The Harmonic Oscillator Approximation to the Density Matrix

Abstract

<p>We derive an approximate form for the angle averaged single particle density matrix for particles in a central potential. It is a Taylor series expansion in the interparticle distance |S| based on the exact density matrix for particles in a harmonic oscillator potential. This form is also shown to be exact for particles in a linear and quadratic potential in one dimension.</p> <p>We show the relation between this approximation and previous ones derived by Negele and Vautherin, Campi and Bouyssy, and Jennings. All these approximations agree with the exact angle averaged density matrix up to the S² coefficient of their respective Taylor series expansion in |S|. We compare the coefficients of S⁴ and higher powers of S within each approximation to show the connections between them.</p> <p>We then check, numerically, the S⁴ coefficients of the various approximations with the exact coefficient. In one dimension we use the Eckart potential and in three dimensions we use the Woods Saxon. All three give reasonably good approximations to the exact S⁴ coefficient, with the exception of the Negele Vautherin form in one dimension.</p> <p>Suggestions for putting the harmonic oscillator approximation into a form amenable to calculating exchange integrals are made.</p>