Mabch 25, 1921] 



SCIENCE 



291 



From this potential energy it is then easy to 

 determine the law of repulsive force. 



The result of this analysis is that we may 

 regard the force between any nucleus of 

 charge Ze and an electron of charge e as con- 

 sisting of two parts which act independently. 

 The first is the Coulomb attractive force Fc 

 given by 



where 



P. ■■ 



(1) 



The second force, which we may call the 

 quantum force is a repulsive force Fq given 

 by 



1 /nh 



(2) 



In these equations r is the distance between 

 the electron and the nucleus, m is the mass of 

 the electron, h is Planck's quantum, and n is 

 an integer denoting the quantum state of the 

 electron. This repulsive force, varying in- 

 versely as the cube of the distance, is re- 

 markable in that it is independent of the 

 charge on the nucleus. It is to be noted 

 especially that an electron which has not 

 undergone any quantum change and for which 

 therefore n = 0, follows Coulomb's law accu- 

 rately. Thus presumably |8-rays in passing 

 through an atom will be acted on only by 

 the usual law. 



It can be readily shown that under the 

 influence of these two forces an electron wUl 

 be in stable equilibrium when it is at a dis- 

 tance from the nucleus equal to 





where a„ is given by 



iir^me' ' 



(3) 



(4) 



This result is identical with that for the 

 radius of the orbit in Bohr's theory, but of 

 course the law of force was chosen to give 

 just this result. 



If W is the total energy of the system with 

 its sign reversed we obtain 



We 



2Zao n^oi? 



(5) 



ft2 



(6) 



Equation (5) has no equivalent in Bohr's 

 theory for it applies to the transitions between 

 stationary states. The first term in the 

 second member represents the Coulomb poten- 

 tial while the second corresponds to the quan- 

 tum potential. 



When an electron has settled down into its 

 position of equilibrium, the value of W be- 

 comes 



W = ■ 



(7) 



This also is identical with the result obtained 

 by Bohr for the total energy in any stationary 

 state. It follows from this that the Eydberg 

 constant, the Balmer series and all other 

 series calculated by Bohr can be obtained by 

 this method without assuming electrons 

 moving about the nucleus. 



If the electron is disturbed from its position 

 of equilibrium it oscillates about this position. 

 From equation 5 the frequency of this oscil- 

 lation is found to be 



n%3 • 



(8) 



This is identical with the frequency of 

 revolution of the electron in Bohr atom. 

 From this we may draw a definite physical 

 picture of the mechanism of the transition 

 between two states, at least when Z is large. 

 Bohr has shown ' that under these conditions 

 the frequency radiated when an electron 

 passes from one circular orbit to the next 

 inner one is the same as the frequency of 

 revolution. According to the present theory, 

 if the quantum number of an electron in a 

 stable position decreases by one unit, the 

 electron is no longer stable but falls towards 

 its new position of equilibrium, and oscillates 

 about this position. It then radiates its 

 energy of oscillation according to the usual 

 laws of electro-dynamics. 



One of the greatest successes of the Bohr 

 theory is that it accounts for certain slight 

 difPerences between hydrogen and helium lines 



