  We theoretically study bound states generated by magnetic impurities within
conventional $s$-wave superconductors, both analytically and numerically. In
determining the effect of the hybridization of two such bound states on the
energy spectrum as a function of magnetic exchange coupling, relative angle of
magnetization, and distance between impurities, we find that quantum phase
transitions can be modulated by each of these parameters. Accompanying such
transitions, there is a change in the preferred spin configuration of the
impurities. Although the interaction between the impurity spins is
overwhelmingly dominated by the quasiparticle contribution, the ground state of
the system is determined by the bound state energies. Self-consistently
calculating the superconducting order parameter, we find a discontinuity when
the system undergoes a quantum phase transition as indicated by the bound state
energies.
