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American Journal of Physics

Equivalence of the δ′ and the Hard Core Potentials

R.E. Schenter, Harry S. Zwibel, R. L. Cassola

Published Dec 1, 1969• 5 citations
Publisher

Abstract

The introduction of the hard core potential into the two-body problem forces the system wave function to vanish, if the interacting particles approach each other more closely than the hard core radius. In this paper it is shown that this “hard core effect” can also be obtained in the two-body problem by replacing the hard core potential with a δ′ interaction of suitable strength.

DOI
10.1119/1.1975289

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Hard-core potentials in quantum mechanics

B. W. Downs, Budh Ram

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American Journal of Physics•Feb 1, 1978•4 citations
The introduction of the hard core potential into the two-body problem forces the system wave function to vanish, if the interacting particles approach each other more closely than the hard core radius. In this paper it is shown that this “hard core effect” can also be obtained in the two-body problem by replacing the hard core potential with a δ′ interaction of suitable strength.
A new method is proposed for treating the hard core potential with a small core radius a. This is an alternative to the method based on Fermi's pseudopotential. Various shortcomings of the latter method are removed with the present method. These shortcomings include the penetrability of hard spheres and incompatibility with the completeness of the unperturbed states, and they result in the invalidity of the pseudopotential method even in applications to effects that are of low order in powers of a. The present method is developed taking account of the three-particle correlation, which is a higher-effect plagued by problems of divergence and the nonanalyticity of the relevant energy as a function of the potential strength. Treatment of the three-particle correlation requires a careful preliminary study of the two-particle interaction in many-body systems, particularly when a hard core potential is involved. In this report, we present basic ideas and obtain improvement over the methods appearing in the literature.
Abstract The wave mechanics of two impenetrable hard core particles in a 1-D box is analyzed. Each particle in the box behaves like an independent entity represented by a macro-orbital (a kind of pair waveform). While the expectation value of their interaction, 〈 V HC(x) 〉, vanishes for every state of two particles, the expectation value of their relative separation, 〈 x 〉, satisfies 〈 x 〉≥λ/2 (or q ≥ π/d, with 2d=L being the size of the box). The particles in their ground state define a close-packed arrangement of their wave packets (with 〈 x 〉= λ/2, phase position separation Δϕ = 2π and momentum |q o| = π/d) and experience a mutual repulsive force (zero point repulsion) f o=h 2/2md 3 which also tries to expand the box. While the relative dynamics of two particles in their excited states represents usual collisional motion, the same in their ground state becomes collisionless. These results have great significance in determining a correct microscopic understanding of widely different many-body systems.
The method of many-body point transforms previously developed by the author is modified to eliminate the interacting particle cluster formalism. The cutoff terms in the transformation, and thus in the Hamiltonian, are therefore removed. For the hard-core many-body problem, as in the previous work, a Hamiltonian is obtained which is Fourier analyzable, Hermitian and amenable to ordinary perturbation and variational techniques while still being equivalent to the original Hamiltonian. No approximations are made. It is demonstrated that the use of the standard zeroth-order approximation of the ground state Bose system, i.e., a constant, for the transformed wavefunction gives a negative expectation value of the energy for the hard core system. This discrepancy arises from the extended range of the new potential generated by the transformation, which necessitates explicit consideration of boundary conditions not satisfied by the above wavefunction.