To explain from where this progress arises, we need to introduce a notion that we will treat much better in chapter 5. We are familiar with the fact that spinning tops keep spinning and can be made to balance on a point, whereas non-spinning tops tend rather to fall over. Thus there is something about the spinning of the top that accounts for that special property. The simplest way to describe it is to grab a non-spinning, and then to grab a spinning top with the hand. The non-spinning top is just an object. It imparts nothing to the hand and takes nothing from it. But the spinning top gives a twisting push to the hand, and the faster the top is spinning or the heavier the top is, or the wider and more circular it is, the stronger that twisting impulse is.
What we are feeling is a property of the motion of the top called angular momentum. We will see later that a momentum is any of the properties of motion that describe the tendency of the motion to ``persist in the same way''. This momentum is called an angular momentum because it describes the tendency of a top to continue to move in a circle. For now it suffices for us, if necessary simply by experimenting with examples, to notice that this angular momentum is bigger when the mass, or the speed, or the circular size of the the spinning object increases, each factor independently of the others. Furthermore, if we measure examples carefully, we find that the relation is a linear proportionality. In other words, in the language of a scaling relation, the dimension that we call angular momentum, and that we recognize as the thing that is ``big'' for spinning tops, is related to our usual set of ``basic'' dimensions as
where L refers to the particular value we would measure for the amoung of angular momentum, m the amount of mass that is spinning in a circle, v is the speed at which it is spinning, and x is how far out it lies from the center about which it spins. Most things that we see in nature, from water in swirling ripples and whirling winds, to things that roll and things that spin, up to and including the earth and even the sun, and whole galaxies, we find to be spinning. So it should come as no particular surprise that very often atoms, and even the electric currents that can be separated from atoms and made to fly through space, are made of parts that spin. These little ``tops'' are not very heavy and they are not very ``large'', so it takes some care to measure the amount of angular momentum that describes their spinning, but it can be done. (which example here? some kind of torsion plate?). And in fact, since all of these objects were first discovered long before it was possible to ``see'' even the biggest of them directly (which has only become possible for atoms within the last five years), the entire notion that they were spinning at all was derived from the fact that they could be measured to have some of this angular momentum when they were ``grabbed'' by various devices. Since angular momentum has always been associated with spinning objects, it is natural to make the same association for these little objects as well. (It is actually good to be aware as we discuss these issues that, in fact, the whole statement that these objects are ``spinning'' should not be taken to describe more than it does. We don't know, especially for the electrons, even whether they have any nonzero physical size, so any notions of ``spinning'' that involve pictures of big tops that go round and round is probably misleading, and may even be wrong. We use the word ``spin'' simply as a mnemonic to remind us that they have angular momentum, though in fact at this level of the discussion we know no more about them than that. And in fact, we will find that trying to read too much into our mnemonics in this case results in very bad descriptions that are not useful and very misleading.)
Once it was determined that these little objects have spin at all, a remarkable thing was observed (by whom, when? Thompson?). Still consistent with the picture not only of atoms, but also of electrons as building blocks, it was found that they could always be ``grabbed'' one at a time, so it was possible to associate a definite amount of angular momentum with each such object. This is equivalent to saying that whenever one grabs a spinning top, there is a meaningful way to say that the top had some definite amount of angular momentum as a result of the way in which it was spinning. The remarkable part of this observation was that every time such an angular momentum was measured, it was found always to be some multiple of a particular given amount. If this experiment had been done with only one kind of object, such as only atoms or only electrons, and it had been found that the amount of the angular momentum was always the same, that discovery could be taken as just another version of the ``identicalness'' of the particles themselves, and thus thrown into the bag of our ignorance about what makes them identical in the first place. But the discovery was more amazing than that. The experiments were done with all kinds of different particles, with electrons flying through space as cathode rays, and with whole atoms of different types (when was it done with light?) The special number was the same in every case, though the actual result measured related to it differently for one kind of particle than for another, and sometimes even from one case to the next of a given particle. It was as if every little particle could be a spinning top, but that it was only allowed to spin in increments of a certain amount.
For instance, when electrons were used, the angular momentum was always found to be 1/2 of this special number, either spinning in a ``right-handed'' or a ``left-handed'' sense, much as a top can spin either clockwise (left-handed) or counterclockwise (right-handed) when viewed from above. Electrons were never found not to be ``spinning'', much unlike tops. For one kind of helium atoms (the kind that weighs four times as much as hydrogen), the angular momentum could be found to be either none (as in, not spinning), or one of the special units, either right-handed or left-handed, or two, etc, depending on how the atoms were prepared and sent into the measuring device. For another kind of Helium, that weighs three times as much as a hydrogen but has the same chemical properties otherwise as the other Helium, the angular momentum was found to be either right- or left- 1/2, like the electron, or in other cases right- or left- 3/2, and so on with half-integers, again depending on how the atoms were prepared. Hydrogen, like the heavier of the two Heliums, can have no angular momentum, or one unit, and so forth.
That such a result should be true seems exceedingly strange, if one views atoms as being made of the building blocks of electrons. It could perhaps be explained away in the simplest experiments with Helium, since that atom has two electrons (we can tell by its chemical properties). So, if the angular momentum is really just a property of the electrons, perhaps it can add or cancel (1/2-1/2 giving 0. 1/2+1/2 giving 1, if we can figure out what kind of rules define the sense of + and - for these angular momenta). But the fact that two kinds of helium atoms that have the same chemistry, and seemingly therefore the same number of electrons, can have different amounts of angular momentum, does not obviously fit into that explanation. Worse yet is the case of Hydrogen (which has only one electron) or the case that can be obtained if one is better at preparing ``more spun'' states of helium, because one can find amounts of angular momentum in the atoms that are greater than all of the 1/2s from the electrons would seem to make possible, but always respecting the 0,1,2,... or 1/2,3/2,... sequence that depends on the type of atom.
The only conclusion that has turned out to make sense is that the angular momentum found ``in'' atoms comes both from the angular momentum intrinsic to the electrons that make them, and from something about the structure of the atoms themselves. This conclusion sounds like just the sort of thing we have been looking for to explain the fact that even if the electrons are ``flying through empty space'' around the nuclei, they only do so in certain very definite and restricted structures, namely those that have ``the right amount'' of angular momentum for that kind of atom. But it is also very confusing, because then whatever characterizes the ``motion'' of electrons through the empty space in atoms and accounts for these amounts of angular momentum also determines something about the intrinsic angular momentum of the electrons themselves, quite independently of whether they are residing in atoms or streaming through empty space as cathode rays.
A natural question to ask at this point is how big is that special unit of angular momentum, and does it happen to be anything we recognize or can identify as familiar? The answer comes from seemingly a totally different direction. Before the start of the 20th century, it had been found by Max Planck that the amount of light of various colors that populated a ``red-hot'' closed box bore a strong resemblence to the behavior of any sort of gas of ``solid matter'' that one might have chosen instead to put into the box (with some important differences). This topic also will be treated in more detail when we discuss the quantum theory. The point we need now is Planck's discovery that, in order to make sense of the fact that different colors of light can share a behavior with heavy things like atoms or molecules of a gas, there must be some way to associate the color of the light with something that behaves like the energy of the heavy atoms. The importance of energy is treated in the chapter on Symmetries, but we have already seen in the last chapter (have we? mention this.) that the dimensions of energy are
Here we may notice a remarkable thing. Before Planck, light had already been recognized as a sort of oscillation of electric and magnetic effects that travels as a wave through empty space, about which much more in a moment. Thus it is natural to discuss it in much of the same language that we use for other waves, such as even the simple waves that travel on a piano string. In the case of a piano string, the material of the string itself can be directly observed, and it has been understood for a long time that the pitch of the tone that is generated by the string is determined entirely by the frequency at which the string vibrates. Thus to every tone there is associated a characteristic time of oscillation, just as for our spring and mass of the last chapter (and in fact, a tuning fork is nothing but such a spring and mass for which the oscillation frequency is fast enough that the resulting disturbances in the air can be heard by human ears). Similarly for light, the color that we perceive, like a kind of ``tone for the eyes'' is the property that corresponds to the frequency of the vibration. More-red colors correspond to lower frequencies and more-blue or -violet colors to higher ones.
Planck found that to account for the behavior of light he had to associate an energy with each color that was proportional to the frequency, as
Since frequency has the inverse dimensions of time, this equation can be used to find the needed dimensions of the proportionality constant, which we will call ``Plancks constant'' after its discoverer. Those dimensions are
So Planck's constant, invented to account for an entirely different set of phenomena, happens to have the same dimensions, energy x time, as those of angular momentum, which is mass x velocity x distance. This might be mere coincidence, except for the staggering surprise that exactly the value of Planck's constant, multiplied by appropriate factors of 0,1,1/2,2,3/2,... gives the angular momenta that were observed for the other particles. And, mentioning for completeness our physical system of starlight that can be emitted and absorbed by atoms and molecules of gas, if the atoms can only be found with certain amounts of angular momentum, and they can emit or absorb light (without gaining or losing electrons, as it happens), one would not be surprised to find that some similar characteristic amounts of angular momentum may be imparted to the light in somewhat the same way that a spinning top ``gives'' angular momentum to the hand that grabs it. Indeed this turns out to be true. So if the same angular momenta that are characteristic of atoms and their building blocks are also characteristic of light, it may not be entirely surprising that the constant that describes them appears in other properties of light as well. That was not, however, the order in which they were discovered.
Note, though, that we have now found the same special number, this ``Planck's constant'', in a large number of very different physical systems, so that while it seems ubiquitous, it also does not seem clearly associated as a ``part'' of any of the single objects themselves, since electrons are not ``part'' of light, nor are atoms, while at the same time experiments can be done with the angular momentum of electrons that have nothing to do with light (though the machinery to do many such experiments has only been built within about the last three decades). And, since we now know that there a forms of matter that consist of torn apart atoms, or do not even include electrons as building blocks, and that they as well as many other mechanisms are capable of creating and absorbing light, we cannot even suppose that light always has this behavior merely because it was created by atoms or by interaction with electrons.
We have discovered our first so-called fundamental dimensionful constant, Planck's constant. We will refer usually to the expression of this number that will be useful to describe things that go around in circles (because of the mathematical conventions used for describing circles). It is usually denoted ``h/ '' , and has the numerical value
In terms of people-sized objects, like spinning car tires, this is an exceedingly small number, which is in part why it took mankind some time to uncover it. From what has been said so far, we understand basically nothing about this number, except that it keeps appearing whenever there is a repeatability of something associated with little ``packets'', like the packets that one uses in describing light if one wants to apply the same math to it (to account for its properties in hot boxes) as one applies to boxes of gas, and in particular to the angular momenta that are measured for single particles (like electrons seem to be), and even for small ``bundles'' of particles like atoms. Much of the third part of this book can be viewed as an explanation of the importance, or ``meaning'' of this fundamental constant. For now we leave our understanding in much the same state it had at the beginning of this century. The number has been discovered, it seems clearly associated with repeatability of structures, its dimensions are known, and it seems to be something more universal than any of the particular cases in which it appears. We leave that understanding here and go on to examine other such numbers that appear in similar mysterious ways and are associated with repeatability.