Stuckelberg Particle in a Coulomb Field: A Non-Relativistic Approximation

We start with the Stuckelberg tensor system of equations for a boson with spin states S = 1 and S = 0 and fixed intrinsic parity, which is transformed to the matrix form, then generalize this matrix system to the generally covariant case with the use of the tetrad method. This equation is detailed in spherical coordinates in the presence of an external Coulomb field. After separation of the variables we derive the system of 11 radial equations. By diagonalizing the space reflection operator, this system is splitted into two system of four and seven equations for the states with the parities P = (−1) j+1 and P = (−1) j respectively. The system for the states with the parities P = (−1) j+1 leads to the known solution and energy spectrum. The system of seven equations for the states with the parities P = (−1) j is solved for the states with the total angular momentum j = 0 in terms of hypergeometric functions. The system of seven equations for the states with the total angular momenta j = 1, 2, 3, ... turns out to be very complicated, the only nonrelativistic approximation has been studied. The derived nonrelativistic equations are solved in terms of confluent hypergeometric functions, and the corresponding energy spectra are found. In addition, the general form of the nonrelativistic equations for the the Stuckelberg particle is derived in the presence of an arbitrary electromagnetic field.