Finite element method for quantum scattering

Abstract

The finite element method is introduced and applied to quantum mechanical scattering problems. In this procedure the space is discretized on a grid with the unknown quantities being the wavefunction values. Local polynomials approximate the wavefunction and no global basis set expansion is required. The scattering solution is constructed by a suitable combination of independent standing wave solutions. These latter solutions are generated numerically by using real, not complex, arithmetic. A one-dimensional barrier crossing is studied as a first example to illustrate finite element discretization and the construction of the scattered wave forms in an uncomplicated situation. A two variable generalization is given next. The method is then sucessfully applied to a model collinear problem which is analytically soluble and to the collinear H + H2 system. Next, a three variable formulation of the co-planar A + BC system is discussed with specific reference to co-planar H + H2. Some comments on the generalization of the technique complete the discussion.