Some structural and numerical aspects of Heisenberg’s matrix mechanics with application to one-dimensional nonpolynomial potentials

The equation-of-motion method based on Heisenberg's matrix mechanics, utilized in this paper, was originally developed within the context of the nuclear many-body problem. The method was later applied successfully to anharmonic oscillators and other problems in quantum mechanics with polynomial potentials in one and two dimensions. Here we apply the method to the one-dimensional quantum system with the nonpolynomial potential lambda sinh(2)x, as an illustration of its applicability to a wider class of problems than has been previously considered. We examine some of the structural and numerical aspects of this one-dimensional quantum system, and show that it is essential to work with sinhx instead of x as the dynamical variable for this nonpolynomial potential problem. This choice is suggested by a preliminary study of the exactly solvable Poschl-Teller potential, -lambda /cosh(2)x, where our methods provide an exact algebraic solution. For the potential lambda sinh(2)x, the same methods yield equations that can be studied numerically by a controlled sequence of nonperturbative approximations.