Solution of the nonrelativistic wave equation in the tridiagonal representation approach
arXiv:1703.01268 · doi:10.1063/1.4993197
Abstract
We choose a complete set of square integrable functions as basis for the expansion of the wavefunction in configuration space such that the matrix representation of the nonrelativistic time-independent wave operator is tridiagonal and symmetric. Consequently, the matrix wave equation becomes a symmetric three-term recursion relation for the expansion coefficients of the wavefunction in this basis. The recursion relation is then solved exactly in terms of orthogonal polynomials in the energy. Some of these polynomials are not found in the mathematics literature. The asymptotics of these polynomials give the phase shift of the continuous energy scattering states and the spectrum for the discrete energy bound states. Depending on the space and boundary conditions, the basis functions are written in terms of either the Laguerre or Jacobi polynomials. The tridiagonal requirement limits the number of potential functions that yield exact solutions of the wave equation. Nonetheless, the class of exactly solvable problems in this approach is larger than the conventional class (see Table 12). We also give very accurate results for cases where the wave operator matrix is not tridiagonal but its elements could be evaluated either exactly or numerically with high precision.
49 pages, 12 tables, 7 figures
References in corpus (4)
Cited by in corpus (31)
- Open problem in orthogonal polynomials
- Establishing correspondence between the reformulation of quantum mechanics without a potential function and the conventional formulation
- Series solutions of Heun-type equation in terms of orthogonal polynomials
- Quantum mechanics with orthogonal polynomials
- Orthogonal polynomials derived from the tridiagonal representation approach
- Series solutions of Laguerre- and Jacobi-type differential equations in terms of orthogonal polynomials and physical applications
- Solution of an Open Problem about Two Families of Orthogonal Polynomials
- The Hahn Quantum System
- Deformed Morse-like potential
- Series solution of a ten-parameter second order differential equation with three regular and one irregular singularities
- Solution of the spin and pseudo-spin symmetric Dirac equation in 1+1 space-time using the tridiagonal representation approach
- Bound states and the potential parameter spectrum
- Construction of potential functions associated with a given energy spectrum -- An inverse problem. II
- Bound states of a short-range potential with inverse cube singularity
- Energy Density Bands Engineering
- Bound-states for generalized trigonometric and hyperbolic Pöschl-Teller potentials
- Bound-state solutions of the Schrödinger equation for two novel potentials
- Finite series representation for the bound states of a spiked isotropic oscillator with inverse-quartic singularity
- Solutions of the scattering problem in a complete set of Bessel functions with a discrete index
- Structural Algebraic Quantum Field Theory
- Exact solvability of two new 3D and 1D nonrelativistic potentials within the TRA framework
- Electric dipole and quadrupole contributions to valence electron binding in a charge-screening environment
- Five-parameter potential box with inverse square singular boundaries
- Extending the class of solvable potentials. IV Inverse square potential with a rich spectrum
- Revisiting the Coulomb problem: A novel representation of the confluent hypergeometric function as an infinite sum of discrete Bessel functions
- Exact scattering and bound states solutions for novel hyperbolic potentials with inverse square singularity
- Electrostatic multipole contributions to the binding energy of electrons
- Progressive approximation of bound states by finite series of square-integrable functions
- New Quantum System of Wilson Orthogonal Polynomial
- L-state solutions of a new four-parameter 1/r^2 singular radial non-conventional potential via asymptotic iteration method
- Potential Function Of the Wilson--Racah Quantum System