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Canadian Journal of Physics

Two-particle potential in field theory

D. Rogers, Mitchel D. Kovarik, Jurij W. Darewych

Published Nov 1, 1986• 1 citation
Publisher

Abstract

The many-particle binding energy in the model [Formula: see text] in 1 + 1 dimensions is obtained using simple nonrelativistic quantum mechanics. We determine an effective two-particle potential from approximate field-theoretical solutions of the two-particle bound state and use it to obtain the many-particle binding energy. The results agree reasonably well with numerical calculations of the three- and four-particle binding energy in the field theory for weak coupling, λ. We also use this nonrelativistic potential to calculate the two-particle scattering amplitude.

DOI
10.1139/p86-268

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The many-particle binding energy in the model [Formula: see text] in 1 + 1 dimensions is obtained using simple nonrelativistic quantum mechanics. We determine an effective two-particle potential from approximate field-theoretical solutions of the two-particle bound state and use it to obtain the many-particle binding energy. The results agree reasonably well with numerical calculations of the three- and four-particle binding energy in the field theory for weak coupling, λ. We also use this nonrelativistic potential to calculate the two-particle scattering amplitude.
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The auxiliary field method is a technique to obtain approximate closed formulae for the solutions of both nonrelativistic and semirelativistic eigenequations in quantum mechanics. For a many-body Hamiltonian describing identical particles, it is shown that the approximate eigenvalues can be written as the sum of the kinetic operator evaluated at a mean momentum p0p_0p0​ and of the potential energy computed at a mean distance r0r_0r0​. The quantities p0p_0p0​ and r0r_0r0​ are linked by a simple relation depending on the quantum numbers of the state considered and are determined by an equation which is linked to the generalized virial theorem. The (anti)variational character of the method is discussed, as well as its connection with the perturbation theory. For a nonrelativistic kinematics, general results are obtained for the structure of critical coupling constants for potentials with a finite number of bound states.
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Alternative lower bounds for the binding energy of a quantum-mechanical system of interacting particles are presented. These bounds are expressed in terms of two-particle quantities and improve the conventional bounds of the Hall-Post type. They are constructed by considering not only the energy in the two-particle system, but also the structure of the pair wave function. We apply the formal results to various numerical examples, and show that in some cases dramatic improvement over the existing bounds is reached.
A nonrelativistic model is investigated that is a certain generalization of quantum mechanics of two particles interacting with a potential. A new point interaction of the particles is introduced, which involves their transition into another particle that has the same quantum numbers as the bound state of the first particles. In the theory thus developed, the mass of the introduced particle and its interaction constant are arbitrary. In the particular case when the mass of the introduced particle is chosen so that its mass exactly equals that of the bound state of the first particles, the theory is identical with ordinary quantum mechanics. The renormalization constants in this case vanish. (auth)