Efficient Finite Difference Solutions to the Time-Dependent Schrödinger Equation

✍ Scribed by Patrick L. Nash; L.Y. Chen

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 244 KB
• Volume: 130
• Category: Article
• ISSN: 0021-9991
• DOI: 10.1006/jcph.1996.5589

✦ Synopsis

explicit and local. Its novel features include the exact evaluation of a major contribution to an approximation to the

The matrix elements of the exponential of a finite difference realization of the one-dimensional Laplacian are found exactly. This evolution operator (Eq. ( )) and a first-order approximamatrix is used to formulate an efficient algorithm for the numerical tion to the exponential of the commutator of the kinetic solution to the time-dependent quantum mechanical scattering of energy operator with the potential energy operator. The a single particle from a time-independent potential in one-space algorithm based on these results yields efficient approxiand one-time dimension. The method generalizes to higher spatial mate finite difference solutions to the Schro ¨dinger equadimensions, as well as to multiparticle problems. ᮊ 1997 Academic Press tion that are accurate to at least O([បͳt/ma 2 ] 3 ), where ͳt is the time step and a is the uniform lattice spacing. This algorithm will provide efficient approximate solutions of