Schr\"odinger Equation for Heavy Mesons Expanded in $1/m_Q$

Abstract

Operating just once the naive Foldy-Wouthuysen-Tani transformation on the Schr\"odinger equation for $Q\bar q$ bound states described by a hamiltonian, we systematically develop a perturbation theory in $1/m_Q$ which enables one to solve the Schr\"odinger equation to obtain masses and wave functions of the bound states in any order of $1/m_Q$. It is shown that positive energy projection with respect to the heavy quark sector of a wave function is, at each order of perturbation, proportional to the 0-th order solution. There appear also negative components of the wave function except for the 0-th order, which contribute also to higher order corrections to masses.