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تسجيل مستخدم جديدTwo Particles in a Trap
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H. S. Kohler
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The Busch-formula relates the energy-spectrum of two point-like particles interacting in a 3-D isotropic Harmonic Oscillator trap to the free scattering phase-shifts of the particles. This formula is used to find an expression for the it shift rm in the spectrum from the unperturbed (non-interacting) spectrum rather than the spectrum itself. This shift is shown to be approximately $Delta=-delta(k)/pitimes dE$, where $dE$ is the spacing between unperturbed energy levels. The resulting difference from the Busch-formula is typically 1/2% except for the lowest energy-state and small scattering length when it is 3%. It goes to zero when the scattering length $rightarrow pm infty$. The energy shift $Delta$ is familiar from a relatedproblem, that of two particles in a spherical infinite square-well trap of radius $R$ in the limit $Rrightarrow infty$. The approximation ishowever as large as 30% for finite values of $R$, a situation quite different from the Harmonic Oscillator case. The square-well results for $Rrightarrow infty$ led to the use ofin-medium (effective) interactions in nuclear matter calculations that were $propto Delta$ and known as the it phase shift approximation rm.Our results indicate that the validity of this approximation depends on the trapitself, a problem already discussed by DeWitt more than 50 years ago for acubical vs spherical trap.
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