Functional integral method for potential scattering amplitude in quantum mechanics

TL;DR

This paper introduces a functional integral approach to calculate potential scattering amplitudes in quantum mechanics, deriving the Green function and cross-sections for specific potentials like Yukawa and Gaussian.

Contribution

It presents a novel functional integral method to obtain scattering amplitudes from the Green function, including high-energy eikonal approximations and specific potential cases.

Findings

Derived the scattering amplitude form from the Green function.

Obtained differential cross-sections for Yukawa and Gaussian potentials.

Applied the method in high-energy, small-angle scattering regimes.

Abstract

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.