Analytic solution of the wave equation for an electron in the field of a molecule with an electric dipole moment

Abstract

We relax the usual diagonal constraint on the matrix representation of the eigenvalue wave equation by allowing it to be tridiagonal. This results in a larger solution space that incorporates an exact analytic solution for the non-central electric dipole potential 2 cos r θ , which was known not to belong to the class of exactly solvable potentials. As a result, we were able to obtain an exact analytic solution of the three-dimensional timeindependent Schrödinger equation for a charged particle in the field of a point electric dipole that could carry a nonzero net charge. This problem models the interaction of an electron with a molecule (neutral or ionized) that has a permanent electric dipole moment. The solution is written as a series of square integrable functions that support a tridiagonal matrix representation for the angular and radial components of the wave operator. Moreover, this solution is for all energies, the discrete (for bound states) as well as the continuous (for scattering states). The expansion coefficients of the radial and angular components of the wavefunction are written in terms of orthogonal polynomials satisfying three-term recursion relations. For the Coulomb-free case, where the molecule is neutral, we calculate critical values for its dipole moment below which no electron capture is allowed. These critical values are obtained not only for the ground state, where it agrees with already known results, but also for excited states as well.

Solutions of the Schrödinger equation in the tridiagonal representation with the noncentral electric dipole plus a novel angle-dependent component

“…For positive parameter E θ , γ must be either greater than zero or less than −1. The reader can refer to [29] for more information in support of this argument. The tridiagonal matrix representation of the θ -angular wave operator in (27) makes the θ -angular equation (14) equivalent to the following three-term recursion relation for the expansion coefficients of the θ -angular component of the wavefunction…”

“…It should be noted that besides Alhaidari's pioneering contribution [27] and our recent work [28], no noncentral electric dipole potential appearing in the literature mentioned above. The major reason behind this fact may be that the noncentral electric dipole potential was believed not to belong to any of the established classes of exactly solvable potentials until Alhaidari, using the tridigonalization program, made it a new member of the exactly solvable potentials [29,30].…”

Analytical arbitrary ℓ‐wave solutions of the manning–rosen potential in the tridiagonalization program

“…The main objective and motivation of this program are to find solutions of new problems that could not be solved by the traditional methods (the diagonal program). For example, the noncentral electric dipole potential V(r,y) ¼ cosy/r 2 [16][17][18], the electric quadrupole potential [19], the hyperbolic single wave potential [3], the screened Coulomb potential with a barrier [20], and the Yukawa potential [21]. Of course, the tridiagonal program must give the traditional solutions automatically [22][23][24].…”

Exact solution of the modified Pöschl–Teller potential in the tridiagonal representation

“…The discrete spectrum is easily obtained by imposing the diagonalization constraint on the recursion relation (17), which requries…”