A new path-integral representation of the T-matrix in potential scattering (original) (raw)
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Exact Path-Integral Representations for the T-Matrix in Nonrelativistic Potential Scattering
Few-Body Systems, 2010
Several path integral representations for the T-matrix in nonrelativistic potential scattering are given which produce the complete Born series when expanded to all orders and the eikonal approximation if the quantum fluctuations are suppressed. They are obtained with the help of "phantom" degrees of freedom which take away explicit phases that diverge for asymptotic times. Energy conservation is enforced by imposing a Faddeev-Popov-like constraint in the velocity path integral. An attempt is made to evaluate stochastically the real-time path integral for potential scattering and generalizations to relativistic scattering are discussed.
Variational approximations in a path integral description of potential scattering
The European Physical Journal A, 2010
Using a recent path integral representation for the T-matrix in nonrelativistic potential scattering we investigate new variational approximations in this framework. By means of the Feynman-Jensen variational principle and the most general ansatz quadratic in the velocity variables-over which one has to integrate functionally-we obtain variational equations which contain classical elements (trajectories) as well as quantum-mechanical ones (wave spreading). We analyse these equations and solve them numerically by iteration, a procedure best suited at high energy. The first correction to the variational result arising from a cumulant expansion is also evaluated. Comparison is made with exact partial-wave results for scattering from a Gaussian potential and better agreement is found at large scattering angles where the standard eikonal-type approximations fail.
Path integrals for potential scattering
Two path integral representations for the T matrix in nonrelativistic potential scattering are derived and proved to produce the complete Born series when expanded to all orders. They are obtained with the help of "phantom" degrees of freedom which take away explicit phases that diverge for asymptotic times. In addition, energy conservation is enforced by imposing a Faddeev-Popov-like constraint in the velocity path integral. These expressions may be useful for attempts to evaluate the path integral in real time and for alternative multiple scattering expansions. Standard eikonal-type high-energy approximations and systematic expansions immediately follow.
Variational Methods for Path Integral Scattering
2009
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.
Scattering theory with path integrals
Journal of Mathematical Physics, 2014
Starting from well-known expressions for the T-matrix and its derivative in standard nonrelativistic potential scattering, I rederive recent path-integral formulations due to Efimov and Barbashov et al. Some new relations follow immediately.
Functional integral method for potential scattering amplitude in quantum mechanics
2021
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.
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Communications in Mathematical Physics, 1998
We study the scattering problem for one-dimensional Schrödinger equations in the semiclassical limit when the energy level is close to the quadratic maxima of the potential. Starting from the formula of the scattering matrix obtained by the exact WKB method, we represent its principal term in a reduced form of the Feynman integral, an absolutely convergent sum of suitably defined probability amplitudes over countably many trajectories on R generated by classical trajectories and tunneling effects.
Path Integrals in Quantum Physics
Lecture Notes in Physics Monographs
These lectures are intended for graduate students who want to acquire a working knowledge of path integral methods in a wide variety of fields in physics. In general the presentation is elementary and path integrals are developed in the usual heuristic, non-mathematical way for application in many diverse problems in quantum physics. Three main parts deal with path integrals in non-relativistic quantum mechanics, manybody physics and field theory and contain standard examples (quadratic Lagrangians, tunneling, description of bosons and fermions etc.) as well as specialized topics (scattering, dissipative systems, spin & color in the path integral, lattice methods etc.). In each part simple Fortran programs which can be run on a PC, illustrate the numerical evaluation of (Euclidean) path integrals by Monte-Carlo or variational methods. Also included are the set of problems which accompanied the lectures and their solutions. Content 0. Contents First, an overview over the planned topics. The subsections marked by ⋆ are optional and may be left out if there is no time available whereas the chapters printed in blue deal with basic concepts. Problems from the optional chapters or referring to "Details" are marked by a ⋆ as well.
Path Integrals in Quantum Physics (English Version)
arXiv:1209.1315v4, 2017
These lectures are intended for graduate students who want to acquire a working knowledge of path integral methods in a wide variety of fields in physics. In general the presentation is elementary and path integrals are developed in the usual heuristic, non-mathematical way for application in many diverse problems in quantum physics. Three main parts deal with path integrals in non-relativistic quantum mechanics, many-body physics and field theory and contain standard examples (quadratic Lagrangians, tunneling, description of bosons and fermions etc.) as well as specialized topics (scattering, dissipative systems, spin \& color in the path integral, lattice methods etc.). In each part simple Fortran programs which can be run on a PC, illustrate the numerical evaluation of (Euclidean) path integrals by Monte-Carlo or variational methods. Also included are the set of problems which accompanied the lectures and their solutions.
2021
The Fourier transform of Cartesian Gaussian functions product is presented in the light of positron scattering. The calculation of this class of integrals is crucial in order to obtain the scattering amplitude in the first Born approximation framework for an ab initio method recently proposed. A general solution to the scattering amplitude is given to a molecular target with no restriction due to symmetry. Moreover, symmetry relations are presented with the purpose of identifying terms that do not contribute to the calculation for the molecules in the D∞h point group optimizing the computational effort. Keywords — Positron and electron scattering, Fourier transform of the Gaussian product theorem, McMurchie-Davidson procedure, Obara-Saika procedure, linear molecules .