Abstract
The small-angle approximation usually encountered in dynamical theories of fast electrons essentially leads to a transformation of the propagation-direction variable z into a time-like parameter [Berry (1971). J. Phys. C, 4, 697-722]. The three-dimensional stationary Schrodinger equation is then approximated by a two-dimensional 'time'-dependent equation which may be solved by using the standard time-perturbation techniques encountered in quantum mechanics. The basic idea of the present approach consists in studying the evolution operator U(z,z0) instead of the wave function. Depending on the choice of bases, the matrix elements of U(z,z0) represent either the transition probabilities of diffraction or the kernel function of the propagation issued from Feynman-path integral theory [Berry & Mount (1972). Rep. Prog. Phys. 35, 315-397; Van Dyck (1975). Phys. Status Solidi, 72, 321-336; Jap & Glaeser (1978). Acta Cryst. A34, 94-102]. Special attention is devoted to the so-called 'Bloch waves' and 'physical-optics' formulations which both correspond to the same perturbation expansion but with two different unperturbed 'Hamiltonians'.
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