Journal of Physics B Atomic Molecular and Optical Physics

Finite matrix computations applied to the scattering of a wavepacket in the Coulomb potential

Ue‐Li Pen, Tsin-Fu Jiang

Published Feb 28, 1995• 3 citations

Abstract

We develop a finite dimensional matrix method to treat wavepacket propagation in potential scattering problems. The method is useful for general quantum mechanical problems. The scattering dynamics of a minimum uncertainty wavepacket in the Coulomb potential is presented.

DOI: 10.1088/0953-4075/28/4/001

Related papers

Wavepacket Scattering Theory for the Coulomb Potential

Scott E. Hoffmann

We treat the nonrelativistic quantum mechanical scattering of a wavepacket (for a single, charged, spinless particle) of well-resolved momentum incident on a fixed Coulomb potential. To implement this, we calculate the incoming and outgoing probability amplitudes for normalized state vectors on the basis of eigenvectors of total momentum magnitude and total angular momentum. We show that the method of partial wave analysis becomes applicable for this long range potential with the use of wavepackets. We find a relation between the finite probability and the differential cross section that is seen to hold sufficiently far from the singularity of the latter in the forward direction. The resulting differential cross section reduces to the familiar Rutherford form for scattering angles sufficiently far from the forward direction. We find that the interaction causes small time shifts in the detection probabilities. For our particular scattering geometry, we see, at low energies, a shadow zone of low probability for small angles around the forward direction, a phenomenon that cannot be predicted by the standard treatment.

Apr 14, 20170 citations

On the classical and quantum Coulomb scattering

Dimitri R. Yafaev

We develop the scattering theory for the Schrödinger operator with the Coulomb potential in the space of an arbitrary dimension d. In particular, we calculate the scattering matrix and show that its spectrum covers the whole unit circle. We also compute the differential cross section and show that it coincides with the classical Coulomb scattering cross section in the dimension d = 3 only.

Journal of Physics A Mathematical and GeneralOct 7, 199721 citations

The method of amplitude functions in two-dimensional scattering theory

Vasily Veniaminovitch Pupyshev

We present a formulation and mathematical justification of the method of amplitude functions. This method allows solving the radial problem of the two-dimensional scattering of a quantum particle by the sum of a Coulomb potential and a certain short-range or long-range central potential.

Theoretical and Mathematical PhysicsApr 1, 20176 citations

Classical orbits and semiclassical wavepacket propagation in the Coulomb potential

I. M. Suarez Barnes, Michael Nauenberg, Mark Douglas Nockleby +1 more

A simple and elegant semiclassical theory for propagating wavepackets is presented. The development of the theory avoids the explicit use of the Green-function formalism. It involves expanding the potential to a quadratic time-dependent form and the use of multiple reference trajectories. The classical analogue in phase space predicts the accuracy of the results. The theory is applied to the calculation of the autocorrelation function in a Coulomb field in one dimension. Remarkable agreement with an exact quantum calculation is obtained in many circumstances in spite of the Coulomb singularity. All the detailed quantum behaviour, including the long-time recurrences and the spectrum, is reproduced.

Journal of Physics A Mathematical and GeneralMay 7, 199420 citations

Propagator for finite range potentials

Ilaria Cacciari, Paolo Moretti

The Schrödinger equation in integral form is applied to the one-dimensional scattering problem in the case of a general finite range, nonsingular potential. A simple expression for the Laplace transform of the transmission propagator is obtained in terms of the associated Fredholm determinant, by means of matrix methods; the particular form of the kernel and the peculiar aspects of the transmission problem play an important role. The application to an array of delta potentials is shown.

Journal of Mathematical PhysicsDec 1, 20066 citations

Computational method for the quantum Hamilton-Jacobi equation: One-dimensional scattering problems

Chia‐Chun Chou, Róbert E. Wyatt

One-dimensional scattering problems are investigated in the framework of the quantum Hamilton-Jacobi formalism. First, the pole structure of the quantum momentum function for scattering wave functions is analyzed. The significant differences of the pole structure of this function between scattering wave functions and bound state wave functions are pointed out. An accurate computational method for the quantum Hamilton-Jacobi equation for general one-dimensional scattering problems is presented to obtain the scattering wave function and the reflection and transmission coefficients. The computational approach is demonstrated by analysis of scattering from a one-dimensional potential barrier. We not only present an alternative approach to the numerical solution of the wave function and the reflection and transmission coefficients but also provide a computational aspect within the quantum Hamilton-Jacobi formalism. The method proposed here should be useful for general one-dimensional scattering problems.

Physical Review EDec 19, 200637 citations