Quantum mechanics without potential function

Abstract

In the standard formulation of quantum mechanics, one starts by proposing a potential function that models the physical system. The potential is then inserted into the Schr\"odinger equation, which is solved for the wave function, bound states energy spectrum and/or scattering phase shift. In this work, however, we propose an alternative formulation in which the potential function does not appear. The aim is to obtain a set of analytically realizable systems, which is larger than in the standard formulation and may or may not be associated with any given or previously known potential functions. We start with the wavefunction, which is written as a bounded infinite sum of elements of a complete basis with polynomial coefficients that are orthogonal on an appropriate domain in the energy space. Using the asymptotic properties of these polynomials, we obtain the scattering phase shift, bound states and resonances. This formulation enables one to handle not only the well-known quantum systems but also previously untreated ones. Illustrative examples are given for two- and there-parameter systems.Comment: 25 pages, 1 table, and 3 figure

Series Solution of a Ten-Parameter Second-Order Differential Equation with Three Regular Singularities and One Irregular Singularity

“…If the basis elements are orthonormal (i.e., n m nm     ) then z E  and 1 n u  but this is not always the case. The recursion coefficients   , , , n n n n u w s t depend on the differential equation parameters and on n but are independent of z and such that 0 n n t s  for all n. Therefore, the solution ( ) n f z of (11) becomes a polynomial of degree n in z modulo an overall factor that depends on z but is independent of n. That is, if we write [ ( )] f z [2,18]. Thus, the solution of the differential equation (3) is equivalent to the solution of the three-term recursion relation (11).…”

Series solutions of Laguerre- and Jacobi-type differential equations in terms of orthogonal polynomials and physical applications

“…The asymptotics ( n   ) of ( , ) n P z   gives the phase shift ( ) arg ( i ) z z      (see, for example, Eq. A4 in Appendix A of [7]). Substituting the physical parameters, we obtain…”

“…are derived from the properties of these polynomials (e.g., their spectrum formula, weight function, asymptotics, etc.) [7][8]. The first ODE is the following 5-parameter Laguerre-type differential equation:…”

“…The polynomials   ( ) n P z are determined explicitly for any degree by the ST 2 R 2 (7) starting with the initial values 0 ( ) P z and 1 ( ) P z . Consequently, the solution of the second order differential equation becomes equivalent to the solution of the ST 2 R 2 (7). Moreover, it is worthwhile making the following remark about the boundedness of the solution series (2…”

Potential function of the Wilson-Racah quantum system

“…For instance, the potential functions might be non-analytic, Nonlocal, energy dependent, or the corresponding differential wave equation is higher second order and so on. As a result of this formulation, new quantum systems that do not belong to the conventional solvable class in quantum mechanics were obtained [1][2][3] However; in [3], we obtained a new quantum system-Wilson-Racah Quantum Systemwithits discrete energy spectrum, scattering phase shift, and discrete wavefunction (Racah quantum system). But for this physical system, we are yet to get the potential function (either analytically or numerically).…”