I promised (threatened) that I would say more about square roots, and so I am. This is me, talking about square roots again. In typical fashion, though, I'm going to start with something else that will seem, for a time, completely unrelated.
Galileo, he of the telescope, the balls rolling down inclined planes (and probably not in actuality from the Tower of Pisa), the sotto voce thumbing of the nose at the Inquisition—Galileo also discovered, or more likely rediscovered, that pendulums mark out roughly even time, no matter how far they swing. It isn't perfectly even time, owing to friction and to the circular track of the pendulum bob (although both of those can be—and were—accounted for, starting with Huygens's employment of cycloid guides). But it's pretty close.
Since the pendulum keeps fairly even time, that must mean that if the pendulum swings in twice as big an arc, it must also be moving twice as fast, in order to keep beating out even time. Now, as it's defined in Newtonian physics, the kinetic energy of the pendulum bob—that is, the energy of the bob due to its motion—goes as the square of its velocity:
KE = ½ mv²
So, twice the arc, twice the velocity, four times the kinetic energy; three times the arc, three times the velocity, nine times the kinetic energy. And so on.
That swinging motion of the pendulum bob is an example of periodic or wave motion, so called by virtue of it swinging back and forth as a water wave swings up and down, if you were to watch it passing by a buoy. Wave motion is primarily characterized by two parameters: its frequency, which is how often it returns to its starting point; and its amplitude, which is how wide it swings. So the arc through which the pendulum bob swings is essentially its amplitude. (Actually, for historical reasons, the amplitude is defined as half of that arc, from the center point of the swing to either of its extremes, but this won't affect our discussion.) So we can say that the pendulum's energy is proportional to the square of its amplitude.
This turns out to be common to many different kinds of waves—including light waves. Light is a wave. (It's also a particle, in many ways, but we'll ignore that for now.) And being a wave, it has an amplitude, which is the extent to which the light oscillates. What is it that's oscillating, anyway? In the case of water waves, it's water, and in the case of sound, it's the molecules in the air. You can't have water waves without the water, and you can't have sound waves without the air; that's why sound doesn't travel in a vacuum. But light does travel in a vacuum, so what's waving the light, so to speak? Well, the answer is that the light itself is waving, or less opaquely (heh heh), the electromagnetic fields that permeate space are waving.
In any event, like other waves, light waves also carry energy that is proportional to the square of the light's amplitude. If you double the amplitude, you quadruple the energy; triple the amplitude, and the energy goes up nine-fold. And so on.
How would light's amplitude be doubled, though? You might imagine that if you put two flashlights, the amplitude of the two together would be twice that of each individual flashlight, and the combined light output—the energy of the two together—would be four times that of each flashlight. But I think, intuitively, we know this to be false, that the combination is only twice as bright as each flashlight. And if you measure the light carefully, in a dark room, this turns out to be perfectly true.
What happened? Light waves, like other waves, have a secondary property, called phase. Two waves of the same frequency are said to be in phase if they swing in the same "direction" (in some not altogether well-defined sense); imagine two pendulums swinging in unison, so that when one swings left, the other does, too. They are out of phase if when one swings left, the other swings right, and vice versa. Or, they may be partly in phase, partly out of phase.
When you combine two light waves of the same frequency and the same amplitude, you get for all intents and purposes a single wave that is the two original waves added together. If they're in phase, the peaks get peakier and the valleys get, err, valleyier, and the amplitude of the waves is in fact doubled. On the other hand, if they're out of phase, the peaks of one get cancelled out by the valleys of the other (and vice versa), and the resultant wave has no amplitude at all.
More typically, though, the two waves are partly in phase and partly out of phase, and the resulting wave's amplitude is somewhere in between zero and two times the original. On average, one can show that the amplitude is the original times √2 . What's more, if you add three waves together at random phases, the amplitude of the sum is the original times √3 . And so on. Aha, the square root!
Galileo, he of the telescope, the balls rolling down inclined planes (and probably not in actuality from the Tower of Pisa), the sotto voce thumbing of the nose at the Inquisition—Galileo also discovered, or more likely rediscovered, that pendulums mark out roughly even time, no matter how far they swing. It isn't perfectly even time, owing to friction and to the circular track of the pendulum bob (although both of those can be—and were—accounted for, starting with Huygens's employment of cycloid guides). But it's pretty close.Since the pendulum keeps fairly even time, that must mean that if the pendulum swings in twice as big an arc, it must also be moving twice as fast, in order to keep beating out even time. Now, as it's defined in Newtonian physics, the kinetic energy of the pendulum bob—that is, the energy of the bob due to its motion—goes as the square of its velocity:
KE = ½ mv²
So, twice the arc, twice the velocity, four times the kinetic energy; three times the arc, three times the velocity, nine times the kinetic energy. And so on.
That swinging motion of the pendulum bob is an example of periodic or wave motion, so called by virtue of it swinging back and forth as a water wave swings up and down, if you were to watch it passing by a buoy. Wave motion is primarily characterized by two parameters: its frequency, which is how often it returns to its starting point; and its amplitude, which is how wide it swings. So the arc through which the pendulum bob swings is essentially its amplitude. (Actually, for historical reasons, the amplitude is defined as half of that arc, from the center point of the swing to either of its extremes, but this won't affect our discussion.) So we can say that the pendulum's energy is proportional to the square of its amplitude.
This turns out to be common to many different kinds of waves—including light waves. Light is a wave. (It's also a particle, in many ways, but we'll ignore that for now.) And being a wave, it has an amplitude, which is the extent to which the light oscillates. What is it that's oscillating, anyway? In the case of water waves, it's water, and in the case of sound, it's the molecules in the air. You can't have water waves without the water, and you can't have sound waves without the air; that's why sound doesn't travel in a vacuum. But light does travel in a vacuum, so what's waving the light, so to speak? Well, the answer is that the light itself is waving, or less opaquely (heh heh), the electromagnetic fields that permeate space are waving.
In any event, like other waves, light waves also carry energy that is proportional to the square of the light's amplitude. If you double the amplitude, you quadruple the energy; triple the amplitude, and the energy goes up nine-fold. And so on.
How would light's amplitude be doubled, though? You might imagine that if you put two flashlights, the amplitude of the two together would be twice that of each individual flashlight, and the combined light output—the energy of the two together—would be four times that of each flashlight. But I think, intuitively, we know this to be false, that the combination is only twice as bright as each flashlight. And if you measure the light carefully, in a dark room, this turns out to be perfectly true.What happened? Light waves, like other waves, have a secondary property, called phase. Two waves of the same frequency are said to be in phase if they swing in the same "direction" (in some not altogether well-defined sense); imagine two pendulums swinging in unison, so that when one swings left, the other does, too. They are out of phase if when one swings left, the other swings right, and vice versa. Or, they may be partly in phase, partly out of phase.
When you combine two light waves of the same frequency and the same amplitude, you get for all intents and purposes a single wave that is the two original waves added together. If they're in phase, the peaks get peakier and the valleys get, err, valleyier, and the amplitude of the waves is in fact doubled. On the other hand, if they're out of phase, the peaks of one get cancelled out by the valleys of the other (and vice versa), and the resultant wave has no amplitude at all.
More typically, though, the two waves are partly in phase and partly out of phase, and the resulting wave's amplitude is somewhere in between zero and two times the original. On average, one can show that the amplitude is the original times √2 . What's more, if you add three waves together at random phases, the amplitude of the sum is the original times √3 . And so on. Aha, the square root!
And since the energy of the final wave is the square of the amplitude, what comes out has two, three, or whatever times the original energy. Which is, of course, exactly what you'd expect. And good thing, too, because if it came out otherwise, we'd have a violation of the conservation of energy. Clearly, it takes n times as much energy to run n flashlights as it does to run one, and if their combined output were something other than n times the original, we'd have to seriously rethink our physics.
You might wonder if there isn't a way to get the waves to line up properly in phase so that the amplitudes do add up in the normal way, and you get a dramatic ramp up in energy. And there is; it's called a laser. A laser essentially gets n individual photons to line up in phase so that what comes out is a sort of super-photon (or super-wave, equivalently) with n² times the energy of any of the input photons. The physics-saving catch is that it takes more energy to line up, or lase, the light than you get as a result.
Nevertheless, that single photon or wave, coordinated as it is, can do things that you couldn't do with the individual photons separately. You can shine a bunch of flashlights at your eye and nothing will happen, other than a rather annoying afterimage and perhaps a headache. But even a modest laser can be used to reshape your cornea and render your eyeglasses superfluous. Of course, it should go without saying that it's not such a great idea to randomly shine lasers into your eye!
Or out, for that matter.
I see in this a kind of metaphor for human nature, and I hasten to say it's only that; as far as I know, one can't really take this and apply it rigorously in any scientific sense. But I think it's a useful metaphor all the same. I like to say that religion, among other things, is a laser of people. What on earth do I mean by that? A single human being can do a certain amount of work (in physics, work is defined as energy applied in furtherance of a force). What happens if you get two human beings together? Well, if they work against each other—if they're out of phase, in other words—less work gets done. Maybe none, if they spend all their time squabbling. Even if they're not exactly out of phase, if they're not particularly coordinated, their combined output is rather less than you might think, like the drunkard making slow and halting progress homeward because he can't put one foot directly in front of the other.
On the other hand, if they cooperate—if they're in phase—they can do twice the work. In fact, maybe they can get even more done, for there's no arguing that a coordinated combination of two people can do things that each individual person couldn't do, even adding their results together. Two people can erect a wall, for instance, that neither person could individually. Maybe, in some sense, those two people can do what it would take four people, working randomly, to achieve. And perhaps three coordinated people can do what it would take nine randomly working people to. And so on.
But it's pretty straightforward to get two or three people to work together, if they're of a mind to. But what about a hundred, or a thousand, or a million? That's where ideologies can be enormously effective; through them, a thousand can achieve what would otherwise require a million. And there may be no ideology better suited for the purpose than religion, although other ideologies—sociological, fiscal, even autocratical—may suffice. That's not to say that all that these various ideologies achieve is beneficial: for every great liberation, there may be a dozen pogroms. But they are part and parcel of a society's capacity for achievement; without them, we get only as far as a drunkard's walk will take us.
I see in this a kind of metaphor for human nature, and I hasten to say it's only that; as far as I know, one can't really take this and apply it rigorously in any scientific sense. But I think it's a useful metaphor all the same. I like to say that religion, among other things, is a laser of people. What on earth do I mean by that? A single human being can do a certain amount of work (in physics, work is defined as energy applied in furtherance of a force). What happens if you get two human beings together? Well, if they work against each other—if they're out of phase, in other words—less work gets done. Maybe none, if they spend all their time squabbling. Even if they're not exactly out of phase, if they're not particularly coordinated, their combined output is rather less than you might think, like the drunkard making slow and halting progress homeward because he can't put one foot directly in front of the other.
On the other hand, if they cooperate—if they're in phase—they can do twice the work. In fact, maybe they can get even more done, for there's no arguing that a coordinated combination of two people can do things that each individual person couldn't do, even adding their results together. Two people can erect a wall, for instance, that neither person could individually. Maybe, in some sense, those two people can do what it would take four people, working randomly, to achieve. And perhaps three coordinated people can do what it would take nine randomly working people to. And so on.
But it's pretty straightforward to get two or three people to work together, if they're of a mind to. But what about a hundred, or a thousand, or a million? That's where ideologies can be enormously effective; through them, a thousand can achieve what would otherwise require a million. And there may be no ideology better suited for the purpose than religion, although other ideologies—sociological, fiscal, even autocratical—may suffice. That's not to say that all that these various ideologies achieve is beneficial: for every great liberation, there may be a dozen pogroms. But they are part and parcel of a society's capacity for achievement; without them, we get only as far as a drunkard's walk will take us.
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