39 



323 



(see page 8), the number of conditions wiiicli fix tiie stationary states will he k^ss flian 

 the number of degrees of IVeedom and equal to the number of fundamental frequencies 

 characterising the motion, hut just as in the case of a non-degenerate system these 

 conditions will be sufficient to fix the value of the total energy of the system, which is 

 determining for the frequencies of the spectral lines. If we, for instance, consider the 

 undisturbed hydrogen atom in space, we have to do with a system of three degrees 

 of freedom, the motion of which is characterised by two fundamental frequencies 

 only. Separation of variables is possible for any set of polar coordinates with the 

 centre at the nucleus, and three quantities /j, I^, may be defined by the formulae 

 (75). There will, however, only be two conditions characterising the stationary states, 

 viz. /j = /ij/i and /.^ -\- 1.^ = nji (or, with the notation of § 2, /, = =- n^h), in 



intimate connection with the fact that the direction in space of the axis of the system 

 of coordinates used for the separation is arbitrary, so that the quantities and /.j 

 themselves naturally must remain undetermined in the stationary states. A very 

 important example ot a degenerate system is further afforded by a system consisting 

 of an electron and a nucleus, the motion of which is governed by Newtonian 

 mechanics; this system will in the following be denoted as the model of a '•simpli- 

 fied hydrogen atom". The motion of this system is simply periodic and its statio- 

 nary states will therefore be characterised by one condition only. Separation of 

 variables may be obtained in an infinite multitude of sets of coordinates, for in- 

 stance in any set of polar coordinates and in any set of parabolic coordinates with 

 the nucleus at the centre. In both of these cases we obtain three quantities /j, I.,, 

 /.„ w^hich coincide with the analogous quantities in § 4, if we take the velocity of 

 light c infinitely large, and with the analogous quantities in 3, if we take the 

 intensity of the electric force F equal to zero. The stationary states will in both 

 cases be fixed by the single condition / = + /.^ 4- ^ where n is a positive 

 integer, in intimate connection with the fact that, due to the arbitrariness in the 

 choice of the set of coordinates used for the separation, the values of I^, I^, /., 

 themselves must remain arbitrary in the stationary stales. We therefore have directly 

 from the formulae (24) and (46i that the energy in the stationary states of the 

 simplified hydrogen atom is given by 



r- n-h- 



The frequency of revolution in these states will according to (11) be given by 

 _ <9E _ 4;:- iVT-'e' m _ 4r=» iV»e* m 



^ ol ~ J- n'lr ' ' 



while the major axis of the Ke|)lerian ellipse described 1)\ the cliclron innv he 

 easily shown to be equal to 



2.^-, 4. ^,-'!;^: . .102, 



2 77' .\ e- m 2 rr- Ne- m 



