December 23, 1922] 



NA TURE 



841 



Letters to the Editor. 



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A Quantum Theory of Optical Dispersion. 



When a theory is framed trying to explain a 

 discrepant system of facts, it is a necessary process 

 of thought to take some branch of the theory as more 

 completely true than the rest, and to adjust the 

 remaining parts in such a way that they will fit in 

 with this base, though they may still conflict with 

 one another. This has certainly been true of the 

 quantum theory ; the speculations connected with 

 it have as their base the law of the conservation of 

 energy. Now a critical examination of fundamentals 

 does not by any means justify this faith. It is, of 

 course, a fact of observation that, in the gross, energy 

 is conserved, but this only means an averaged energy ; 

 and as pure dynamics has failed to explain many 

 atomic phenomena, there seems no reason to maintain 

 the exact conservation of energy, which is only one 

 of the consequences of the dvnamical equations. 

 Indeed it is scarcely too much to say that had the 

 photoelectric effect been discovered a century ago it is 

 probable that no one would ever have suggested that 

 the status of the first law of thermodynamics was 

 in any way different from that of the second. On the 

 other hand, Bohr's theory, and especially Sommerfeld's 

 extension of it, have given great encouragement to 

 the belief that in dynamics lay the way to the com- 

 plete truth, so that in consequence of the triumphs 

 of that theory there has been little thought in other 

 directions. Another impediment is that our whole 

 ideas are saturated with the principle of energy, 

 so that denying it leaves scarcely any foundation from 

 which to start. 



Now there is another field of phenomena which 

 forms a consistent whole, but at present only fits 

 into the quantum theory with a good deal of difficulty 

 — that is, the wave theory of light. Interference 

 and diffraction are completely explained by a wave 

 theory, and it would seem almost impossible to devise 

 any really different alternative which would account 

 for them. Here is a base which seems to be free 

 from the objections which attach to energy, and I 

 have therefore been examining the consequences of 

 fitting it in with those parts of the Bohr theory 

 which seem to be most completely established. The 

 result is what I believe to be a satisfactory theory 

 of dispersion — one of the weakest points in the 

 quantum theory ' — and a great promise of future 

 extensions in other directions. 



We shall assume then that the wave theory gives 

 a correct account of events outside matter, and it is 

 convenient to take over the terminology of the 

 electromagnetic theory, provided we remember that 

 " electric force " is only to mean " light vector," 

 and that we are not prescribing how the electric 

 force will affect the behaviour of atoms or electrons. 

 The assumption brings with it of course the exact 

 conservation of energy in the arther ; it is in inter- 

 changes with matter that it need not be conserved. 

 When a wave passes over matter there is a mutual 

 influence, and without any inquiry into what happens 

 to the matter, we can say that it is inconceivable 

 that the effect on the a?ther should be anything 



1 The difficulty is that the standard theory indicates a dispersion formula 

 involving the frequency of the electron's motion in the atom which is quite 

 different from its absorption frequency. 



but in the form of an expanding spherical wave. 

 Every such wave can be described in terms of 

 spherical harmonics, and the simplest is the one 

 corresponding to the harmonic of zero order. In 

 this the electric force vanishes at two poles and is 

 elsewhere along the lines of longitude and proportional 

 to the cosine of the latitude, while the magnetic 

 force lies in the circles of latitude. This is the type 

 of wave given in the classical theory by a Hertzian 

 doublet vibrating in a line, and it proves unnecessary 

 for our theory to postulate that any more complicated 

 type is emitted by the atom. If x is the direction 

 of the pole of the wave, then at x, v, z, at a great 

 distance r from the atom, the wave is given by : 



E,= - 



E„ = 



E 2 = 



%A*-rlc) 



(1) 



Next, borrowing from the Bohr theory, we shall 

 assume that when an atom is struck by a wave, 

 there is a certain chance that the atom should emit 

 a secondary wave of the above type. With these 

 assumptions it is possible to argue inductively from 

 the observed fact that if incident waves are superposed 

 the result can be found by an addition of their 

 effects and from the known form of the dispersion 

 formula. There is no need to give the argument, 

 but only its final result. The complete statement 

 of this for unpolarised waves is rather more com- 

 plicated, but the essential points of the theory are 

 fully represented in what follows. 



When a wave, polarised so that the electric force 

 is along x, strikes an atom at the origin there is a 

 chance A,(rE,/' t)dt that in the time dt it will excite 

 the atom to emit a spherical wave of the type (i) 

 with/of the form a„e~ x "' cos k„t. Here A„, a„, \„ and k„ 

 depend only on the nature of the atom and not at all 

 on the incident force ; A„ is supposed to be small. 

 Of course c~E x jct may be negative ; in this case we 

 shall suppose that there is a chance A„( - 

 for the emission of a wave -/. We shall be able 

 to treat both cases together and need not make the 

 distinction. The subscript „ indicates that we 

 suppose there are several different ways in which 

 the atom may be excited, each with a separate 

 chance for it. 



Consider a simple case, a monochromatic wave 

 polarised along x and advancing along z, which 

 strikes a group of N atoms at the origin. Let the 

 wave be E x =H„ = F cos p(t - zjc). The number excited 

 in the interval dt will be NA„( - Fp sin pt)dl. Con- 

 sider the secondary wave crossing the point 

 x, y, z, at the time t + r/c. This is due to all the 

 atoms which were excited before the time t. The 

 number excited in the interval ds at a time t - s is 

 - NA n Fp sin p{t - s)ds and each of these will at the 

 time t be giving a wave typ mec l by f =a„e- A - s cos k n s. 

 So the total effect will be a wave which at the time 

 t +rjc at x, y, z has an .r-component 



/ .,2 _ ,.2\ foo 



s)ds. 



! cos k n s 



= NA„«„ 



-„ cos pt, 



provided that \„ is taken as small. The averaging 

 has entirely blotted out the Ereqi tie atoms 



and left only that of the incident wave. Now on 

 the classical theory, if there is a group of N„ electrons 



NO. 2773, VOL. I IO] 



