SOME CONTEMPORARY ADVANCES IN PHYSICS— X 125 



is an integer multiple of li/'Zw. Thus it would f<jrni an adequate 

 model for a s>'stem of Stationary States in which e\ery group of 

 terms in every sequence was a triad; but this is not a doublet system, 

 nor even a triplet system, nor any other observed system whatever. 

 To make this long story short; it is impossible to simulate any of the 

 eight groupings of terms set forth in the eight lines of Table I by 

 assuming that Py, Pr and Pa are all integer multiples of h/2Tr (or 

 of any other common factor). 



It is in fact necessary to put Py equal, not to h/2Tr and to 2h/2w 

 and to 3/z/27r, but to \ {h/2Tr) and to f {h/2Tr) and to f {h/2iv), for the 

 s and p and d states, respectively. This use of "half quantum num- 

 bers" makes it possible to produce an adequate model for an atom 

 possessed of a doublet system, by assuming that the angular momen- 

 tum Pr of its residue is always ///27r, and that its two w^hirling parts 

 must always be so inclined to one another that the angular momentum 

 of the entire atom is an integer multiple of h/2Tr. 



For (to w^ork out one example, and one only) when we make 

 PR = h/ 2Tr and Pv = \ Qi/2Tr), then the greatest possible resultant 

 that can be obtained by combining these vectorially is f (/j/27r) and 

 the least possible one is 4- {h/2ir) ; these two extreme values being at- 

 tained when the two component vectors are parallel and when they are 

 anti-parallel,^ respectively. If we permit for the resultant only such 

 values as are integer multiples of h/2ir, then there is only 07ie that is 

 permitted : the value h/2Tr — for this is the only such value lying within 

 the possible range. Next, put Pr = ]iI2-k and Py = \ {h/2-K). All pos- 

 sible values of the resultant lie between f {h/2Tv) and \ {h/2Tr) ; within 

 this range there are hvo of the integer multiples of /i/27r which are the 

 sole permitted ones. Next, put Pr^Ji /2Tr and Pv = i {h/2Tr). All 

 possible values of the resultant lie between | {h/2T) and f (h/2ir), and 

 this range again includes two permitted values. Thus the model de- 

 scribes properly the grouping of the terms in a doublet system. I 

 leave it to the reader to show that by putting PR^2h/2T, or 3^/27r, 

 or 4/i/27r, and treating Py as in this foregoing case, he can reproduce 

 the groupings of terms in quartet, or sextet, or octet systems, respec- 

 tively, as Table I, describes them.^ 



A more drastic use of "half quantum numbers" is required to obtain 

 an adequate model for atoms showing singlet and triplet and other 



^ This convenient word is used in German to describe vectors pointing in opposite 

 senses along the same direction. 



' The diagrams with arrows, offered by Sommerfeld in the fourth edition of his 

 classic book, are very helpful in studying these models. Incidentally Sommerfeld's 

 alternative way of arriving at the groupings of multiplet terms by compounding 

 vectors is instructive. 



