In 1744 de Maupertuis, president of the Prussian Academy in the reign of Frederick the Great, enunciated the principle of least action which subsumes the laws of classical mechanics in a single statement, just as Fermat's principle does for geometrical optics. The naïve belief of de Maupertuis that his principle was based on the attributes of the Supreme Intelligence was not inappropriate, for it has turned out to be deeper and wider in its scope than any other in the whole range of physical science. For a system with one degree of freedom it may be stated in the form where q is the positional coordinate and p the corresponding momentum. The symbol 6 represents a small variation for a given energy, E, just as in (49) it represents a small variation for a given period T. If we adopt the corpuscular theory of light, still held in the time of de Maupertuis. it is not difficult to show that (50) actually coincides with Fermat's principle. In order to appre ciate the wave mechanics of de Broglie and Schroedinger it is very important to realize that classical mechanics rests on the principle expressed by equation (5o) or a simple generalization of it, namely when we are concerned with systems with several degrees of free dom, in just the same way as geometrical optics rests on equation (49) or better (49a). It will be noticed that p and i/X are analo gous things in the two formulae. The principle of action can be ex pressed in other ways besides that of de Maupertuis, The most im portant of these alternative forms is that due to Sir W. Hamilton and known as Hamilton's principle. We shall only remark about Hamilton's principle that it brings out a parallelism between E, the energy of a particle and in the corresponding optical problem, of precisely the same kind as we have found between p the momentum of the particle and i/X.
De Broglie supposed every ultimate particle to be the manifesta tion of some kind of "phenomene periodique" and to be associated with a wave or a group of waves of a narrow range of frequencies. There is a well known distinction between the velocity with which a group of waves travels and the phase velocity or the velocity of the crests and troughs of the individual waves. According to de Broglie the velocity v of the particle should be identified with that of the group. This view is supported by the circumstance that the velocity of a group is given by The parallelism we have recognized between E and 1/7 and be tween p and 1/X suggests that we should investigate the quotient dE and we find, in fact, on substitutinglme for E and my for p, dp This is also the result we arrive at if we use the more elaborate relativistic expressions for E and p.
We now reach the crux of the wave mechanics. The way in which classical mechanics, and classical physical theory generally breaks down when we have to deal with sufficiently small scale phenomena, is just like the way in which geometrical optics fails when applied to paths which are not long compared with the wave length of the light.
By way of illustration we shall study the case of a rigid body spinning about a fixed axis and not in a field of force. It will help us if we think of it as a thin uniform ring of mass n: turning about an axis through the centre of the ring and perpendicular to its plane. Apart from a constant, the energy of the ring is identical with its kinetic energy t. We shall use the letter q to represent the distance a mark on the ring has travelled from some chosen zero position, and R to represent the radius of the ring which is, of course, constant. In this example the energy E and the momentum
p of the ring remain constant during its motion, and the simplest kind of de Broglie wave we can associate with it may be expressed in the form where A, X and r are constants. We shall postpone the enquiry about the significance of tk. This equation represents a wave of travelling in the direction of increasing q with a velocity X/r and it may be written in the form where, so far, h may be any quantity, but if we agree that it shall be identical with Planck's constant, and remember the parallelism between E and 1/7- and between p and r/X we are almost forced that the state of undulation is a superposition of vibrations with the frequencies: Let us now examine the simple Bohr atom in the light of wave mechanics. It is convenient to start out by representing the position of the electron by rectangular coordinates for which we shall use the letters x, y and z instead of and For sim plicity we shall ignore the motion of the nucleus, as we have done already, and suppose it to be situated at the origin. The wave equation is now The same process which led to equation (54) may be applied here, and so we get for the wave equation 2where we have replaced V, the potential energy, by Ze r r being the distance of the electron from the nucleus or origin. If E is negative, that is if the electron is travelling in an elliptic orbit, the requirements that must be a one-valued function of posi tion and finite restricts E to the values where n is an integer, and the state of undulation round about the nucleus consists of a superposition of vibrations the frequencies of which are expressed by We recognise here the spectroscopist's spectral terms. The func tion is usually written in a form equivalent to where i=V—i and all the other quantities are real. If we employ IT to represent a corresponding expression in which A/ — t is re placed by — it will be seen that the product • may be expressed in the followine form Therefore • IT represents a state of undulation, in the region surrounding the atom, which is a superposition of vibrations the frequencies of which are differences of spectral terms. This state of undulation must be identified with light waves or vibrations, and Schroedinger has made the suggestion that the value of 1k at any place is identical with the electric density at that place. This, of course, involves giving up the old picture of an electron revolving in an elliptic or other orbit about the nucleus in favour of one which depicts a complicated distribution of negative electricity, the density of which falls off asymptotically to zero at great distances from the nucleus. This new picture of an atom has the great advantage of close conformity to Clerk Maxwell's electromagnetic theory of light. It also provides us with a means of fixing what has hitherto been indeterminate about the func tion We must lay down that the quantity e which we have hitherto described as the charge on the electron, shall be identical with where dv is a small element of volume and the summation indi cated by the symbol I is extended over all space. It must be pointed out, however, that there is another interpretation of 2,/, which has much to be said for it, and which does not require us to give up the picture of an electron in orbital motion, namely, one which regards the value of • ITc/v at any given time as a measure of the probability that the electron is within the volume dv at that time.