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Journal of Mathematical Physics

Time-dependent scattering theory for infinite delta function potentials in one dimension

John D. Dollard

Published Apr 1, 1978• 3 citations
Publisher

Abstract

Existence of the Mo/ller wave operators is proved for a system of n quantum mechanical particles interacting through infinite delta function potentials in one dimension.
DOI
10.1063/1.523725

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A general framework is set up for the description of quantum-mechanical scattering processes for the case that both the free hamiltonian H 0 (t)=H 0 +H 1 (t) and the full hamiltonian H(t)=H 0 (t)+V are time-dependent. In particular the existence of the various evolution operators is considered and the wave operators Omega +or- (s), which now depend on the initial time s, are defined. The consequences of a periodic time dependence are studied and for H 1 (t) of the type H 1 (t)=H 1 cos( omega t+ delta ) it is shown that for mod omega mod to infinity the frequency-dependent wave operators converge to the wave operators which pertain to the case that H 1 (t) is absent.
A phase-shift analysis of quantum mechanical scattering problems in one dimension is developed. Many of the techniques and results of three dimensions remain even in one dimension, while the discussion is vastly simplified, and thus accessible to the student much earlier, due to the absence of special functions (in particular, Bessel and Legendre functions) with their complicated asymptotic formulas and orthogonality relations. A “partial wave” analysis leads to the familiar relation fl∼eiδl sinδl between the scattering amplitude in the lth partial wave and the phase shift δl. The optical theorem is evident as an immediate consequence of the conservation of probability. The method is applied to an analysis of the well-known square-well scattering resonances.
The problem of NNN particles interacting in one dimension by a repulsive delta-function potential is solved, when the wave function $\ensuremath{\psi}$ transforms like any irreducible representation ${R}_{\ensuremath{\psi}}$ of SN{S}_{N}SN​ for which the Young tableau consists of a finite number of either rows or columns.
Analytical solutions to the time-dependent Shrödinger equation in one dimension are developed for time-independent potentials, one consisting of an infinite wall and a repulsive delta function. An exact solution is obtained by means of a convolution of time-independent solutions spanning the given Hilbert space with appropriately chosen spectral functions. Square-integrability and the boundary conditions are satisfied. The probability for the particle to be found inside the potential well is calculated and shown to exhibit non-exponential decay decreasing at large times as t−3t^{-3}t−3. The result is generalized for all square-integrable solutions to this problem.