The Analysis of Matter

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," [Pg 33] 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[Pg 34] 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: [Pg 35] 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. [Pg 36] 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[Pg 37] 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.[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