( a) the effective interaction experienced by a particle is small, even though

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the interparticle potential may have large values;

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(b) the details of the interparticle potential are unimportant, because a

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particle that is spread out in space sees only an averaged effect of the potential.

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In the quantum theory of scattering it is known that at low energies the

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scattering of a particle by a potential does not depend on the shape of the potential, but depends only on a single parameter obtainable from the potential -the scattering length a. The total scattering cross section at low energies is 4'1Ta 2 Hence roughly speaking a is the effective diameter of the potential. We may also say that at low energies the scattering from a potential looks like that from a hard sphere of diameter a. This makes it plausible that at extremely low temperatures it is possible to describe an imperfect gas solely in terms of the three parameters A, v1/ 3 , and a. Our problem is to formulate a method by which all the thermodynamic functions of the imperfect gas can be obtained to lowest order in the small parameters a/A and a/vI/3 . We first show that, for the purpose of calculating the low-lying energy levels of an imperfect gas, the Hamiltonian of the system may be replaced by an effective Hamiltonian in which only scattering parameters, such as the scattering length, appear explicitly. The partition function of the imperfect gas can then be calculated with the help of the effective Hamiltonian. This method, first introduced by Fermi,* is known as the method of pseudopotentials. Consider first a system of two particles interacting through a finite-ranged potential which has no bound state. The object of the method of pseudopotentials is to obtain all the energy levels of the system in terms of the scattering phase shifts of the potential. For the sake of concreteness we first assume that the potential is the hard-sphere potential with diameter a. The wave function for the two particles may be written in the form (10.100)

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"E. Fermi, Ricerca Sci. 7, 13 (1936). Our presentation follows that of K. Huang and C. N. Yang, Phys. Rev. 105,767 (1957).

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STATISTICAL MECHANICS

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where

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Hrl + r)

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r=r 2 - rl

(10.101)

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and P is the total momentum vector. The Schrodinger equation in the center-ofmass system is

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+ k 2 )l/;(r)

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t/;(r) = 0

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(r> a) (r.$;a)

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(10.102)

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The hard-sphere potential is no more than a boundary condition for the relative wave function t/;(r). It is understood that some boundary condition for r --+ 00 is specified, but what it is is irrelevant to our considerations. The number k is the relative wave number, and (10.102) presents an eigenvalue problem for k. When the allowed values of k are known, the energy eigenvalues of the system are given by p2 E(P,k) = 2M + where M is the total mass and JL the reduced mass of the system. The aim of the method of pseudopotentials is to replace the hard-sphere boundary condition by an inhomogeneous term for the wave equation. Such an idea is familiar in electrostatics, where to find the electrostatic potential in the presence of a metallic sphere (with some given boundary condition at infinity) we may replace the sphere by a distribution of charges on the surface of the sphere and find the potential set up by the fictitious charges. We can further replace the surface charges by a collection of multipoles at the center of the sphere with appropriate strengths. If we solve the Poisson equation with these multipole sources, we obtain the exact electrostatic potential outside the sphere. In an analogous way, the method of pseudopotentials replaces the boundary condition on t/;(r) by a collection of sources at the point r = O. Instead of producing electrostatic multipole potentials, however, these sources will produce scattered 8 waves, p waves, D waves, etc. Let us first consider spherically symmetric (8 wave) solutions of (10.102) at very low energies (k --+ 0). The equations (10.102) become

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:(r2:~)

t/;(r)

(r> a) (r.$;a)

(10.103)