St¨uckelberg particle in external magnetic field. Nonrelativistic approximation. Exact solutions

The St¨uckelberg equation for a particle with two spin states, 
S = 1 and S = 0, is studied in the presence of an external 
uniform magnetic field. In relativistic case, the particle is described 
by an 11-component wave function. On the solutions 
of the equation, the operators of energy, the third projection 
of the total angular momentum, and the third projection of the 
linear momentum along the direction of the magnetic field are 
diagonalized. After separation of variables, we derive a system 
for 11 functions depending on one variable. We perform 
the nonrelativistic approximation in this system. For this we 
apply the known method of deriving nonrelativistic equations 
from relativistic ones, which is based on projective operators 
related to the matrix Γ0 of the relativistic equation. The 
nonrelativistic wave function turns out to be 4-dimensional. 
We derive the system for 4 functions. It is solved in terms 
of confluent hypergeometric functions. There arise three series 
of energy levels with corresponding solutions. This result 
agrees with that obtained for the relativistic St¨uckelberg 
equation.