Analytic solution of the Schrodinger 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 cos(theta)/r^2, 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 time-independent Schrodinger 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.

arXiv comment20 pages, 1 figure, 4 tables

References in corpus (16)

• Self-adjoint extensions of operators and the teaching of quantum mechanics
• Renormalization of the Inverse Square Potential
• Anomalies in Quantum Mechanics: the 1/r^2 Potential
• Quantum Anomaly in Molecular Physics
• Regularization of the Singular Inverse Square Potential in Quantum Mechanics with a Minimal length
• The singular inverse square potential, limit cycles and self-adjoint extensions
• An extended class of L2-series solutions of the wave equation
• Scattering and bound states for a class of non-central potentials
• Effective theories of scattering with an attractive inverse-square potential and the three-body problem
• Relativistic extension of the complex scaling method
• L2 series solution of the relativistic Dirac-Morse problem for all energies
• Representation reduction and solution space contraction in quasi-exactly solvable systems
• Exact L2 series solution of the Dirac-Coulomb problem for all energies
• Regularized Green's Function for the Inverse Square Potential
• Self-Adjoint Extensions of the Pauli Equation in the Presence of a Magnetic Monopole
• Density function associated with nonlinear bifurcating map

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