The Instantaneous Approximation to the Transition Matrix Elements between Two Bound States

Under the framework of the Bethe-Salpeter (B.S.) wave functions and the Mandelstam formalism as well, to make ``instantaneous approximation’’ to a transition matrix element (a current operator sandwiched between two bound-states of double heavy quarks) is described. By taking the typical concerned decays as examples, such as $B_c$ meson decaying to $J/\psi + (\bar l \nu)$, the advantages of the approach and its limitations are illustrated. Finally, potential applications to various processes for possible double heavy flavoured systems, such as those of $(Q^{\mathrm{\prime }}\bar{Q})$ and $(Q^{\mathrm{\prime }}Q)$ ($m_Q, m_{Q’}\gg \Lambda_{QCD}$), are discussed.