Physics
Scientific paper
Jul 2003
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2003njph....5..103l&link_type=abstract
New Journal of Physics, Volume 5, Issue 1, pp. 103 (2003).
Physics
6
Scientific paper
I formulate a 'pseudo-paradox' in the theory of a dilute Bose gas with repulsive interactions: the standard expression for the ground state energy within the Gross Pitaevskii (GP) approximation is lower than that in the Bogoliubov approximation, and hence, by the standard variational argument, the former should prima facie be a better approximation than the latter to the true ground state—a conclusion which is of course opposite to the established wisdom concerning this problem. It is shown that the pseudo-paradox is (unsurprisingly) resolved by a correct transcription of the two-body scattering theory to the many-body case; however, contrary to what appears to be a widespread belief, the resolution has nothing to do with any spurious ultraviolet divergences which result from the replacement of the true interatomic potential by a delta-function pseudopotential. Rather, it relates to an infrared divergence which has the consequence that (a) the most obvious form of the GP 'approximation' actually does not correspond to any well-defined ansatz for the many-body wavefunction, and (b) that the 'best shot' at such a wavefunction always produces an energy which exceeds, or at best equals, that calculated in the Bogoliubov approximation. In fact, the necessity of the latter may be seen as a consequence of the need to reduce the Fock term in the energy, which is absent in the two-particle problem but dominant in the many-body case; it does this by increasing the density correlations, at distances less than or approximately equal to the correlation length xi, above the value extrapolated from the two-body case. As a by-product I devise an alternative formulation of the Bogoliubov approximation which does not require the explicit replacement of the true interatomic potential by a delta-function pseudopotential.
Leggett Anthony J.
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