New method for the solution of the two-body Dirac equation for the positronium bound states

A new theoretical method is developed to solve the two-body bound-state Dirac equation for positronium. Only Coulomb potential was included in the Dirac Hamiltonian. It is shown that the two-body Dirac Hamiltonian can be written in the Hermitian matrix form of the 4$\times$4 size and diagonalized in the momentum-state representation. Numerical results for the energy spectrum of the para- and ortho-positronium ground states performed within the variational method using the harmonic oscillator basis functions are in good agreement with a high-precision finite-element method of T.C. Scott et al. After the Fourier transformation into the coordinate-state representation the bound state wave functions of the para-Ps and ortho-Ps do not contain any singularity at the origin in contrast to the method mentioned above. The weights of the large-small and small-large components of the ground state wave functions are estimated to be of order 10$^{-6}$, while the weight of the small-small component is of order 10$^{-12}$.