Physics – Quantum Physics
Scientific paper
2000-11-27
J. Phys. A: Math. Gen. 34, 3897 (2001)
Physics
Quantum Physics
36 pages, 16 figures. This updated version includes important references to existing work on uniform approximations, such as O
Scientific paper
10.1088/0305-4470/34/18/316
Eigenvalues and eigenfunctions of Mathieu's equation are found in the short wavelength limit using a uniform approximation (method of comparison with a `known' equation having the same classical turning point structure) applied in Fourier space. The uniform approximation used here relies upon the fact that by passing into Fourier space the Mathieu equation can be mapped onto the simpler problem of a double well potential. The resulting eigenfunctions (Bloch waves), which are uniformly valid for all angles, are then used to describe the semiclassical scattering of waves by potentials varying sinusoidally in one direction. In such situations, for instance in the diffraction of atoms by gratings made of light, it is common to make the Raman-Nath approximation which ignores the motion of the atoms inside the grating. When using the eigenfunctions no such approximation is made so that the dynamical diffraction regime (long interaction time) can be explored.
O'Dell Duncan H. J.
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