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Quantal-Classical Correspondence Impulse Theory

M. R. Flannery, D. Vrinceanu

2000
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Physical Review Letters
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The quantal impulse cross section is derived in a novel form appropriate for direct classical correspondence. The classical impulse cross section is then uniquely defined and yields the first general classical expression for nᐉ 2 n 0 ᐉ 0 collisional transitions. The derived cross sections satisfy the optical theorem and detailed balance. Direct connection with the classical binary encounter approximation is also firmly established. The unified method introduced is general in its application to
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... arious collision and recombination processes and enables new directions of enquiry to be pursued quite succinctly. PACS numbers: 03.65.Sq, 31.15.Gy, 32.80.Cy, 34.60. + z The quantal impulse approximation (QIA) is an established and productive cornerstone in general scattering theory [1] . It is based on the assumption that an inelastic collision of incident particle i with a two-particle subsystem ͑2, 3͒ results from each binary ͑i 2 j͒ scattering under the two-body interaction V ij alone, leaving the "spectator" particle k unaffected. The internal interaction V jk between the "active" particle j and the spectator particle k is ignored during the ͑i 2 j͒ collision except insofar as it generates a distribution r͑p jk ͒dp jk over the relative momentum p jk of particles j and k. This approach has been invaluable in nuclear physics for high-energy neutron scattering by complex nuclei [1] and in atomic physics for atom-Rydberg atom collisions at thermal energies [2-7] and for electron (ion)-atom and atomatom collisions [8, 9] at high energies. It is also valuable, for example, in the study of three-body recombination, Li 1 Li 1 Li ! Li 2 1 Li which limits the density and lifetimes of Bose-Einstein condensates at ultralow energy, and e 1 1p 1 e 1 !H 1 e 1 for the formation of antihydrogen. A classical binary-encounter approximation (BEA) has also been formulated for Rydberg collisions [3] and for high-energy collisional ionization [8] [9] [10] . Although QIA and BEA are conceptually connected, the formal interrelationship between them is quite complex and has never been firmly established. Previous studies [6, 8, 11] have depended on fairly complicated theoretical analysis in an effort to reduce QIA to BEA. In this Letter, the quantal impulse approximation is presented in a new form which provides quite naturally "the royal road" to classical correspondence. A classical impulse approximation can then be uniquely defined. The novel expression yields the first general classical cross section for nᐉ 2 n 0 ᐉ 0 transitions and the standard result for nᐉ 2 n 0 transitions. The derived cross sections satisfy the optical theorem (which implies probability conservation) and detailed balance. The method introduced then permits BEA formulas to be derived quite succinctly and in a unified way. There is substantial renewed [7, [10] [11] [12] [13] [14] [15] interest in the power of classical dynamics in almost all fields of modern physics, attributed to the desire [15] to obtain a more thorough understanding of the classical-quantal correspondence. This present development provides a profound and important quantal-classical connection in collision physics. The new formulation also provides a natural origin for development of new semiclassical methods. The transition T-matrix for the free-free (p ij ! p 0 ij ) scattering under the two-particle (i 2 j) potential V ij alone is T ij ͑p ij , p 0 ij ͒ ͗exp͑ip 0 ij ? r ij ͞h͒ jV ij ͑r ij ͒jC͑p ij , r ij ͒͘ r ij , where C͑p ij , r ij ͒ is the exact wave function for i 2 j scattering in the (i 2 j) center-of-mass frame CM͑i, j͒.

doi:10.1103/physrevlett.85.1
pmid:10991144
fatcat:fqky477wozf6jg56vb55o3u4ne