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Next: References Up: A remark on quantum Previous: Numerical experiments

Relations with other numerical methods

The usual Schrödinger evolution tex2html_wrap_inline1146 needs a numerical integration like tex2html_wrap_inline1148 , where n is so large that tex2html_wrap_inline1152 is smaller than the desired accuracy tex2html_wrap_inline1154 . Any such truncation produces high-frequency noise after a relatively small number of time steps. An other method used in quantum dynamics is to diagonalize a finite dimensional approximation tex2html_wrap_inline1156 of L and to evolve its eigenfunctions (see for example [14]). It is quantitatively not clear, how well a finite dimensional Galerkin cut-off respects the actual dynamics. Moreover, in dimension d, the number of grid points N has to be so small that an eigensystem of a tex2html_wrap_inline1164 matrix can be found.

While numerical approximations of tex2html_wrap_inline1146 are not unitary, the evolution (2.1) is conjugated to a unitary evolution. Other discrete unitary time evolutions have been considered in [4]. The problem to preserve unitarity is similar to numerical integration problems for ODE's, where for example symplecticity should be preserved during the discretisation of a Hamiltonian system.

If tex2html_wrap_inline906 has compact support, then tex2html_wrap_inline1064 has this property too. This leads to a finite propagation speed as in a relativistic set-up. This fact has computational advantages. For example, we know exactly, after which time, boundary effects begin to influence the value of a wave at some point.

The discrete evolution preserves the (not necessarily closed) algebraic field in which L is defined. For example, if L is an operator defined over the rationals tex2html_wrap_inline1176 and if the coordinates of tex2html_wrap_inline906 are rational, then tex2html_wrap_inline532 is rational and tex2html_wrap_inline1182 can be determined exactly.

The evolution (2.1) can be defined on all bounded sequences tex2html_wrap_inline1184 and not only on tex2html_wrap_inline1186 . For example, the evolution leaves almost periodic configurations invariant, elements tex2html_wrap_inline1188 , for which the closure of all translated sequences tex2html_wrap_inline1190 is compact in the uniform topology tex2html_wrap_inline1192 ). This is useful, because solutions of (2.1) define generalized eigenfunctions tex2html_wrap_inline1194 of an operator K defined on space-time.

The unitary operator V=- i U solves tex2html_wrap_inline1200 which is a discretisation of the Schrödinger equation tex2html_wrap_inline1202 . Since tex2html_wrap_inline1204 , the evolutions U and V are essentially the same. The discrete evolution tex2html_wrap_inline1210 is a second order approximation to tex2html_wrap_inline1212 in the sense that tex2html_wrap_inline1214 and tex2html_wrap_inline1216 agree up to second order. This second order approximation is more efficient than the second order but computationally more expensive Cayley method tex2html_wrap_inline1218


next up previous
Next: References Up: A remark on quantum Previous: Numerical experiments

Oliver Knill, Tue Aug 18, 1998