International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 5.2, pp. 552-553   | 1 | 2 |

Section 5.2.4. Evolution operator

A. F. Moodie,a J. M. Cowleyb and P. Goodmanc

aDepartment of Applied Physics, Royal Melbourne Institute of Technology, 124 La Trobe Street, Melbourne, Victoria 3000, Australia,bArizona State University, Box 871504, Department of Physics and Astronomy, Tempe, AZ 85287-1504, USA, and cSchool of Physics, University of Melbourne, Parkville, Australia 3052

5.2.4. Evolution operator

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Equation (5.2.3.1)[link] is a standard and much studied form, so that many techniques are available for the construction of solutions. One of the most direct utilizes the causal evolution operator. A recent account is given by Gratias & Portier (1983[link]).

In terms of the `Hamiltonian' of the two-dimensional system, [-{\bf H}(z) \equiv {1\over 2k_{z}} (\nabla_{x, \,  y}^{2} + K_{0}^{2}) + \sigma \varphi,] the evolution operator [{\bf U}(z, z_{0})], defined by [\psi(z) = {\bf U}(z, z_{0}) \psi_{0}], satisfies [i {\partial\over \partial z} {\bf U}(z, z_{0}) = {\bf H}(z) {\bf U}(z, z_{0}), \eqno(5.2.4.1a)] or [{\bf U}(z, z_{0}) = 1 - i \textstyle\int\limits_{z_{0}}^{z} {\bf U}(z, z_{1}) {\bf H}(z_{1})\;\hbox{d}z_{1}. \eqno(5.2.4.1b)]

References

Gratias, D. & Portier, R. (1983). Time-like perturbation method in high energy electron diffraction. Acta Cryst. A39, 576–584.








































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