The correct way to difference Schr¨ dinger’s equation [1,2] is to use Cayley’s o form for the finite-difference representation of e−iHt , which is second-order accurate and unitary: e−iHt In other words, n+1 n 1 + 1 iH∆t ψj = 1 − 1 iH∆t ψj 2 2 | Initial Value Problems in Multidimensions 853 The correct way to difference Schrodinger s equation 1 2 is to use Cayley s form for the finite-difference representation of e--iHt which is second-order accurate and unitary c-iHt 1 2 iHAt 1 2 iHAt In other words 1 2 HCrr 1 - 1 iHAt fn On replacing H by its finite-difference approximation in x we have a complex tridiagonal system to solve. The method is stable unitary and second-order accurate in space and time. In fact it is simply the Crank-Nicholson method once again CITED REFERENCES AND FURTHER READING Ames . 1977 Numerical Methods for Partial Differential Equations 2nd ed. New York Academic Press Chapter 2. Goldberg A. Schey . and Schwartz . 1967 American Journal of Physics vol. 35 pp. 177-186. 1 Galbraith I. Ching . and Abraham E. 1984 American Journal of Physics vol. 52 pp. 6068. 2 Initial Value Problems in Multidimensions The methods described in and for problems in 1 1 dimension one space and one time dimension can easily be generalized to N 1 dimensions. However the computing power necessary to solve the resulting equations is enormous. If you have solved a one-dimensional problem with 100 spatial grid points solving the two-dimensional version with 100 x 100 mesh points requires at least 100 times as much computing. You generally have to be content with very modest spatial resolution in multidimensional problems. Indulge us in offering a bit of advice about the development and testing of multidimensional PDE codes You should always first run your programs on very small grids . 8 x 8 even though the resulting accuracy is so poor as to be useless. When your program is all debugged and demonstrably stable then you can increase the grid size to a reasonable one and start looking at the results. We have actually heard someone protest my program would be unstable for a crude grid but I am sure the instability will go away on a larger grid. That is nonsense of a .
