Metamagical Themas, page 60
In that year, Max Planck invented a sort of hybrid formula that looked like a mathematical splicing-together of two different components, one pertaining to long wavelengths and the other to short wavelengths. At the longer wavelengths, the formula agreed with the classical prediction and also with the measured data. At the shorter wavelengths, Planck's formula diverged from the classical prediction but stayed in agreement with the data. The long and the short of it was that Planck's equation seemed right on the money for all wavelengths and temperatures-but it had not been derived from the first principles. It was a lucky guess, although much more than luck was involved, since Planck's intuition had guided him like a bloodhound to this formula.
Planck himself was particularly baffled by the fact that he'd had to throw a strange quantity he called "the elementary quantum of action", h, into his formula. What h represented physically was unclear. It was just a constant that, with a suitable value, would make the formula exactly reproduce the observed spectrum. It seemed therefore to be a universal constant of nature.
But what in the world was it doing in this equation? What did it mean? Einstein was the first to postulate a physical reason for the appearance of Planck's constant h in the equation. Einstein began with the concept that the energy content of light waves is deposited in tiny "lumps"--photons-whose size has to do with h and their wavelength. For example, if the light is red, the photons carry always 3.3 X 10-12 erg of energy. Green photons carry 4 X 10-12 erg. AM radio-wave photons carry somewhere between 3 X 10221 and 9 X 10-21 erg (depending on what station you're listening to). The amount of energy per photon was postulated to be invariant, given its color that is, its wavelength).
In the water-wave analogy, you can try to envision ripples that, when they reach the shore, suddenly disappear and are replaced by frogs who hop up the bank where the waves, had they landed, would have lapped. The longer the wavelength of the ripple, the tinier the frog that jumps out, and conversely: delicate ripples with very short wavelengths, when they reach the shore, suddenly become thundering monster-frogs who knock eucalyptus trees down and send boulders crashing into the lake (this is the infamous phrogo-eucalyptic effect, so yclept by reason of its analogy with the famous photoelectric effect, in which incoming photons of sufficient energy knock electrons out of a metal surface).
Einstein's interpretation of Planck's formula implied that a frog's energy -or rather, a photon's energy-and its wavelength must be inversely proportional. The equation linking them is:
E = hc / λ
Here, E is the photon's energy, h is Planck's newly discovered constant, c is the speed of light, and λ is the photon's wavelength. E and λ are the only variables. This mixing of wave and particle viewpoints was one of the most baffling aspects of quantum mechanics, and it has continued to plague the intuitions of physicists ever since, although mathematically it was greatly cleared up by the blossoming of the field in the 1920's and 1930's
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The next step en route to Heisenberg's uncertainty principle came in 1924, when Prince Louis-Victor de Broglie was reflecting on the mysterious particle-like nature of light waves. He asked himself: Why should only light waves be particle-like? Why not the reverse? That is, mightn't particles also have wavelike properties? De Broglie's intuition was more or less as follows: If you want to generalize Einstein's equation so that it holds for particles other than photons, you have to get rid of the one direct reference in it to light, namely c. Hence de Broglie thought about how he might most elegantly and relativistically recast the equation in a c-less form.
This proved to be not too hard, because by then it was known that photons have both energy E and momentum p, and that they are related by the equation E =pc. If you combine the two equations, you can cancel out the c , and the result is:
p = h / λ.
Mathematically speaking, this equation of de Broglie's is new, but physically speaking, its content is no different from that of Einstein's original equation -at least when it is applied to photons. De Broglie's conceptual bravery was to propose-without any experimental evidence for it-that this equation should be universal. It should apply to all matter: not just photons, but also electrons, protons, atoms, billiard balls, people-even frogs! Thus Kermit the Frog would have a quantum-mechanical wavelength whose value would depend on how fast he's hopping.
What would this mean physically? What can a hopping frog's wavelength mean? Well, if you calculate it, you will find that Kermit's wavelength comes out far shorter than the radius of a proton-yet Kermit himself is considerably bigger than a proton. If Kermit were very, very small-small enough that his wavelength and his own size were comparable-then his wavelength would make him diffract around objects the way water waves and sound waves do. But since Kermit is macroscopic, his having a microscopic wavelength is all but irrelevant.
For electrons, though, it is entirely another matter. They are smaller than their own wavelengths. (In fact, as far as anyone knows, electrons are perfect point particles, with zero radius.) Shortly after de Broglie's suggestion, experiment and theory thoroughly confirmed his notion. Electron waves were soon being diffracted in laboratories around, the world, just like light waves. But now there arises a puzzle. Are electrons spread out in space in the way waves must be, or are they localized? If they are truly points, how can they be diffracted? If they are truly waves, where is their electric charge carried?
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Experiments have shown that even a single electron can be diffracted. Richard Feynman, in his little book The Character of Physical Law, describes it beautifully. In an idealized experiment, one electron is released in the direction of a barrier with two slits in it. On the far side of the barrier is a detecting screen. The electron follows some trajectory and hits the screen somewhere. One such event simply results in one dot being made on the screen. Suppose we repeat the experiment many times, each time releasing just one electron. We get a buildup of dots on the screen. Intuition, building on our experience with such things as bullets fired from a gun, tells us clearly to expect the dots to be clustered directly behind each of the two slits, with their distribution tailing off with distance from the center of each cluster. In other words, we would expect to find two clusters of dots and no other kind of distribution. (See Figure 20-1a.)
But if the de Broglie wavelength of the electron is close to the distance between the slits, the pattern on the screen after thousands of arrivals will look very different. It will be a complex regular structure characteristic of waves interfering with each other. In fact, it will reproduce the intensity ,pattern created by a wave that splits itself into two pieces, which pass through the two slits and interfere with each other on the far side of the
barrier. (See Figures 20-1b and 20-Ic.) It must be inferred that each electron, as it flew in its trajectory from source to screen,, somehow "sensed" both slits and interfered with itself in the manner of a wave and yet deposited itself froglike (that is, in a point) on the screen without a trace of its schizophrenia.
The dilemma is, then, that electrons act as if they are both spread out and localized-as if they were both waves and particles. This kind of wishy-washiness is inconceivable in the macroscopic realm. Most of us have no trouble distinguishing between, say, ripples on a pond, and frogs. For those who do, however, it might be useful to clip out the following handy frog-ripple distinguisher:
Test 1: Is the candidate solid, tangible, and above all, always somewhere?
If your answers to these three questions are yes, you are probably dealing with a frog.
Test 2: Is the candidate massless, intangible, and spread out?
If your answers to these three questions are yes, it is probably a ripple
If you are hungry for frog's legs and want to know where a frog is, you can just look around, and as soon as you sense some froglike photons entering your eyes, you will have found it. Those photons bounced off the frog and into your eyes. But suppose the frog somehow grew smaller and smaller. After it got down to the size of a mitochondrion in a living cell, its diameter would be about the wavelength of frog-green light. Then it would diffract light, and you would not be able to find it so easily. If it grew even smaller, something terrible would begin to happen. The individual photons hitting it would, with their momentum and energy, begin to jostle it around. The particle-like quality of photons would start to enter the picture. Indeed, a frog the size of an electron would probably be very hard to find. So if you were starved for frog's legs, you would do better to look around for a bigger one.
Unfortunately, though, no matter how starved you might be for electron's legs, you cannot find a bigger electron! To find an electron, you cannot do anything but bombard it with other particles or with photons. Since particles and photons have both particle-like and wavelike aspects, either bombardment will lead to similar consequences. If you want to pinpoint a particle, you need waves whose wavelength is about the size of that particle (or shorter). To understand this intuitively, think of the way water waves would be affected by a floating piece of wood. If they have a very long wavelength, they will not even "notice" the wood. Only if their wavelength gets down to the size of the object will they begin to be affected by it.
Consequently, in order to find our electron, we need photons of very short wavelength. But wavelength is inversely proportional to momentum.
That is the deadly import of de Broglie's equation. You pay for your short wavelength by having a lot of momentum. And so, as you try to diffract waves ever so gently off your particle, hoping not to move it, you will not -be able to do so without transmitting momentum to it. Either you are gentle (using long-wavelength photons) and do not see the electron well, or you are violent (using short-wavelength photons) and throw the electron completely off its course.
Heisenberg made a careful study of this perversity, which follows from de Broglie's equation, and, to the bewilderment of epistemology lovers the world over, he discovered that to know the position of a particle perfectly is to give up any hope of knowing its momentum, and that to know the momentum is to give up any hope of knowing its position. And knowing either one imprecisely still imposes bounds on the precision with which you could know the other. The principle can even be summarized in an inequality, which Heisenberg deduced. If you are trying to determine the location of the particle, there will be an uncertainty, conventionally denoted Ax. There will also be an uncertainty in the value of the momentum, -denoted Op. Heisenberg's uncertainty principle is the following inequality:
AxAp > h/4π.
There are a couple of things to point out here. First, note the presence of h, Planck's mysterious constant. This tells you that the effect is due to the wave-particle duality of matter (and of photons), and has nothing to do with ,the notion of an observer disturbing the thing under observation. Second,
FIGURE 20-1. Three related two-slit experiments, two classical and one quantum-mechanical. [Drawing by David Moser, after Richard Feynman. ]
In (a), a wildly swinging machine gun sprays bullets toward a wall with two holes in it. Occasionally, a bullet will pass through one of the two holes, and will hit the backstop and make a mark. Eventually, the buildup of marks looks as shown. It has two peaks, one for each hole.
In (b), a bobbing buoy creates ripples that spread out toward a jetty with two breaks in it. When the ripples hit the jetty, new circular ripples emanate from each of the two breaks, and those ripples, crisscrossing each other, interfere constructively at some points and destructively at others. On a vertical barrier parallel to the jetty, areas of highly constructive interference are dark, and areas of highly destructive interference are white. This characteristic interference pattern is due to two facts: first, that any ripple passes through both holes, rather than just one, and second, that the phases at the two holes are correlated.
In (c), a wildly swinging electron gun sprays electrons toward a wall with two holes in it. Beyond the wall there is a backstop made of some material that emits a flash whenever an electron hits it. There is no classical way to describe what happens to any electron en route, but that what is certain is that, when it comes in for a landing on the backstop, its local spot of arrival is clearly visible, just as in (a) (thus reminding us of the corpuscular, or bullet-like, nature of electrons); and yet, if those flashes are tallied up over a period of time, they are found to be distributed in an interference pattern just like the one formed in (b) (thus reminding us of the undulatory, or ripple-like, nature of electrons). Any attempt to ascertain which of the two holes the electrons pass through ends up in destruction of the interference pattern.
notice that even with this epistemological restriction, arbitrarily accurate measurement of either position or momentum is possible; you just can't get both.
In short, it is a total misinterpretation of Heisenberg's uncertainty principle to suppose that it applies to macroscopic observers making macroscopic measurements. For example, it does not follow from Heisenberg's principle that psychologists studying the phenomena of human cognition are somehow limited in principle by the fact that the conscious human beings they are observing are capable of the same kind of observation. What psychologists are limited by is their knowledge of the human brain, their ingenuity, and, of course, their funding.
If you wanted to know more about grammatical anomalies in the speech of woman W, there are all sorts of ways that you could, in principle, go about it without making her self-conscious. For just a few thousand dollars, for instance, you could secretly install a bug in her home and monitor all her conversations. For a few hundred thousand dollars, you could have tiny radio transmitters manufactured and secretly sewn into all her lapels. For, say, a few million dollars, you might be able to convince her she needed minor surgery of some sort, and then while she was anesthetized you could open up her skull and have harmless electrodes implanted in her brain to monitor her speech areas-all without her knowing. If you fear that such blatant physical interference with her brain might disturb her grammatical habits, then you may have to wait a while longer until we figure out how neural activity can be examined remotely. These possibilities are clearly extravagant, even ridiculous, but the point is that, in principle, we can study macroscopic phenomena with an arbitrary degree of precision.
To recapitulate: The uncertainty principle states not that the observer always interferes with the observed, but rather that at a very fine grain size, the wave-particle duality of the measuring tools becomes relevant. It is a consequence of the fact that Planck's constant is not zero, rather than an epistemological law about observation that would have been discovered with or without the discovery of quantum mechanics.
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The uncertainty principle is not an axiom of quantum physics; it is a deduced principle, just as Einstein's most famous equation E = mc2 was deduced from the more fundamental equations of special relativity-a fact that most non-physicists do not appreciate. Both equations are useful (and famous) because they are so pithy. For example, the uncertainty principle is often applied by physicists as a rule of thumb. If you want to estimate the approximate momentum a neutron will have when it is emitted by a nucleus decaying from an excited state, a seat-of-the-pants estimate is given by p = h/d, where d is on the order of the dimensions of the confining nucleus. You can think of the confinement within the nucleus as making the position uncertainty very small, so that the neutron is bouncing around inside its ``cage" with a compensatingly large momentum uncertainty. When it escapes, a rough estimate of the momentum it will have is given by the uncertainty value.
When you examine the foundations of quantum mechanics, it becomes clear that the uncertainty principle is more than an epistemological restriction on human observers; it is a reflection of uncertainties in nature itself. Quantum-mechanical reality does not correspond to macroscopic reality. It's not just that we cannot know a particle's position and momentum simultaneously; it doesn't even have definite position and momentum simultaneously!
In quantum mechanics, a particle is represented by a so-called wave function describing the probabilities that the particle is here, there, or somewhere else; that the particle is heading east, west, north, or south; and so on. For each point in space, there is what is called a probability amplitude of finding the particle there, and this number is given by the wave function. Alternatively, one can read the wave function through different "mathematical glasses" and obtain a probability amplitude for each possible value of momentum. All the facts about the particle are wrapped up in its wave function. In more modern terminology, the term "state" is often used instead of "wave function".
In classical physics, quantities such as x and p-position and momentum directly enter the equations governing a particle's behavior. The values of x and p are definite at any one moment, and they change according to :the forces that are acting on the particle. With such equations of motion, physicists can plot in advance the positions and momenta of particles in -simple, stable systems with incredible accuracy. An example is the motions of the planets, which even the ancients learned to predict with considerable accuracy. A more contemporary example is provided in computer space games, where rockets and planets are affected by a star's gravity and can go into orbit right before your eyes, swinging about in perfect ellipses on a screen. The underlying equations of such motion are differential equations, and one obvious property they have-we take it for granted-is that the motions they describe are smooth. Planets and rocket ships do not jump out of their orbits. There are no sudden discontinuities in their motion.
In quantum mechanics; x and p do not enter into the equations of motion as they do in classical mechanics. Instead, it is the wave function (in nonrelativistic quantum mechanics) that evolves in time according to a differential equation: Schrödinger’s equation, named for Heisenberg's contemporary, the quantum-mechanical pioneer Erwin Schrödinger. As time progresses, the values of the wave function ripple through space just the way a water wave ripples on a lake's surface. This would seem to imply that quantum phenomena, like nonquantum ones, proceed smoothly and with no jumps. In one sense, that is right. A well-known example is the smooth precession of a spinning charged particle in a magnetic field. It is a kind of electromagnetic analogue to the precession of a spinning top on a table. The parameters that characterize the state of the spinning top or spinning particle do indeed change smoothly, without any jumps.
