The Analysis of Matter

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 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): "The theory here described must be regarded as still incomplete. The real geometrical or kinematical meaning of the fundamental assumption (5)[19] has not yet been made completely clear. In particular, there is a serious difficulty in the fact that the time apparently has a different rôle from the space co-ordinates, and is formally differently treated. The formal character of the time co-ordinate in the mathematical structure of the theory is made particularly evident by the fact that in the theory hitherto the question of the temporal course of a process has no immediate meaning, and that the concept of earlier and later can hardly be defined exactly. Nevertheless, we need not consider these difficulties as an objection to the theory, since the appearance of just such difficulties was to be expected from the nature of the space-time relations that hold for atomic systems." 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.[Pg 46] 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[Pg 47] 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