[Paper Review] Variational Methods for Path Integral Scattering

Julien Carron|arXiv (Cornell University)|Mar 2, 2009

Quantum and Classical Electrodynamics10 references3 citations

TL;DR

This master's thesis introduces a novel variational method for non-relativistic potential scattering using path integral representations of the T-matrix, applying the Feynman-Jensen variational principle to derive classical equations of motion in both real and complex trajectory forms. The method yields accurate approximations that recover leading and next-to-leading order terms of the eikonal expansion and show substantial numerical improvements over existing methods in challenging scattering regimes.

ABSTRACT

In this master thesis, a new approximation scheme to non-relativistic potential scattering is developed and discussed. The starting points are two exact path integral representations of the T-matrix, which permit the application of the Feynman-Jensen variational method. A simple Ansatz for the trial action is made, and, in both cases, the variational procedure singles out a particular one-particle classical equation of motion, given in integral form. While the first is real, in the second representation this trajectory is complex and evolves according to an effective, time dependent potential. Using a cumulant expansion, the first correction to the variational approximation is also evaluated. The high energy behavior of the approximation is investigated, and is shown to contain exactly the leading and next-to-leading order of the eikonal expansion, and parts of higher terms. Our results are then numerically tested in two particular situations where others approximations turned out to be unsatisfactory. Substantial improvements are found.

Motivation & Objective

• To develop a new variational approximation scheme for non-relativistic potential scattering using path integral formulations of the T-matrix.
• To apply the Feynman-Jensen variational principle to exact path integral representations, yielding classical equations of motion for the scattering trajectory.
• To evaluate the first correction to the variational approximation using a cumulant expansion and assess its high-energy behavior.
• To numerically test the method in cases where prior approximations fail, particularly for strong or rapidly varying potentials.
• To ensure unitarity and consistency with known expansions such as the eikonal approximation.

Proposed method

• The method starts from two exact path integral representations of the T-matrix, one using the eikonal representation and the other the ray representation.
• A trial action is introduced with a simple Ansatz, and the Feynman-Jensen variational principle is applied to derive a variational equation for the classical trajectory in integral form.
• In the eikonal representation, the trajectory is real and governed by a time-dependent effective potential; in the ray representation, the trajectory is complex and evolves under a modified potential.
• The first correction to the variational approximation is computed via a cumulant expansion, with explicit expressions derived for Gaussian potentials.
• High-energy asymptotics are analyzed, showing that the method reproduces the leading and next-to-leading terms of the eikonal expansion exactly.
• Numerical implementation uses adaptive integration (DCUHRE) and linear interpolation for variational trajectories, with analytical simplifications for the second cumulant in the ray representation.

Experimental results

Research questions

• RQ1 Can the Feynman-Jensen variational principle be successfully applied to path integral representations of the T-matrix in potential scattering?
• RQ2 Does the resulting variational approximation recover known high-energy limits such as the eikonal expansion?
• RQ3 How does the method perform numerically in regimes where standard approximations fail, such as strong or non-slowly varying potentials?
• RQ4 What is the structure of the first correction to the variational approximation, and how is it computed using cumulant expansions?
• RQ5 Can the method maintain unitarity and provide accurate differential cross sections in challenging scattering configurations?

Key findings

• The variational procedure yields a classical equation of motion in integral form, with real trajectory in the eikonal representation and complex trajectory in the ray representation.
• The high-energy expansion of the approximation reproduces exactly the leading and next-to-leading order terms of the eikonal expansion.
• Numerical tests show substantial improvements over existing approximations in cases where they previously failed, particularly for strong or rapidly varying potentials.
• The first correction to the variational approximation is computed via a cumulant expansion, with explicit analytical expressions derived for Gaussian potentials.
• Adaptive integration and interpolation techniques were essential for accurate evaluation of the second cumulant, especially in the ray representation, where standard Gauss-Legendre integration failed.
• An analytical approximation for the second cumulant, based on a linearized trajectory, showed good agreement with full numerical integration up to the point where adaptive integration failed.