VI.THE THEORY OF QUANTA

THE theory that the energy of a body cannot vary continuously, but only by a certain finite amount, or exact multiples of this amount, was not originally derived from a study of the atom or the spectroscope, but from the study of the radiation of heat. The theory was first suggested by Planck in 1900, thirteen years before Bohr applied it to the atom. Planck showed that it was necessary in order to account for the laws of temperature radiation; roughly speaking, if bodies could part with their warmth continuously, and not by jumps, they ought to grow colder than they do, when they are not exposed to a source of heat. It would take us too far from our subject to go into Planck’s reasoning, which is somewhat abstruse. A good account of it in English will be found in Jean’sReport on Radiation and the Quantum-Theory, published for the Physical Society of London (1914).

Planck’s principle in its original form is as follows. If a bodyis undergoing any kind of vibration or periodic motion of frequency (i.e. the body goes through its whole period times in a second), then there is a certain fundamental constantsuch that the energy of the body owing to this periodic motion is or some exact multiple of.That is to say,is the smallest amount of energy that can exist in any periodic process whose frequency is,and if the energy is greater thanit must be exactly twice as great, or three times as great, or four times as great, or etc. The energy was at first supposed to exist in atoms or little indivisible parcels, each of amount.There might be several parcels together, but there could never be a fraction of a parcel. We shall see that this principle has been modified as it has been applied in new fields, so that in its present form it can no longer be stated as involving indivisible parcels of energy. But it is as well to understand its original form before considering the more recent statements of the principle.

The quantity,which is called Planck’s “quantum,” is of course very, very small, so small that in all the large-scale processes observable by means of our senses there is an appearance of continuity. It is in fact so small that one unit of it is involved in one revolution of the electron in its minimum orbit round the hydrogennucleus. It is difficult to express very large numbers in words, particularly as the word “billion” is sometimes used to mean a thousand million, and sometimes to mean a million million. If we use it to mean a million million, we may say that a billion billion timeswould be a quantity just appreciable without instruments of precision. Taking the electron in its smallest orbit,is exactly obtained by multiplying the circumference of the orbit by the velocity of the electron and multiplying the result by the mass of the electron.[3]In the second orbit the result of this multiplication is,in the third,,and so on.

Planck’s principle in its original form applies only to certain kinds of systems, and if rashly generalized it gives wrong results. The right way to generalize it has been discovered by Sommerfeld, but unfortunately it is very difficult to express in non-mathematical language. It turns out that the principle, in its general form, cannot be stated as involving little parcels of energy; this only seemed possible because Planck was dealing with a special case. The general form requires a method of stating the principles of dynamics which is due to Hamilton. In this form, if the state of some material systemis determined at any moment when we know, at that moment, how large certain quantities are (as for example the position of an aeroplane is known if we know its latitude and longitude and its height above the ground), then these quantities are called “coordinates” of the system. Corresponding to each coordinate, the system has at each moment a certain characteristic which may be called the corresponding “impulse-coordinate.” In simple cases this reduces to what is ordinarily called momentum; in a generalized sense, it may itself be called the “momentum” corresponding to the coordinate in question. It is possible to choose our coordinates in such a way that the momentum corresponding to a given coordinate at a given moment shall not involve any other coordinate. When the coordinates have been chosen in this way, the generalized quantum principle is applicable. We shall assume that such a choice has been made. When such a choice has been made, the coordinates are said to be “separated.”

The quantum-principle is only applicable to motions that are periodic, or what is called “conditionally periodic.” The motion of a system is periodic if, after a certain lapse of time, its previous condition recurs, and if this goes on and on happening after equal intervalsof time. The motion of a pendulum is periodic in this sense, because, when it has had time to move from left to right and from right to left, it is in the same position as before, and it goes on indefinitely repeating the same motion. Wave-motions are periodic in the same sense; so are the motions of the planets. Any motion is periodic if it can be described by means of a quantity which increases up to a maximum, then diminishes to a minimum, then increases to a maximum again, and so on, always taking the same length of time from one maximum to the next. One “period” of a periodic process is the time taken to complete the cycle from one maximum to the next, or from one minimum to the next—for example, from midnight to midnight, from New Year to New Year, from the crest of one wave to the crest of the next, or from a moment when the pendulum is at the extreme left of its beat to the next moment when it is at the extreme left.

A system is called “conditionally periodic” when its motion is compounded of a number of motions, each of which separately is periodic, but which do not have the same period. For example, the earth has a motion compounded of rotation round its axis, which takes a day, and revolution round the sun, which takes a year. There are not anexact number of days in a year, if a year is taken in the astronomical and not in the legal sense; that is why we need a complicated system of leap-years to prevent errors from piling up. Thus when we take account of both rotation and revolution, the motion of the earth is “conditionally periodic.” We shall find later that the motions of electrons in their orbits, when we take account of niceties, are, strictly speaking, conditionally periodic and not simply periodic. The quantum-theory in its general form applies to motions that are conditionally periodic in terms of “separated” coordinates.

We can now state the generalized quantum-principle. Take some one coordinate of the system, and imagine the motion of the system throughout one period of this coordinate divided into a great number of little bits. In each little bit, take the generalized momentum corresponding to the coordinate in question, and multiply it by the amount of change in the coordinate during that little bit. Add up all these for all the little bits that make up one complete period. Then, in the limit, when the bits are made very small and very numerous, the result of the addition for one complete period will be exactlyororor some other exact multiple of.[4]No oneknows in the least why this should be the case; all we can say is that it is so, in all the cases that can be tested.

In later developments we shall have occasion to consider the principle in its general form. For the present, we are only concerned with its application to the electron revolving in a circle round the hydrogen nucleus. In this case, the generalized momentum is the same thing that is called “angular momentum” in elementary dynamics; in the case of circular motion, which is the case that concerns us, it is got by multiplying the mass by the radius and the velocity. As these are all constant, there is no difficulty about obtaining the sum of little bits for a complete cycle; each little bit consists of the angular momentum multiplied by a little angle, and the sum of all the little bits consists of the angular momentum multiplied by four right angles; that is to say, it is obtained by multiplying the mass of the electron by the circumference (instead of the radius) of its orbit and by the velocity. By the generalized quantum-principle, this has to beororor etc. In the minimum orbit it is;that is whyno smaller orbit is possible. In the next orbit, which is four times as large, it is;in the third orbit, which is nine times as large, it is;and so on. In virtue of the quantum-principle, these are the only orbits that are possible.

We can now understand how Bohr’s theory explains the lines of the hydrogen spectrum. When the electron jumps from a larger to a smaller orbit, it loses energy. A little very elementary mathematics[5]shows that the kinetic energy in the second orbit is a quarter of that in the first; in the third it is a ninth; in the fourth, a sixteenth; and so on. It is also very easy to show that (apart from a constant portion which may be ignored) the total energy in any orbit (potential and kinetic together) is numerically equal to the kinetic energy, but with the opposite sign. Therefore the loss of total energy in passing from a larger to a smaller orbit is equal to the gain of kinetic energy. It follows that, if we callthe kinetic energy in the smallest orbit, the loss of energy in passing from the second orbit to the smallest is,the loss in passing from the third orbit to the first is,the loss in passing from the third to the second is,i.e.;and so on. It will be noticed that the numbers that come here are the same as those thatoccurred in connection with Rydberg’s constant in the preceding chapter.

The energy which is lost by the atom in one of these jumps is turned into a light-wave. What sort of light-wave it is to become is determined by the theory of quanta. A light-wave is a periodic process, and if its frequency is,its period is aof a second. The generalized quantum-principle shows that, if the period of a wave is,the energy of the wave multiplied bymust beor an exact multiple of;in fact, so far as observation goes, it appears to be always.Sinceis aof a second (whenis the frequency), it follows that the energy of the wave is.Also, by the principle of the conservation of energy, the energy of the wave is equal to the energy that the atom has lost.

This shows that, if e is the kinetic energy of the electron in the smallest orbit, the wave caused by a transition from the second orbit to the first will have a frequencygiven by the equationFor a transition from the third orbit to the first,For a transition from the third orbit to the second,and so on. Comparing these results with the empirical results set forth inChapter IV, we see that they will agree if Rydberg’s constant is equal todivided byand the velocity of light. (We have to divide by the velocity of light, because in this chapter we have been speaking of frequencies, while inChapter IVwe were speaking of wave-numbers.) Nowis easily calculated since we know the charge on a hydrogen nucleus and on an electron, the mass of an electron, and the radius of the minimum orbit; alsoand the velocity of light are known. It is found that the calculated value of Rydberg’s constant, from these data, agrees closely with the observed value; this was, from the first, a powerful argument in favour of Bohr’s theory.

For different kinds of light, the frequencyis different; in the visible parts of the spectrum, it determines the colour, being smallest for red and greatest for violet. By measuring the frequencies of the different lines in the hydrogen spectrum, and multiplying each by,we find out how much energy the above loses in the differenttransitions from orbit to orbit that are possible. The terms in the spectrum are proportional to the energies in the different possible orbits, and the frequencies of the lines are proportional to the loss of energy in passing from one orbit to another. We can calculate what the different possible orbits should be from the fact that their energies must differ by an amount,whereis the frequency of some line in the hydrogen spectrum. We can also calculate the possible orbits from the fact that the mass of the electron multiplied by the circumference of an orbit multiplied by the velocity in that orbit must be an exact multiple of.These two methods lead to the same result, and thus confirm our theory.

There are, however, certainminutiæof the hydrogen spectrum which cannot be explained by Bohr’s theory in its original form. All these, down to the smallest particular, are explained by the generalized form of the theory which is due in the main to Sommerfeld. We shall explain this generalized theory in the next chapter.

The quantity,Planck’s quantum, has been found to be involved in all the very minute phenomena that can be adequately studied. It is one of the fundamental constants to which science is led: for the present, it represents a limit of explanations, since no one knows why thereis such a constant or why it is just the size it is. The limits of our explanations in any given stage of science are, while that stage lasts, brute facts; and so Planck’s quantum, for the present, is a brute fact. It is involved in all very small periodic processes; but why this should be the case we do not know.

[3]Expressed in the usual C.G.S. units,.Its dimensions are those of action or angular momentum.

[4]For the mathematical statement of the principle, see Sommerfeld’sAtomic Structure and Spectral Lines, 3rd ed., translated by Henry L. Brose M. A. (Dutton, New York) Chap. IV. and Appendix 7.

[5]See Appendix.

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