86 



intensities of the components ot the fine structure and the Stark effect of the 

 hydrogen lines. 



A procedure quite analogous to that applied above may be used to treat the 

 problem of the effect of a homogeneous magnetic field on the hydrogen spectrum. 

 also when the relativity modifications are taken into account and when the atoms 

 at the same time are exposed to a small external field of force of constant potential, 

 which possesses axial symmetry round an axis through the nucleus parallel to the 

 magnetic force; because also in this case we can obviously make direct use of 

 Larmor's theorem. We shall not. however, proceed in this way. but shall come 

 back to these questions when we have shown how, by a simple modification 

 of the general considerations of perturbed periodic systems given in .^ 2. it is pos- 

 sible to represent the theory of the stationary states of the hydrogen atom in the 

 presence of a small magnetic field on a form, which allows to discuss the effect on 

 the hydrogen spectrum also if the atom is exposed to a magnetic field which is 

 not homogeneous, or to discuss the efi'ect of a homogeneous magnetic field if elec- 

 tric forces, which do not possess axial symmetry round an axis through the nucleus 

 parallel to the magnetic field, are acting on the atom at the same time. 



In order to examine the general problem of the secular perturbations of the 

 orbit of the electron in the hydrogen atom which take place if the atom is ex- 

 posed to small external forces which, entirely or partly, are of magnetic origin, 

 we shall, as in the usual theory of planetary perturbations, take our starting point 

 in the equations of motion in their canonical form. Now the equations of motion 

 of an electron of charge — e. which besides by an electric field of potential V is 

 acted upon by a magnetic field of vector potential it i defined by div it = and 

 curl 31 = Ö, where ö is the magnetic force considered as a vectori. can be written 

 in the Hamiltonian form given by (4), if, just as in the absence of the magnetic 

 field, E is taken equal to the sum of the kinetic energy T of the electron and its 

 potential energy — eV relative to the electric field, while the momenta which are 

 conjugated to the positional coordinates q,, q,, g, of the electron in space are 

 defined by the equations ^J 



P, = P^-l^-Tr-^, '^■= 1>2, 3) (83) 



where the p's are the momenta defined in the usual way i compare page 10 . and 

 where {d21) represents the scalar product of the velocity of the electron n and the 

 vector potential 31, considered as a function of the q's and of the generalised velo- 

 cities q., q,, q,. If we now assume that the effect of the magnetic forces on the 

 motion of the electron is so small compared with the effect of the electric forces, 

 that in the calculations we may look apart from all terms proportional to .§-. it 

 is simply seen that the energy function E in (4), obtained by introducing the 

 momenta defined by (83), will differ from the corresponding function, holding in 



^j See f. insL G. A. Schott: Electromagoetic Radiation, App. F (Cambridge, 1912). 



