672 BELL SYSTEM TECHXIC.iL JOURXAL 



hut ilicre is one among them for which the angular momentum is 

 equal to /; 2-k. And in general for the wth stationary state of energy- 

 value — Rh ir. there are « elliptical (including one circular) orbits 

 which would give the same cnerg>-\alue and ;; \alues of angular 



9., 



16a, 



Fig. 4a — Diagram to show the proportional dimensions of ellipses with identical 

 total quantum-number « = / 7; and different azimuthal quantum-numbers * = 1, 

 2 .... n ! rom left to right we have the cas?s « = 1, 2, 3, 4, on scales varying as 

 indicated by the subjoined arrows. 



momentum equal rcspecti\eK' to nli 'Itt, {ii — l)li 2ir Ii'lir. 



These, as the reader can show from (.")2). are distingtiished by the 

 following values of t: 



E- =k n 



k = \. 



(o3) 



Thus if we desire to regard the equalit\- of angular momentum with 

 an integer multiple of h 2-k as being essential to the permissible orbits, 

 we can keep, along with the circles, some of the other elliptical orbits 

 compatible with tiie prescribed energ>-\ allies; btit except for these 



•ig. 4b — The same ellipses as appear in Fig. 4a, drawn confocalK as I hey should 

 appear, instead of concentrically 



few, the inlinil\' of I'liipiical ori)iis will remain tnia\'ailaiile. There 

 is additional reason for liking to do this; for it amounts to a quite 

 natural generalization of the condition imposed on the angular mo- 

 mentimi, which as we saw it is highh' desirable to generalize if possible. 

 The angular momentum »ir-{d<t> dl), which I shall hereafter call p<i> 

 instead of simply p, stands on an equal footing with the radial 

 momentum pr = m{dr.dt) of the electorn; in the Hamiltonian equa- 

 tions for the motion of the particle, these two quantities stand side 

 by side. Now the condition imposed upon the angular momentum 



