Schrodinger equation potential scattering

In this section we present several numerical techniques that are commonly used to solve the Schrodinger equation for scattering processes. Because the potential energy functions used in many chemical physics problems are complicated (but known to reasonable precision), new numerical methods have played an important role in extending the domain of application of scattering theory. Indeed, although much of the formal development of the previous sections was known 30 years ago, the numerical methods (and computers) needed to put this formalism to work have only been developed since then. 

Schrodinger Equation with Absorbing Potential for Quantum Scattering Calculations. 

The simplest form of scattering theory is for a single particle moving in a local external potential. If we ignore relativistic effects and return to the Schrodinger equation... 

Numerical integration of the radial Schrodinger equation is generally necessary if realistic interaction potentials are selected. For this purpose, efficient computer codes exist based, for example, on Cooley s method of integration. Other widely available codes written to compute scattering... 

Earlier studies of positron-molecule elastic scattering did not involve such detailed descriptions of the scattering process as do the variational and R-matrix formulations. Instead, the interaction between the positron and the molecule was represented by a relatively simple model potential, and the positron wave function F(r 1) was assumed to satisfy the equivalent single-particle Schrodinger equation... 

We begin our discussion with the simple case of a spinless particle of mass m and kinetic energy E = h2k2/2m in a spherical, time-independent potential V(r), so that the Schrodinger equation can be decomposed into uncoupled partial waves l. For a particular l, the scattering matrix or the S matrix is defined as S(k) = exp[2/5(A )] in terms of the phase shift 8(k). Here and in the following, the subscript l on the S matrix and the phase shift is suppressed. The asymptotic form of the time-dependent radial wavefunction is expressible as... 

In the exact quantum calculation of the dynamics of a reaction on a particular potential-energy surface, it is necessary to solve the Schrodinger equation for the scattering from all occupied internal states of the reagents to all possible internal states of the products as a function of the relative collision energy. This is a multichannel process in that many final product vibrational and rotational states may be populated from a... 

This problem is important in formal scattering theory. A cubic box of side L is represented by a potential F(r), which has a constant value (say zero) inside the box and is infinite at the box boundary. This means that the particle cannot be found outside the box. The coordinate representation of the Schrodinger equation is... 

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