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arXiv:0903.0273

Variational Methods for Path Integral Scattering

Julien Carron

Published Mar 2, 2009• 5 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.0903.0273

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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.
Abstract.: 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
Two path integral representations for the TTT 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.
A new formulation of nonrelativistic scattering theory is developed which expresses the SSS matrix as a path integral. This formulation appears to have at least two advantages: (1) A closed formula is obtained for the SSS matrix in terms of the potential, not involving a series expansion; (2) the energy-conserving $\ensuremath{\delta}$ function can be explicitly extracted using a technique analogous to that of Faddeev and Popov, thereby yielding a closed path-integral expression for the TTT matrix. The introduction of the concept of the classical interaction picture provides considerable physical insight into this formulation. This formulation also suggests a successionof improvements to the eikonal approximation, the first of which is discussed explicitly.
Previous work on the application to scattering theory of the Feynman path-integral formulation of nonrelativistic quantum mechanics is extended. By using the regularization $\mathrm{Tr}({e}^{\ensuremath{-}\ensuremath{\beta}H}\ensuremath{-}{e}^{\ensuremath{-}\ensuremath{\beta}{H}^{0}})$ of the divergent partition function $\mathrm{Tr}{e}^{\ensuremath{-}\ensuremath{\beta}H}$, where H0{H}^{0}H0 is the free-particle Hamiltonian, H=H0+VH={H}^{0}+VH=H0+V, and VVV is a short-range potential of the form $V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ which cannot support a bound state, we obtain a variational upper bound on the scattering length. This bound is expressed in terms of path integrals containing a trial functional $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}$, and reduces to the natural analog of the Rayleigh-Ritz result when $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}$ is of the standard form $\ensuremath{-}\ensuremath{\int}\frac{[{E}_{K}+\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}]dt}{\ensuremath{\hbar}}$, with EK{E}_{K}EK​ the kinetic energy and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ a trial potential. Since $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}$ need not be of this form, the bound contains new features.
New nonperturbative methods for dealing with Hamiltonian systems are introduced. The derivation of these methods requires identities for rewriting exponentials of sums of operators which are different from the usual Campbell-Hausdorff formula. These identities allow one to derive approximations to ${e}^{\ensuremath{-}\ensuremath{\delta}H}$ which are correct to higher order in $\ensuremath{\delta}$ and which contain fewer terms than the Campbell-Hausdorff formula. This allows one to generate path-integral actions which are more accurate for finite-size steps in time and which can be exploited to improve the rate of convergence of Monte Carlo calculations. To show that these methods allow one to include effects which show up in the stationary-phase approximation to the path integral, e.g., instantons, solitons, etc., I not only derive the connection between the Hamiltonian and path-integral formalism but the relationship between a specific stationary-phase approximation and the corresponding Hamiltonian calculation. My focus, however, is the direct application of these new identities to the study of nonperturbative Hamiltonian dynamics. I show that these methods are easier to apply and give better results than those based upon the naive ttt-expansion, block mean-field, or real-space renormalization ideas. Comparison of these older methods with the computational tools introduced in this paper are discussed in the context of simple examples. It is shown that the new methods allow one to extrapolate answers to finite ttt without the use of Pad\'e approximants.
This chapter shows how scattering problems are formulated in the framework of path integrals. In quantum mechanics, the state of an isolated system evolves under the action of a unitary operator, as a consequence of the conservation of probabilities and, thus, of the norm of vectors in Hilbert space. Quantum evolution (that is, in real time) is introduced, after which a path integral representation of the scattering matrix is constructed. From this S matrix, the standard perturbative expansion in powers of the potential is recovered. Even the evolution of a free quantum particle is slightly non-trivial; in general, one observes a spreading of wave packets. Scattering is then characterized by the asymptotic deviations at infinite time from this free evolution and this leads to the definition of a scattering or S-matrix. An S-matrix is defined in the example of bosons and fermions. Various other semi-classical approximation schemes are then discussed.