Part II Of Other Things 14. A Piece Of The Action
When my book 1, Robot was reissued by the estimable gentlemen of Doubleday amp; Company, it was with a great deal of satisfaction that I noted- certain reviewers (posses sing obvious intelligence and good taste) beginning to refer to it as a "classic."
"Classic" is derived in exactly the same way, and has precisely the same meaning, as our own "first-class" and our colloquial "classy"; and any of these words represents my own opinion of 1, Robot, too; except that (owing to my modesty) I would rather die than admit it. I mention it here only because I am speaking confidentially.
However, "classic" has a secondary meaning that dis pleases me. The word came into its own when the literary men of the Renaissance used it to refer to those works of the ancient Greeks and Romans on which they were model ing their own efforts. Consequently, "classic" has come to mean not only good, but also old.
Now 1, Robot first appeared a number of years -ago and some of the material in it was written... Well, never mind. The point is that I have decided to feel a little hurt at being considered old enough to have written a classic, and therefore I will devote this chapter to the one field where "classic" is rather a term of insult.
Naturally, that field must be one where to be old is, almost automatically, to be wrong and incomplete. One may talk about Modem Art or Modern Literature or Modem Furniture and sneer as one speaks, comparing each, to their disadvantage, with the greater work of earlier ages. When one speaks of Modem Science, however, one removes one's hat and places it reverently upon the breast.
In physics, particularly, this is the case. There is Modern Physics and there is (with an offhand, patronizing half smile) Classical Physics. To put it into Modern Terrninol ogy, Modern Physics is in, man, in, and Classical Physics is like squaresvhle.
What's more, the division in physics is sharp. Everything after 1900 is Modern; everything before 1900 is Classical.
That looks arbitrary, I admit; a strictly parochial twentieth-century outlook. Oddly enough, though, it is per fectly legitimate. The year 1900 saw a major physical theory entered into the books and nothing has been quite the same since.
By now you have guessed that I am going to tell you about it.
The problem began with German physicist Gustav Robert Kirchhoff who, with Robert Wilhelm Bunsen (popularizer of the Bunsen burner), pioneered in the de velopment of spectroscopy in 1859. Kirchhoff discovered that each element, when brought to incandescence, gave off certain characteristic frequencies of light; and that the vapor of that element, exposed to radiation from a source hotter than itself, absorbed just those frequencies it itself emitted when radiating. In short, a material will absorb those frequencies which, under other conditions, it will radiate; and will radiate those frequencies which, under other conditions, it will absorb.'
But su Ippose that we consider a body which will absorb all frequencies of radiation that fall upon it-absorb them completely. It will then reflect none and will therefore ap pear absolutely black. It is a "black body." Kirchhoff pointed out that such a body, if heated to incandescence, would then necessarily have to radiate all frequencies of radiation' Radiation over a complete range in this manner would be "black-body radiation."
Of course, no body was absolutely black. In the 1890s, however, a German physicist named Wilhelm Wien thought of a rather interesting dodge to get around tiat.
Suppose you had a furnace with a small opening. Any radiation that passes through the opening is either ab sorbed by the rough wall opposite or reflected. The re 175 flected radiation strikes another wall and is again partially absorbed. What is reflected strikes another wall, and so on. Virtually none of the radiation survives to find its way out the small opening again. That small opening, then, absorbs the radiation and, in a manner of speaking, reflects none. It is a black body. If the furnace is heated, the radia tion that streams out of that small opening should be black-body radiation and should, by Kircbhoff's reasoning, contain all frequencies.
Wien proceeded to study the characteristics of this black-body radiation. He found that at any temperature, a wide spread of frequencies was indeed included, but the spread was not an even one. There was a peak in the mid dle. Some intermediate frequency was radiated to a greater extent than other frequencies either higher or lower than that peak frequency. Moreover, as the temperature was increased, this peak was found to move toward the higher frequencies. If the absolute temperature were doubled, the frequency at the peak would also double.
But now the question arose: Why did black-body radia tion distribute itself like this?
To see why the question was puzzling, let's consider infrared light, visible light, and ultraviolet light. The fre quency range of infrared light, to begin with, is from one hundred billion (100,000,000,000) waves per second to four hundred trillion (400,000,000,000,000) waves per second. In order to make the numbers easier to handle, let's divide by a hundred billion and number the frequency not in individual waves per second but in hundred-billion wave packets per second. In that case the range of infrared would be from 1 to 4000.
Continuing to use this system, the range of visible licht would be from 4000 to 8000; and the range of ultraviolet light would be from 8000 to 300,000.
Now it might be supposed that if a black body absorbed all radiation with equal ease, it ought to give off all radia tion with equal case. Whatever its temperature, the energy it had to radiate might be radiated at any frequency, the particular choice of frequency being purely random.
But suppose you were choosing numbers, any numbers with honest radomness, from I to 300,000. If you did this repeatedly, trillions of times, 1.3 per cent of your numbers would be less than 4000; another 1.3 per cent would be between 4000 and 8000 ' and 97.4 per cent would be between 8000 and 300,000.
This is like saying that a black body ought to radiate
1.3 per cent of its energy in the infrared, 1.3 per cent in visible light, and 97.4 per cent in the ultraviolet. If the temperature went up and it had more energy to radiate, it ought to radiate more at every frequency but the relative amounts in each range ought to be unchanged.
And this is only if we confine ourselves to nothing of still higher frequency than ultraviolet. If we include the x-ray frequencies, it would turn out that just about nothing should come off in the visible light at any temperature.
Everything would be in ultraviolet and x-rays.
An English physicist, Lord Rayleigh (1842-1919), worked out an equation which showed exactly this. The radiation emitted by a black body increased steadily as one went up the frequencies. However, in actual practice, a frequency peak was reached after which, at higher fre quencies still, the quantity of radiation decreased again.
Rayleigh's equation was interesting but did not reflect reality.
Physicists referred to this prediction of the Rayleigh equation as the "Violet Catastrophe"-the fact that every body that bad energy to radiate ought to radiate practically all of it in the ultraviolet and beyond.
Yet the whole point is that the Violet Catastrophe does not take place. A radiating body concentrated its radiation in the low frequencies. It radiated chiefly in the infrared at temperatures below, say, 1000' C., and radiated mainly in the visible region even at a temperature as high as
6000' C., the temperature of the solar surface.
Yet Rayleigh's equation was worked out according to the very best principles available anywhere in physical theory-at the time. His work was an ornament of what we now call Classical Physics.
Wien himself worked out an equation which described the frequency distribution of black-body radiation in the bigh-frequency range, but he had no explanation for why it worked there, and besides it only worked for the high frequency range, not for the low-frequency.
Black, black, black was the color of the physics mood all through the later 1890s.
Bt4t then arose in 1899 a champion, a German physicist, Max Karl Ernst Ludwig Planck. He reasoned as fol -lows...
If beautiful equations worked out by impeccable reason ing from highly respected physical foundations do not de scribe the truth as we observe it, then either the reason ing or the physical foundations or both are wrong.
And if there is nothing wrong about the reasoning (and nothing wrong could be found in it), then the physical foundations had to be altered.
The physics of the day required that all frequencies of light be radiated with equal probability by a black body, and Planck therefore proposed that, on the contrary, they were not radiated with equal probability. Since the equal probability assumption required that more and more light of higher and higher frequency be radiated, whereas the reverse was observed, Planck further proposed that the probability of radiation ought to decrease as frequency increased.
In that case, we would now have two effects. The first effect would be a tendency toward randomness which would favor high frequencies and increase radiation as frequency was increased. Second, there was the new Planck effect of decreasing probability of radiation as frequency went up. This would favor low frequencies and decrease radiation as frequency was increased.
In the low-frequency range the first effect is dominant, but in the high-frequency range the second effect increas ingly overpowers the first. Therefore, in black-body tadia tion, as one goes up the frequencies, the amount of radia tion first increases, reaches a peak, then decreases again exactly as is observed.
Next, suppose the temperature is raised. 'ne first effect can't be changed, for randomness is randomness. But sup 178 pose that as the temperature is raised, the probability of emitting high-frequency radiation increases. The second effect, then, is steadily weakened as the temperature goes up. In that case, the radiation continues to increase with increasing frequency for a longer and longer time before it is overtaken and repressed by the gradually weakening second effect. The peak radiation, consequently, moves into higher and higher frequencies as the temperature goes up-precisely as Wien had discovered.
On this basis, Planck was able to work out an equation that described black-body radiation very nicely both in the low-frequency and high-frequency range.
However, it is all very well to say that the higher the frequency the lower the probability of radiation, but why?
There was nothing in the physics of the time to explain that, and Planck had to make up something new.
Suppose that energy did not flow continuously, as physicists had, always assumed, but was given off in pieces.
Suppose there were "energy atoms" and these increased in size as frequency went up. Suppose, still further, that light of a particular frequency could not be emitted unless enough energy had been accumulated to make up an "energy atom" of the size required by that frequency.
The higher the frequency the larger the "energy atom" and the smaller the probability of its accumulation at any given instant of time. Most of the energy would be lost as radiation of lower frequency, where the "energy atoms" were smaller and more easily accumulated. For that rea son, an object at a temperature of 400' C. would radiate its heat in the infrared entirely. So few "energy atoms" of visible light size would be accumulated that no visible glow would be produced.
As temperature went up, more energy would be gen erally available and the probabilities of accumulating a high-frequency "energy atom" would increase. At 6000' C. most of the radiation would be in "energy atoms" of visible light, but the still larger "energy atoms" of ultraviolet would continue to be formed only to a minor extent.
But how big is an "energy atom"? How much energy does it contain? Since this "how much" is a key question, Planck, with admirable directness, named the "energy atom" a quantum, which is Latin for "how much?" the plural is quanta.
For Planck's equation for the distribution of black-body radiation to work, the size of the quantum had to be directly proportional to the frequency of the radiation. To express this mathematically, let us represent the size of the quantum, or the amount of energy it contains, by e (for energy). The frequency of radiation is invariably repre sented by physicists by means of the Greek letter nu (v).
If energy (e) is proportional to frequency (v), then e must be equal to v multiplied by some constant. This con stant, called Planck's constant, is invariably represented as h. The equation, giving the size of a quantum for a par ticular frequency of radiation, becomes: e = hv (Equation 1)
It is this equation, presented to the world in 1900, which is the Continental Divide that separates Classical Physics from Modern Physics. In Classical Physics, energy was considered continuous; in Modern Physics it is con sidered to be composed of quanta. To put it another way, in Classical Physics the value of h is considered to be 0; in Modern Physics it is considered to be greater than 0.
It is as though there were a sudden change from con sidering motion as taking place in a smooth glide, to mo tion as taking place in a series of steps.
There would be no confusion if steps were long ga lumphing strides. It would be easy, in that case, to dis tinguish steps from a glide. But suppose one minced along in microscopic little tippy-steps, each taking a tiny frac tion of a second. A careless glance could not distinguish that from a glide. Only a painstaking study would show that your head was bobbing slightly with each step. The smaller the steps, the harder to detect the difference from a glide.
In the same way, everything would depend on just how big individual- quanta were; on how "grainy" energy was.
The size of the quanta depends on 'the size of Planck's constant, so let's consider that for a while.
If we solve Equation I for h, we get: h = elv (Equation 2) Energy is very frequently measured in ergs (see Chapter 13). Frequency is measured as "so many per second" and its units are therefore "reciprocal seconds" or "I/second."
We must treat the units of h as we treat h itself. We get h by dividing e by v; so we must get the units of h by dividing the units of e by the units of v. When we divide ergs by I/second we are multiplying ergs by sec onds, and we find the units of h to be "erg-seconds." A unit which is the result of multiplying energy by time is said, by physicists, to be one of "action." Therefore, Planck's constant is expressed in units of action.
Since the nature of the universe depends on the size of Planck's constant, we are all dependent on the size of the piece of action it represents. Planck, in other words, had sought and found the piece of the action. (I understand that others have been searching for a piece of the action ever since, but where's the point since Planck has found it?)
And what is the exact size of h? Planck found it had to be very small indeed. The best value, currently ac cepted, is: 0.0000000000000000000000000066256 erg seconds,or 6.6256 x 10-2" erg-seconds.
Now let's see if I can find a way of expressing just how small this is. The human body, on an average day, con sumes and expends about 2500 kilocalories in maintaining itself and performing its tasks. One kilocalorie is equal to 1000 calories, so the daily supply is 2,500,000 calories.
One calorie, then, is a small quantity of energy from the human standpoint. It is 1/2,500,000 of your daily store. It is the amount of energy contained in 1/113,000 of an ounce of sugar, and so on.
Now imagine you are faced with a book weighing one pound and wish to lift it from the floor to the top of a bookcase three feet from the ground. The energy expended in lifting one pound through a distance of three feet against gravity is just about 1 calorie,.
Suppose that Planck's constant were of the order of a calorie-second in size. The universe would be a very strange place indeed. If you tried to lift the book, you would have to wait until enough energy had been accumu lated to make up the tremendously sized quanta made necessary by so large a piece of action. Then, once it was accumulated, the book would suddenly be three feet in the air.
But a calorie-second is equal to 41,850,000 erg-seconds, and since Planck's constant is 'Such a minute fraction of one erg-secoiid, a single calorie-second equals 6,385,400, 000,000,000,000,000,000,000,000,000 Planck's constants, or 6.3854 x 10:[email protected]' Planck's constants, or about six and a third decillion Planck's constants. However you slice it, a calorie-second is equal to a tremendous number of Planck's constants.
Consequently, in any action such as the lifting of a one pound book, matters are carried through in so many tril lions of trillions of steps, each one so tiny, that motion seems a continuous glide.
When Planck first introduced his "quantum theory 91 in 1900, it caused remarkably little stir, for the quanta seemed to be pulled out of midair. Even Planck himself was dubious-not over his equation describing the dis tribution of black-body radiation, to be sure, for that worked well; but about the quanta he had introduced to explain the equation.
Then came 1905, and in that year a 26-year-old theo retical physicist, Albert Einstein, published fivo separate scientific papers on three subjects, any one of which would have been enough to establish him as a first-magnitude star in the scientific heavens.
In two, he worked out the theoretical basis for "Brown ian motion" and, incidentally, produced the machinery by which the actual size of atoms could be established for the first time. It was one of these papers that earned him his Ph.D.
In the third paper, he dealt with the "photoelectric effect" and showed that although Classical Physics could not explain it, Planck's quantum theory could.
This really startled physicists. Planck had invented quanta merely to account for black-body radiation, and here it turned out to explain the photoelectric effect, too, something entirely different. For quanta to strike in two different places like this, it seemed suddenly very reason able to suppose that they (or something very like them) actually existed.
(Einstein's fourth and fifth papers set up a new view of the universe which we call "The Special Theory of Rela tivity." It is in these papers that he introduced his famous equation e = MC2; see Chapter 13.
These papers on relativity, expanded into a "General Theory" in 1915, are the achievements for which Einstein is known to people outside the world of physics. Just the same, in 1921, when he was awarded the Nobel Prize for Physics, it was for his work on the photoelectric effect and not for his theory of relativity.)
The value of h is so incredibly small that in the ordinary world we can ignore it. The ordinary gross events of everyday life can be considered as though energy were a continuum. This is a good "first approximation."
However, as we deal with smaller and smaller energy changes, the quantum steps by which those changes'must take place become larger and larger in comparison. Thus, a flight of stairs consisting of treads 1 millimeter high and 3 millimeters deep would seem merely a slightly roughened ramp to a six-foot man. To a man the size of an ant, how ever, the steps would seem respectable individual obstacles to be clambered over with difficulty. And to a man the size of a bacterium, they would be mountainous precipices lin the same way, by the time we descend into the world within the atom the quantum step has become a gigantic thing. Atomic physics cannot, therefore, be described in Classical terms, not even as an approximation.
The first to realize this clearly was the Danish physicist Niels Bohr. In 1913 Bohr pointed out that if an electron absorbed energy, it had to absorb it a whole quantum at a time and that to an electron a quantum was a large piece of en 'ergy that forced it to change its relationship to the rest of the atom drastically and all at once.
Bohr pictured the electron as circling the atomic nucleus in a fixed orbit. When it absorbed a quantum of energy, it suddenly found itself in an orbit farther from the nucleus - there was no in-between, it was a one-step proposition.
Since only certain orbits were possible, according to Bohr's treatment of the subject, only quanta of certain size could be absorbed by the atom-only quanta large enoug to raise an electron from one permissible orbit to another.
When the electrons dropped back down the line of per missible orbits, they emitted radiations in quanta. They emitted just those frequencies which went along with the size of quanta they could emit in going from one orbit to another.
In this way, the science of spectroscopy was rational ized. Men understood a little more deeply why each ele ment (consisting of one type of atom with one type of energy relationships among the electrons making up that type of atom) should radiate certain frequencies, and cer tain frequencies only, when incandescent. They also under stood why a substance that could,absorb certain frequen cies should also emit those same frequencies under other circumstances.
In other words, Yirchhoff had started the whole problem and now it had come around fuil-circle to place his em pirical discoveries on a rational basis.
Bohr's initial picture was oversimple; but he and other men gradually made it more complicated, and capable of explaining finer and finer points of observation. Finally, in 1926, the Austrian physicist Erwin Schri3dinger worked out a mathematical treatment that was adequate to an alyze the workings of the particles making up the interior of the atom according to the principles of the quantum theory. This was called "quantum mechanics," as opposed to the "classical mechanics" based on Newton's three laws of motion and it is quantum mechanics that is the founda- tion of Modern Physics.
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