1 research outputs found
Analytic solution of the Schrodinger equation for an electron in the field of a molecule with an electric dipole moment
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.Comment: 20 pages, 1 figure, 4 table