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 $\ensuremath{\lambda}{\mathrm{sinh}}^{2}x,$ 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 $\mathrm{sinh}x$ 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, $\ensuremath{-}\ensuremath{\lambda}/{\mathrm{cosh}}^{2}x,$ where our methods provide an exact algebraic solution. For the potential $\ensuremath{\lambda}{\mathrm{sinh}}^{2}x,$ the same methods yield equations that can be studied numerically by a controlled sequence of nonperturbative approximations.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 $\ensuremath{\lambda}{\mathrm{sinh}}^{2}x,$ 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 $\mathrm{sinh}x$ 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, $\ensuremath{-}\ensuremath{\lambda}/{\mathrm{cosh}}^{2}x,$ where our methods provide an exact algebraic solution. For the potential $\ensuremath{\lambda}{\mathrm{sinh}}^{2}x,$ the same methods yield equations that can be studied numerically by a controlled sequence of nonperturbative approximations.

Physical Review AApr 18, 20012 citations

Solvable quantum many-body problems in two dimensions

Mu-Lin Yan, Bao‐Heng Zhao

Physics Letters AAug 1, 19927 citations

Exactly solvable noncentral potentials in two and three dimensions

Avinash Khare, Rajat K. Bhaduri

We show that the list of analytically solvable potentials in nonrelativistic quantum mechanics can be considerably enlarged. In particular, we show that those noncentral potentials for which the Schrödinger equation is separable are analytically solvable provided the separated problem for each of the coordinates belongs to the class of exactly solvable one dimensional problems. As an illustration, we discuss in detail two examples, one in two and the other in three dimensions. A list of analytically solvable noncentral potentials in spherical polar coordinates is also given. Extension of these ideas to other standard orthogonal coordinate systems as well as to higher dimensions is straightforward.We show that the list of analytically solvable potentials in nonrelativistic quantum mechanics can be considerably enlarged. In particular, we show that those noncentral potentials for which the Schrödinger equation is separable are analytically solvable provided the separated problem for each of the coordinates belongs to the class of exactly solvable one dimensional problems. As an illustration, we discuss in detail two examples, one in two and the other in three dimensions. A list of analytically solvable noncentral potentials in spherical polar coordinates is also given. Extension of these ideas to other standard orthogonal coordinate systems as well as to higher dimensions is straightforward.

arXiv:hep-th/9310104American Journal of PhysicsNov 1, 199457 citations

\mathrm{sinh}x

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, $\ensuremath{-}\ensuremath{\lambda}/{\mathrm{cosh}}^{2}x,$ where our methods provide an exact algebraic solution. For the potential $\ensuremath{\lambda}{\mathrm{sinh}}^{2}x,$ the same methods yield equations that can be studied numerically by a controlled sequence of nonperturbative approximations.