Iterative evaluation of the effect of long-range potentials on the solution of the Schrodinger equation

作者：Rawitscher George^*

来源：Physical Review A, 2013, 87(3): 032708.

DOI：10.1103/PhysRevA.87.032708

There are cases where the potentials present in the Schrodinger equation are of long range and have measurable effects as, for instance, for the interaction between atoms at low temperatures or for the calculation of atomic three-body collisions. In these cases, the solution of the Schrodinger equation for the wave functions by finite-difference or finite-element techniques may not achieve the desired accuracy. An iterative method is presented, based on the Lippmann-Schwinger integral equation, that is similar in spirit to the Born approximation but is applied only in the region of the potential tails. This procedure extends the numerical solution obtained for short distances to large distances without loss of accuracy. Numerical examples are presented for atomic van der Waals potentials C-n/r(n). For C-6/r(6), the size of the radial interval, for which an accuracy of 10(-10) is achieved, is similar or equal to [100,1000] atomic units a(0). For the case of C-3/r(3), the required interval for the same level of accuracy is [4000,50 000], which, because of its large size, has to be subdivided into smaller partitions. The wave numbers k chosen for these examples correspond to atomic collision energies in the micro-Kelvin range. The larger the wave number k, the faster the rate of the convergence, and the limit k -%26gt; 0 is also investigated. A criterion is given for determining whether the iterations converge in that limit. DOI: 10.1103/PhysRevA.87.032708

• 出版日期2013-3-18