.SOMIi C()A/7:.U/'()A'./A'l' .ll>r.l\CI:S l\ I'liySICS- IS U7\ 



(larijirrous. for \vi> h.ivo i(lrtitit"K'<l i-iTi.iin distinclivr ft-atiirfs of the 

 jicrmissiMc circular orbits which ma\' Ik- OM-uti.il; and ihcsi- features 

 may not Ik.' iransfcralile to tlu' ellipses. Let us test them. 



The second and the third of tiie three distinctive features which I 

 cititl are transferable — that is they can be extended to the totality 

 of all ellii>si's having one or another of the enern\-values — Rli tir,' 

 am! they differentiate these from all other ellipses. I-'or it can be 

 shown. !>>• integrating the kinetic energy A" (the first term on the right 

 hand side of {'M)) ) around an elliptical orbit, that 



7=1 JA'r// = 27r \/aine/-: 5 1 



depending onl\' on the major a.xis a of the oriiil. Now we have 

 shown that I = nh Un the «th of the permissible circles; hence for each 

 ellipse ha\ing the same major axis as the Hth permissib.e circle, in 

 other words for each ellijise of energy-\alue — Rh n'-. we haw 



/ = ///; 



and the second of the distinctive features is transferable to the ellipses. 

 It is the same for the third ; for 7' is b\- (42) tlependeiu on a onl\-, and so 



Lim w = Liiu v. 



But it is otherwise with the first. 



In the first place it was shown that the angular momentum of the 

 electron in the circle of diameter eE Rh is equal to h/2ir. Obviously 

 this cannot l>e true of all the ellipses of major axis eE Rh. For ac- 

 cording to (37), the angular momentum of the electron in such an 

 ellipse is 



p = VeEma{l-t-) (52) 



defK'nding on the eccentricit\'. This is e(iual to ("vw/a, which by 

 (12) is equal to // 2;r, only if «=U. The circle therefore is the only 

 orbit for which the energv-value and the angular momentum of the 

 atom are simultaneously equal to —Rh and to /; 2ir respectively. 

 If we admit the ellipses to equal value with the circle, we concede 

 that the equality of the angular momentum with /; 2;r is of no sig- 

 nificance. 



There is a partial escape from this conclusion for the remaining 

 stationary- states. Take for instance the second, of energy-value 

 — Rh, 4. The circular orbit of diameter 4eE Rh, for which the atom 

 possesses this energy-value, is distinguished by the angular momen- 

 tum 2/j 2t. For each of the infinity of ellipses possessing the same 

 major axis 4eE/Rh there is a different value of the angular momentum; 



