294 BELL SYSTEM TECHNICAL JOURNAL 



whole, it substitutes Vl/2-3/2 and V3 /2 -5/2 for the factors 1/2 and 3/2. 

 In order that these new values of /' and j" may be resultants of 5 and /, 

 it is necessary that the angular momenta of orbital motion and of spin 

 should be neither quite parallel nor quite anti-parallel to one another — 

 the two permitted values of the angle between them must differ from 

 0° and 180° ; and this also is admitted by quantum mechanics. The last 

 two clauses of the foregoing paragraph remain, however, unchanged. 



Then why not say from the outset that the spin of the electron has 

 the magnitude Vl/2 •3/2(^/27r), and that all the other angular momenta 

 occurring inside an atom have the magnitudes assigned to them by 

 quantum mechanics? Inertia of habit, sanctified by years of earlier 

 theories, is itself a mighty obstacle : after so long a time of saying that 

 the electron-spin is 1/2, the world of physics could scarcely get ac- 

 customed to saying that really it is Vl/2-3/2 (in terms of hjlir as unit). 

 There is also a difficulty of printing to be considered: these numbers 

 have often to be used as subscripts; it is bad enough to print 1/2 or 

 3/2 as a subscript, without resorting to their quantum-mechanical 

 substitutes. The most serious reason, however, is, that the original 

 1/2 and 3/2 (and, of course, their analogues in the many other kinds of 

 atomic states) lend themselves uniquely well to stating how man>- 

 permitted orientations there are. Thus in the present example, I 

 quoted 2 and 4 respectively as the number of permitted orientations 

 in a magnetic field, of certain P states for which the angular momenta 

 were designated asj(h/2ir) andj had the values 1/2 and 3/2 respectively. 

 I was reminded of those numbers by the rule that they are equal to 

 (2j +1). Had I kept in mind only the magnitudes •V3/4 andVl5/4 

 assigned by quantum mechanics to these angular momenta, the rule 

 would not have been available. 



One should therefore keep in mind both the "marker" or "quantum- 

 number" of an angular momentum, of which the foregoing 1/2 and 1 

 and 3/2 are examples; and the numerical value assigned by quantum- 

 mechanics to the magnitude of that angular momentum. Fortunately 

 this is rendered easy by the fact that there is a general formula for the 

 latter in terms of the former, with which we can now make acquaint- 

 ance in the course of taking a deeper plunge into the lush notation of 

 spectroscopy; as follows: 



The angular momentum of electron-spin has the quantum-number 

 5 and the magnitude ^ls{s +1), and i" is always equal to 1/2. 



The angular momentum of the orbital motion of an electron has the 

 quantum-number / and the magnitude V/(/ + 1), and / may have the 

 various values 0, 1, 2, 3 • • •. 



The angular momentum of the atom-as-a-whole has the (|uantum- 



