The Analysis of Matter

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.[Pg 38] 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. [Pg 39] 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[Pg 40] 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,[Pg 41] 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[Pg 42] 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.[Pg 43] 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.[Pg 44] 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): [Pg 45] where represents the matrix whose