NONTRIVIAL EXACTLY SOLVABLE POTENTIALS WITH LINEAR-EQUATIONS OF MOTION

Based on an equations-of-motion method in the framework of Heisenberg' s matrix mechanics, we investigate the conditions under which a one-di mensional quantum mechanical system becomes exactly solvable. By linea rizing the equation of motion which is a double-commutation relation o f some appropriately chosen function of the position operator with the Hamiltonian, we obtained a set of nontrivial exactly solvable potenti als in one dimension. These potentials not only can be solved analytic ally in closed forms but also contain both the Morse potential and the Poschl-Teller potential as their limiting cases. They may thus be val uable for some potential model calculations as well as for testing var ious approximation schemes. We also examine these potentials in the fr amework of supersymmetric quantum mechanics, which is particularly use ful for studying exactly solvable potentials.