93 



reference to the invariance of the a-priori probabihly of the stationary states during 

 such a transformation (see Part I, page 9 and 27), we must llierefore conclude that, 

 also in the case of a hydrogen atom in the presence of a magnetic field, no stationary 

 states exist for wliich the angular momentum round the axis would he equal to 

 zero, although these states in mechanical respect do not exhibit singularities from 

 which we might anticipate that they are physically unrealisable.') 



In case we consider the general problem of the effect on a hydrogen atom of 

 a small electric or magnetic field, which do not possess axial symmetry round an 

 axis through the nucleus, or of the simultaneous effect of two such fields, which 

 do not possess such symmetry round a common axis, we must expect that the 

 secular perturbations of the orbit of the electron will in general not be of condi- 

 tionally periodic type. In such a case we cannot obtain a complete fixation ot the 

 stationary states, and we may conclude that the presence of the external forces will 

 not give rise to the splitting up of the hydrogen lines into a number of sharp 

 components but to a diffusion of these lines. With a simple example, in which the 

 secular perturbations of the atom seem not to be of conditionally periodic type, we 

 meet if we consider the simultaneous effect on the hydrogen spectrum of 

 an external homogeneous electric field and a homogeneous magnetic 

 field, the directions of which make an angle with each other. If 



') Note added during the proof. In a dissertation whicli has just appeared, J. M. Buhgers (Het 

 Atoommodel van Rutherford-Bohr, Haarlem 1918) has given a very interesting general survey of the 

 applications of the quantum tlieory to the problem of the constitution of atoms, and has in this con- 

 nection entered upon several of the questions discussed in the present paper; for instance on the 

 question of the relation between the spectrum of an atomic system, deduced by application of relation 

 (1) from the values of the energy in the stationarj' states, and the frequencies of the harmonic vibra- 

 tions into which the motion in these states can be resolved; and on the question of the determination 

 of the relative values for the a-priori probability of the different stationary states of an atomic sj'steni 

 by means of Ehrenfest's principle of the invariance of these values during a continuous transforma- 

 tion ot the system. As an illustration of the latter considerations, Burgers has deduced an expression 

 for the relative values of the a-priori probability of the different stationär)' states of the undisturbed 

 hydrogen atom, by means of an enumeration of the states, determined by the conditions (22) when 

 applied in connection with a separation of variables in polar coordinates, which correspond to a sta- 

 tionary state of the undisturbed atom, characterised bj' a given value of ;i in the condition I ^ nb. 

 Excluding only such states for which the total angular momentum of the electron round the nucleus 

 would be equal to zero, Burgers (loc. cit. p. 259) finds in this wa)' for the value of tlie a-priori pro- 

 bability in question {n + 1)^ — 1. In connection with the analogous consideration, given in the Note on 

 page 76 of the present paper, which leads to a different result, it may be of interest to remark that 

 the necessary conformity between the relative values for the a-priori probabilit)' of the different sta- 

 tionary states of the undisturbed hydrogen atom, deduced from an enumeration of the stationary states 

 of the atom which appear in the presence of a small external electric field or in the presence of a 

 small magnetic field respectively, cannot be obtained if in both cases we would exclude only such 

 states in which the angular momentum of the electron round the nucleus is always equal to zero. In 

 fact, while in case of a magnetic field this would give (n + 1)-— 1 different states corresponding to 

 a given value of n, it would in case of an electric field give only (n -- 1)- — 2 such states. On the other 

 hand, if the possible stationary states are selected in the manner explained in the text, the conformity 

 in question will obviously be obtained. 



n. K. I). Vldcnsk.Selsk.Skr.. nalurvidensk. og nuilhem. AM . 8. Riekke, IV. 1. 13 



