The matrix representation of Schroedinger's equation and its implications for the quantum mechanical inversion problem.

摘要

For an operator A and scalar λ, the equation Af = λf is a typical expression of the eigenvalue problem, where f and λ are respectively the eigenfunctions and eigenvalues of A. When A is completely specified, Af = λf is solved for f and λ. Using this formulation of the eigenvalue problem, the following inversion problem is considered. Let A represent the secular (Hamiltonian) matrix arising from the Schrödinger equation for a one-dimensional harmonic oscillator, where the elements of A are given as functions of a suitably parameterized potential energy function. Assume the eigenfunctions, f, are expanded within a specified, finite orthonormal basis set. If a set of eigenvalues λ_i, and the corresponding projections of the eigenfunctions on a particular basis set element are known, can this data be inverted to determine the potential energy. This formulation of the algebraic eigenvalue problem provides a method for deriving systems of algebraic equations in which the potential energy parameters and the basis set projections (eigenvector components) occur as unknowns. Gaussian elimination and Gröbner bases methods are applied to these systems equations to determine the parameters of the potential energy function.