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American Journal of Physics

Off-the-mass shell scattering amplitude in a two-particle potential model

Francisco Antônio Bezerra Coutinho

Published Jan 1, 1982• 0 citations
Publisher

Abstract

In the textbooks of elementary quantum mechanics, two-particle scattering is treated by introducing the concept of off-the-energy shell scattering amplitude. On the other hand, in relativistic quantum field theory, the need to treat momentum and energy on the same footing forces one to use an off-the-mass shell scattering amplitude. We believe the introducing the off-the-mass shell extrapolation of the scattering amplitude in nonrelativistic quantum mechanics should be illuminating. We try to show this in this paper.

DOI
10.1119/1.12985

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In the textbooks of elementary quantum mechanics, two-particle scattering is treated by introducing the concept of off-the-energy shell scattering amplitude. On the other hand, in relativistic quantum field theory, the need to treat momentum and energy on the same footing forces one to use an off-the-mass shell scattering amplitude. We believe the introducing the off-the-mass shell extrapolation of the scattering amplitude in nonrelativistic quantum mechanics should be illuminating. We try to show this in this paper.
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.
The functional integral method can be used in quantum mechanics to find the scattering amplitude for particles in the external field. We obtained the potential scattering amplitude from the complete Green function in the corresponding external field through solving the Schrodinger equation, after being separated from the poles on the mass shell, which takes the form of an eikonal (Glauber) representation in the high energy region and the small scattering angles. Considering specific external potentials such as the Yukawa or Gaussian potential, we found the corresponding differential scattering cross-sections.
Abstract Potential scattering is the scattering of two collision partners whose mutual interaction is described by a static potential. This potential depends in general on the structure of the collision partners, in particular, on their size and shape (usually taken as spherical), but once the potential is specified and the centre-of-mass motion is separated out, the scattering problem in the remaining relative coordinate reduces to the description of the motion of a point-particle with a mass equal to the reduced mass of the two collision partners in the static field of the interaction potential. Only elastic scattering can be described in this way.
Momentum space Ward identities are derived for the amputated n-point Green’s functions in 3 + 1 dimensional non-relativistic conformal field theory. For n = 4 and 6 the implications for scattering amplitudes (i.e. on-shell amputated Green’s functions) are considered. Any scale invariant 2-to-2 scattering amplitude is also conformally invariant. However, conformal invariance imposes constraints on off-shell Green’s functions and the three particle scattering amplitude which are not automatically satisfied if they are scale invariant. As an explicit example of a conformally invariant theory we consider non-relativistic particles in the infinite scattering length limit. 1 Poincaré invariant theories that are scale invariant usually have a larger symmetry group called the conformal group1. A similar phenomena happens for 3+1 dimensional nonrelativistic systems. These are invariant under the extended Galilean group, which consists of 10 generators: translations (4), rotations (3), and Galilean boosts (3). The largest space-time symmetry group of the free Schrödinger equation is called the Schrödinger or nonrelativistic
The functional integral method can be used in quantum mechanics to find the scattering amplitude for particles in the external field. We will obtain the potential scattering amplitude form the complete Green function in the corresponding external field through solving the Schrodinger equation, after being separated from the poles on the mass shell, which takes the form of an eikonal (Glauber) representation in the high energy region and the small scattering angles. Consider specific external potentials such as the Yukawa or Gaussian potential, we will find the corresponding differential scattering cross-sections.