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Europhysics Letters (EPL)

Pseudopotential method and dilute hard “sphere” Bose gas in dimensions 2, 4 and 5

Chen Ning Yang

Published Oct 16, 2008• 13 citations
Publisher

Abstract

The ground-state energy for a dilute hard sphere Bose gas in various dimensions is studied theoretically through the pseudopotential method. Earlier analysis has not been mathematically rigorous so the current treatment uses new technologies and powerful computers. The 1D and 2D systems have been studied experimentally and qualitative and quantitative features exhibited for these and other dimensions.

DOI
10.1209/0295-5075/84/40001

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arXiv:math-ph/0412080•Physical Review A•May 13, 2005•54 citations
The ground-state energy for a dilute hard sphere Bose gas in various dimensions is studied theoretically through the pseudopotential method. Earlier analysis has not been mathematically rigorous so the current treatment uses new technologies and powerful computers. The 1D and 2D systems have been studied experimentally and qualitative and quantitative features exhibited for these and other dimensions.
The method of point transformations is used to provide a Hamiltonian for strongly interacting particles that is equivalent to the usual Hamiltonian. The new Hamiltonian is Hermitian, Fourier analyzable, velocity dependent, and free of the difficulties of pseudopotentials. The present work is restricted to the case that two-body encounters are dominant. The theory is expressed in the quantized wave formalism so that number-violating approximations can be applied. The well known results for the spectrum and ground-state energy of the dilute hard-sphere gas are then derived as an elementary application.
Recent developments in the physics of low-density trapped gases make it worthwhile to verify old, well-known results that, while plausible, were based on perturbation theory and assumptions about pseudopotentials. We use and extend recently developed techniques to give a rigorous derivation of the asymptotic formula for the ground-state energy of a dilute gas of NNN fermions interacting with a short-range, positive potential of scattering length aaa. For spin-1∕21∕21∕2 fermions, this is $E\ensuremath{\sim}{E}^{0}+({\ensuremath{\hbar}}^{2}∕2m)2\ensuremath{\pi}N\ensuremath{\varrho}a$, where E0{E}^{0}E0 is the energy of the noninteracting system and $\ensuremath{\varrho}$ is the density. A similar formula holds in two dimensions (2D), with $\ensuremath{\varrho}a$ replaced by $\ensuremath{\varrho}∕\ensuremath{\mid}\mathrm{ln}(\ensuremath{\varrho}{a}^{2})\ensuremath{\mid}$. Obviously this 2D energy is not the expectation value of a density-independent pseudopotential.