts with something.

But an electron is also a Newtonian object. It has a well-defined mass and electric charge. At any given instant it has a location, a linear momentum, and an energy, though their values are subject to Heisenbergian indeterminacy. We could, in concept, determine the force imposed on an electron by an electromagnetic field, calculate the consequent acceleration, and devise a description of the electron’s trajectory. We don’t actually do that because the concepts of force and acceleration don’t acknowledge the wave nature of the electron’s motion and thus miss phenomena like the self-interference that occurs when the electron encounters the twin slits in our razor-blade barrier.

Doesn’t that latter statement invalidate my claim that the electron is a Newtonian object? Yes, we have no problem conceiving a particle’s trajectory as being more like a smoke trail than an infinitely thin thread. But the wave aspect of the particle doesn’t match anything that Isaac Newton put into his dynamic geometry. That wave aspect, though, doesn’t invalidate the description of the electron as a Newtonian object; it augments it. Imagine an invisible surfer: he has a definite mass, but his location, momentum, and energy depend on the wave he’s riding and we can’t know those with any precision. Because of that indeterminacy, we can’t know where on the beach he will land.

The surfer didn’t stop being a Newtonian object when he caught the wave. Even though his destination is indeterminate, he still responds to forces by changing his momentum in accordance with Newton’s second law of motion. It’s true to say that we can’t identify or describe the forces acting on the surfer as he rides the wave, but they exist nonetheless, even though we must use a probabilistic calculation to describe their effect on the surfer.

Forces also act on our electron, but we can’t describe them in proper detail, much less use them in calculations. Because we can’t describe the electron’s location in space and time with infinite precision, we must use in our calculations momentum and energy, which are subject to conservation laws and, thus, are more amenable to use in quantum calculations.

That we have a quantum theory at all comes from the fact that the Universe does not acknowledge any action smaller than Planck’s constant. We see that fact reflected most clearly in Heisenberg’s indeterminacy principle, which tells us that the indeterminacies in certain pairs of variables must multiply out to a value equal to or greater than Planck’s constant. On scales that are small relative to Planck’s constant, essentially the atomic scale and smaller, particle trajectories become indistinct. We see a kind of fuzziness in the particles’ motions. As the actions involved in events become larger, the fuzziness shrinks down to insignificance and becomes effectively nonexistent. We see this happening at the scale of molecules and larger.

If an electron were to grow into a baseball, its quantum nature would remain part of it but would diminish to such a degree that we could ignore it and use only the formulations of classical Newtonian dynamics to describe its motion. It would still carry the self-interference possibility, but that possibility would be so diminished that no instrument in the Universe could detect it. But we don’t need to resort to such fantasies to conceive an idea of the quantum-to-classical transition.

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