The analysis of matter

THE atomicity of matter is a hypothesis as old as the Greeks, and in no way repugnant to our mental habits. The theory that matter is composed of electrons and protons is beautiful through its successful simplicity, but is not difficult to imagine or believe. It is otherwise with the form of atomicity introduced by the theory of quanta. This might possibly not have surprised Pythagoras, but it would most certainly have astonished every later man of science, as it has astonished those of our own day. It is necessary to understand the general principles of the theory before attempting a modern philosophy of matter; but unfortunately there are still unsolved physical problems connected with it, which make it improbable that a satisfactory philosophy of the subject can yet be constructed. Nevertheless, we must do what we can.

As everyone knows, the quantum was first introduced by Planck in 1900 in his study of black-body radiation. Planck showed that, when we consider the vibrations which constitute the heat in a body, these are not distributed among all possible values according to the usual law of frequency which governs chance distributions, but on the contrary are tied down by a certain law. If is the energy of a vibration, and its frequency, then there is a certain constant ,[7] known as Planck's constant, such that is or , or , or some other small integral multiple of h. Vibrations with other amounts of energy do not occur. No reason is known for their non-occurrence, which remains so far of the nature of a brute fact. At first, it was an isolated fact. But now Planck's constant has been found to be involved in various other kinds of phenomena; in fact, wherever observation is sufficiently minute to make it possible to discover whether it is involved or not.

A second field for the quantum theory was found in the photo-electric effect. This effect is described as follows by Jeans:[8]

The explanation of this phenomenon in terms of the quantum was first given by Einstein[9] in 1905. When light of frequency falls on the conductor, it is found that the amount of energy absorbed by an electron which the light separates from its atom is about five-sixths of , where is Planck's constant. It may be supposed that the other one-sixth is absorbed by the atom, so that atom and electron together absorb exactly one quantum . When the light is of such low frequency that is not enough to liberate an electron, the photo-electric effect does not take place. Explanations not involving the quantum have been attempted, but none seem able to account for the data.

Another field in which the quantum hypothesis has been found necessary is the specific heat of solids at low temperatures. According to previous theories, the specific heat (at constant volume) multiplied by the atomic weight ought to have the constant value 5·95. In fact, this is found to be very approximately correct for high temperatures, but for low temperatures there is a falling off which increases as the temperature falls. The explanation of this fact offered by Debye is closely analogous to Planck's explanation of the facts of black-body radiation; and as in that case, it seems definitely impossible to obtain a satisfactory theory without invoking the quantum.[10]

The most interesting application of quantum theory is Bohr's explanation of the line spectra of elements. It had been found empirically that the lines in the hydrogen spectrum which were known had frequencies obtained from the difference of two "terms," according to the formula: where is the frequency, is "Rydberg's constant," and are small integers, and are what are called "terms." After the formula had been discovered, new lines agreeing with it were sought and found. Certain lines formerly attributed to hydrogen, and not agreeing with the above formula, were attributed by Bohr to ionized helium; they are given by the formula: Bohr's theoretical grounds for attributing these lines to helium were afterwards confirmed experimentally by Fowler. It will be seen that they fit into the formula (1) when is substituted for , a fact which Bohr's theory explains, as well as the more delicate fact that, to make the formula exact, we have to substitute, not exactly , but a slightly smaller quantity.

The form of the equation (1) suggested to Bohr that a line of the hydrogen spectrum is not to be regarded as something which the atom emits when it is in a state of periodic vibration, but as produced by a change from a state connected with one integer to a state connected with another. This would be explained if the orbit of the electron were not just any orbit possible on Newtonian principles, but only an orbit connected with an integral "quantum number"—i.e. with a multiple of .

The way in which Bohr achieved a theory on these lines is as follows. He supposed that the electron can only revolve round the nucleus in certain circles, these being such that, if is the moment of momentum in any orbit, we shall have: where is, as always, Planck's constant, and is a small whole number. (In theory might be any whole number, but in practice it is never found to be much larger than 30, and that only in certain very tenuous nebulæ.) The reason why the quantum principle assumes just this form will be explained presently.

Now if is the mass of the electron, the radius of its orbit, and its angular velocity, we have: But, on grounds of the usual theory, since the radial acceleration of the electron is and the force attracting it to the nucleus is we have:

From equations (3) and (4) we obtain:

The possible orbits for the electron are obtained by putting = 1, 2, 3, 4, ... in the above formulæ for . Thus the smallest possible orbit is: and the other possible orbits are , , , etc.

For the energy in an orbit of radius we have, since the potential energy is double the kinetic energy with its sign changed:[11] in virtue of (5). Thus when the electron falls from an orbit whose radius is to one whose radius is , there is a loss of energy:

It is assumed that this energy is radiated out in a light-wave whose energy is one quantum of energy , where is its frequency. Hence we obtain the frequency of the emitted light by the equation: This agrees exactly with the observed lines if [see equation (1)]: where is Rydberg's constant. On inserting numerical values, it is found that this equation is verified. This striking success was, from the first, a powerful argument in favour of Bohr's theory.

Bohr's theory has been generalized by Wilson[12] and Sommerfeld so as to allow also elliptic orbits: these have two quantum numbers, one corresponding, as before, to angular momentum or the moment of momentum (which is constant, by Kepler's second law), the other depending upon the eccentricity. Only certain eccentricities are possible; in fact, the ratio of the minor to the major axis is always rational, and has as its denominator the quantum number corresponding to the moment of momentum. In order to explain the Zeeman effect (which arises in a magnetic field) we used a third quantum number, corresponding to the angle between the plane of the magnetic field and the plane of the electron's orbit. In all cases, however, there is a general principle, which must now be explained. This will show, also, why, in Bohr's theory, the quantum equation (2) takes the form it does.[13]

The first thing to observe is that the quantum principle is really concerned with atoms of action, not of energy: action is energy multiplied by time. Suppose now that we have a system depending upon several co-ordinates, and periodic in respect of each. It is not necessary to suppose that each co-ordinate has the same period: it is only necessary to suppose that the system is "conditionally periodic"—i.e. that each co-ordinate separately is periodic. We must further assume that our co-ordinates are so chosen as to allow "separation of variables" (as to which, see Sommerfeld, op. cit., pp. 559-60). We then define the "momentum" (in a generalized sense) associated with the co-ordinate as the partial differential of the kinetic energy with respect to —i.e. calling the generalized momentum , we put: where is the kinetic energy. The quantum condition is to apply to the integral of over a complete period of —i.e. we are to have: where the integration is taken through one complete period of . Here will be the quantum number associated with the co-ordinate . The above is a general formula of which all known cases of quantum phenomena are special cases. This is its sole justification.

The above principle is exceedingly complicated—more so, even, than it appears in our summary account, which has omitted various difficulties. It is possible that its complication may be due to the fact that quantum dynamics has had to force its way through the obstacles which the classical system put in its way; it is possible also that quantum phenomena may turn out to be deducible from classical principles. But before pursuing this line of thought, it may be well to say a few words about the developments of Bohr's theory by Sommerfeld and others.

In its original form, in which circular orbits were assumed, Bohr's theory accounted for the main facts concerning the line spectra of hydrogen and ionized helium. But there were a number of more delicate facts which required the hypothesis of elliptic orbits: with this hypothesis, together with some niceties derived from relativity, the most minute agreement has been obtained between theory and observation. But perhaps this great success has made people think that more was proved than really was proved. The great advantage obtained from admitting elliptic orbits is that they provide a second quantum number. In the emission of light by atoms, what we have is essentially as follows. The atom is capable of various states, characterized by whole numbers (the quantum numbers). There may be more or fewer quantum numbers, according to the degrees of freedom of the system. The loss or gain of energy when an atom passes from a state characterized by one set of values of the quantum numbers to a state characterized by another set is known. When energy is lost (without the loss of an electron or of any part of the nucleus of the atom), it passes out as a light-wave, whose energy is equal to what the atom has lost, and whose energy multiplied by the time of one vibration is . Energy is what is conserved, but action is what is quantized.

Let us revert, in illustration, to the circular orbits of Bohr's original theory, which remain possible, though not universal, in the newer theory. If we call the kinetic energy when the electron is in the smallest possible orbit, the kinetic energy in the orbit is . (The measure of the total energy is the kinetic energy with its sign changed.) We do not know what determines the electron to jump from one orbit to another; on this point, our knowledge is merely statistical. We know, of course, that when the atom is not in a position to absorb energy the electron can only jump from a larger to a smaller orbit, while the converse jump occurs when the atom absorbs energy from incident light. We know also, from the comparative intensities of different lines in the spectrum, the comparative frequencies of different possible jumps, and on this subject a theory exists. But we do not know in the least why, of a number of atoms whose electrons are not in minimum orbits, some jump at one time and some at another, just as we do not know why some atoms of radio-active substances break down while others do not. Nature seems to be full of revolutionary occurrences as to which we can say that, if they take place, they will be of one of several possible kinds, but we cannot say that they will take place at all, or, if they will, at what time. So far as quantum theory can say at present, atoms might as well be possessed of free will, limited, however, to one of several possible choices.[14]

However this may be, it is clear that what we know is the changes of energy when an atom emits light, and we know that in the case of hydrogen or ionized helium these changes are measured by . It seems almost unavoidable to infer that the previous state of the atom was characterized by the integer and the later one by the integer . But to assume orbits and so on, though proper as a help to the imagination, is hardly sufficiently justified by the analogy of large-scale processes, since the quantum principle itself shows the danger of relying upon this analogy. In large-scale occurrences there is nothing to suggest the quantum, and perhaps other familiar features of such occurrences may result merely from statistical averaging.

It may be worth while to consider briefly the elliptical orbits which are possible.[15] This will also illustrate the application of the quantum principle to systems with more than one co-ordinate.

Taking polar co-ordinates, the kinetic energy is: The two generalized momenta are therefore: We have thus two quantum conditions: By Kepler's second law, is constant; call it . Thus: The other integration is more troublesome, but we arrive at the result that, if and are the major and minor axes of the ellipse,

A little further calculation leads to the result that the energy in the orbit which has the quantum numbers , is: This is exactly the same as in the case of circular orbits, except that replaces . If this were all, the line spectrum of hydrogen would be exactly the same whether elliptic orbits occurred or not, and there would be no empirical means of deciding the question.

However, by introducing considerations derived from the special theory of relativity we are able to distinguish between the results to be expected from circular and elliptic orbits respectively, and to show that the latter must occur to account for observed facts. The crucial point is the variation of mass with velocity: the faster a body is moving, the greater is its mass. Therefore in an elliptic orbit the electron will have a greater mass at the perihelion than at the aphelion. From this it is found to follow that an elliptic orbit will not be accurately elliptic, but that the perihelion will advance slightly with each revolution.[16] That is to say, taking polar co-ordinates , , the co-ordinate increases by slightly more than between one minimum of and the next. The system is thus "conditionally periodic"—i.e. each separate co-ordinate changes periodically, but the periods of the two do not coincide. The result[17] is that the equation is replaced by: being the velocity of light, and , as before, the angular momentum. It will be seen that is very nearly 1, because is large.

The formula for the energy associated with the quantum numbers , now becomes much more complicated; its great merit is that it accounts for the fine structure of the hydrogen line spectrum. It must be felt that this minuteness of agreement between theory and observation is very remarkable. But it is still the case that the only empirical evidence concerns differences of energy in connection with different quantum numbers, and that the theory of actual orbits, proceeding, during steady motion, according to Newtonian principles, must inevitably remain a hypothesis—a hypothesis which, as we shall see, has disappeared from the latest form of the quantum theory.

The fact of the existence of the quantum is as strange as it is undeniable, unless it should turn out to be deducible from classical principles. It seems to be the case that quantum principles regulate all interchange of energy between matter and the surrounding medium. There are grave difficulties in reconciling the quantum theory with the undulatory theory of light, but we shall not consider these until a later stage. What is much to be wished is some way of formulating the quantum principle which shall be less strange and ad hoc than that due to Wilson and Sommerfeld. For practical purposes, it amounts to something like this: that a periodic process of frequency has an amount of energy which is a multiple of , and, conversely, if a given amount of energy is expended in starting a periodic process, it will start a process with a frequency such that the given amount of energy shall be a multiple of . When a process has a frequency and an energy , the amount of "action" during one period is . But we cannot say: In any periodic process the amount of action in one period is or a multiple of . Nevertheless, some formulation analogous to this might in time turn out to be possible. As has appeared from the theory of relativity, "action" is more fundamental than energy in physical theory; it is therefore perhaps not surprising that action should be found to play an important part. But the whole theory of the interaction of matter and the surrounding medium, at present, rests upon the conservation of energy. Perhaps a theory giving more prominence to action may be possible, and may facilitate a simpler statement of the quantum principle.

In Bohr's theory and its developments, there is a lacuna and there is a difficulty. The lacuna has already been mentioned: we do not know in the least why an electron chooses one moment rather than another to jump from a larger to a smaller orbit. The difficulty is that the jump is usually regarded as sudden and discontinuous: it is suggested that if it were continuous, the experimental facts in the regions concerned would become inexplicable. Possibly this difficulty may be overcome, and it may be found that the transition from one orbit to another can be continuous. But it is as well to consider the other possibility, that the transition is really discontinuous. I have emphasized how little we really know about what goes on in the atom, because I wished to keep open the possibility of something quite different from what is usually supposed. Have we any good reason for thinking that space-time is continuous? Do we know that, between one orbit and the next, other orbits are geometrically possible? Einstein has led us to think that the neighbourhood of matter makes space non-Euclidean; might it not also make it discontinuous? It is certainly rash to assume that the minute structure of the world resembles that which is found to suit large-scale phenomena, which may be only statistical averages. These considerations may serve as an introduction to the most modern theory of quantum mechanics, to which we must now turn our attention.[18]

In the new theory inaugurated by Heisenberg, we no longer have the simplicity of the Rutherford-Bohr atom, in which electrons revolve about a nucleus like separate planets. Heisenberg points out that in this theory there are many quantities which are not even theoretically observable—namely, those representing processes supposed to be occurring while the atom is in a steady state. In the new theory, as Dirac says: "The variable quantities associated with a stationary state on Bohr's theory, the amplitudes and frequencies of orbital motion, have no physical meaning and are of no physical importance" (4, p. 652). Heisenberg, in first introducing his theory, pointed out that the ordinary quantum theory uses unobservable quantities, such as the position and time of revolution of an electron (1, p. 879), and that the electron ought to be represented by measurable quantities such as the frequencies of its radiation (1, p. 880). Now the observable frequencies are always differences between two "terms," each of which is represented by an integer. We thus arrive at a representation of the state of an atom by means of an infinite array of numbers—i.e. by a matrix. If and are two "terms," an observable frequency (in theory) is , where: It is such numbers as (of which there is a doubly infinite series) that characterize the atom, so far as it is observable.

Heisenberg sets out this view as follows (5, p. 685). In the classical theory, given an electron with one degree of freedom, in harmonic oscillation, the elongation at time can be represented by a Fourier series: where is a constant and is the number of the harmonic. The single terms of this series, namely: would contain the quantities which have been signalized as directly observable—namely, frequency, amplitude, and phase. But in virtue of the fact that, in atoms, frequencies are found to be the differences of "terms" we shall have to replace the above by: and the collection (not the sum) of such terms represents what was formerly the elongation . The sum of all these terms has no longer any physical significance. Thus the atom comes to be represented by the numbers , arranged in an infinite rectangle or "matrix."

It is possible to construct an algebra of matrices, which differs formally from ordinary algebra in only one respect, namely, that multiplication is not commutative.

A new operation is defined which, when the quantum numbers become large, approximates to differentiation. By using this operation, Hamilton's equations of motion can be preserved in a form which is applicable equally to periodic and to unperiodic motions, so that it is no longer necessary to distinguish a certain sphere of quantum phenomena, to which different laws are applied from those applied to the phenomena amenable to classical dynamics: "A distinction between 'quantized' and 'unquantized' motions loses all meaning in this theory, since in it there is no question of a quantum condition which selects certain motions from a great number of possible ones; in place of this condition appears a quantum-mechanical fundamental equation ... which is valid for all possible motions, and is necessary in order to give a definite meaning to the problem of motion" (3, p. 558). The fundamental equation alluded to in the above is as follows: Let be a Hamiltonian co-ordinate, and the corresponding (generalized) momentum, both being matrices. It will be remembered that multiplication is not commutative for matrices; in fact, we have as the fundamental equation in question (2, p. 871): where represents the matrix whose diagonal consists of 's, and whose other terms are all zero. The above is the sole fundamental equation containing (Planck's constant), and it is true for all motions.

Heisenberg does not claim that the new theory solves all difficulties. On the contrary, he says (5, p. 705):

In a more or less popular exposition (6), Heisenberg has set forth some of the consequences of his theory. Electrons and atoms, he says, do not have "the degree of immediate reality of objects of sense," but only the sort of reality which one naturally ascribes to light quanta. The troubles of the quantum theory have come, he thinks, from trying to make models of atoms and picture them as in ordinary space. If we are to retain the corpuscular theory, we can only do it by not assigning a definite point of space at each time to the electron or atom. We substitute a well-defined physical group of quantities which represent what was the place of the electron. They are the observable radiation quantities, each of which is associated with two "terms," so that we obtain a matrix. The distinction of inner and outer electrons in an atom becomes meaningless. "It is, moreover, in principle impossible to identify again a particular corpuscle among a series of similar corpuscles" (p. 993).

The matrix theory of the electron is too new to be amenable, as yet, to the kind of logical analysis which it is our purpose to undertake in this Part. It is clear, however, that it affects a scientific economy by substituting for the merely hypothetical steady motions of Bohr's atoms a set of quantities representing what we really know—namely, the radiations that come out of the region in which the atom is supposed to be. It is clear, also, that there is an immense logical progress in the construction of a dynamic which destroys the distinction between quantized and unquantized motions, and treats all motions by means of a uniform set of principles. And the greater abstractness of the Heisenberg atom as compared with the Bohr atom makes it logically preferable, since the pictorial elements in a physical theory are those upon which least reliance can be placed.

An apparently different quantum theory, due to de Broglie[20] and Schrödinger,[21] has been found to be formally the same as Heisinger's theory, although at first sight very different. This is described by de Broglie as "the new wave theory of matter," in which "the material point is conceived as a singularity in a wave."[22] Here, also, the radiations which we think of as coming out of the atom have more physical "reality" than the atom itself. One of the merits of the theory is that it diminishes the difficulties hitherto existing in the way of a reconciliation of the facts of interference and dispersion with the facts which led to the hypothesis of light quanta.

Meanwhile, there remains the possibility that all the quantum phenomena may be deducible from classical principles, and that the apparent discontinuities may be only a question of sharp maxima or minima. The most successful theory known to me on these lines is that of L. V. King.[23] He assumes that electrons rotate with a certain fixed angular velocity, the same for all; he makes a similar assumption as regards protons. Consequently there is a magnetic field which introduces conditions that are absent if electrons and protons have no spin. There will be electromagnetic radiation of frequency , where: being Planck's constant, the invariant mass of the electron, and its velocity. (The identity of with Planck's constant is obtained by adjusting the hypothetical constants.) From this formula he deduces many of the phenomena upon which the quantum theory is based, and promises to deduce others in a later paper. An article by Mr R. H. Fowler ("Spinning Electrons," Nature, Jan. 15, 1927) discusses Mr King's theory without arriving at a verdict for or against. Presumably it will not be long before a definite answer as to the adequacy of Mr King's theory is possible. If it is adequate, the quantum theory ceases to concern the philosopher, since what remains valid in it becomes a deduction from more fundamental laws and processes which are continuous and involve no atomicity of action. For the moment, until the physicists have arrived at a decision, the philosopher must be content to investigate both hypotheses impartially.