Comparison of classical and quantum continuum expectation values

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Abstract
Exact continuum diagonal matrix elements of interest in scattering theory have been compared with both a classical approximation and a WKB approximation at a range of energies and angular momenta. The matrix elements of the interactions R-n and exp (-αR)/R 2 are calculated with scattering wavefunctions describing motion in Lennard-Jones (12, 6) potentials supporting a varying number of bound states. The classical approximation is as simple to calculate as the classical angle of deflection and is much more easily calculated than the WKB approximation or the exact result. Apart from a few limited regions of little physical significance where quantum mechanical tunnelling is important, the classical approximation is very accurate provided the potential supports three or more vibrational states.

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