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SIAM Journal on Applied Mathematics

Analytical Treatment of a Periodic $\delta $-Function Potential in the Path-Integral Formalism

Marc J. Goovaerts, Fernand Broeckx

Published Jun 1, 1985• 8 citations
Publisher

Abstract

A method for evaluating the density matrix, written as a path-integral, is proposed and applied to a periodic delta function potential. It is the first time that a model which gives rise to a band structure is analytically treated in the framework of the Feynman path-integral formalism. It is shown that the results for the energy spectrum and the wave functions completely coincide with those deduced in the Schrödinger formalism.

DOI
10.1137/0145028

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A method for evaluating the density matrix, written as a path-integral, is proposed and applied to a periodic delta function potential. It is the first time that a model which gives rise to a band structure is analytically treated in the framework of the Feynman path-integral formalism. It is shown that the results for the energy spectrum and the wave functions completely coincide with those deduced in the Schrödinger formalism.
An expansion method in the path-integral formulation of quantum mechanics, as proposed in a previous paper, is extended to account for states with odd parity. The method is tested on the case of a one-dimensional attractive delta-function potential and the well-known solutions for both bound and scattering states are obtained analytically by an exact summation of the expansion series. The applicability of the method is discussed and it is shown how the energy spectrum could be determined by means of Stieltjes theory of moments.
A straightforward analytical calculation of the s-like energy spectrum of the hydrogen atom is performed entirely within Feynman's path integral formalism. For this purpose the integral transform W = ∫d3rβK(rβ,0), where K(rβ,0) is the density matrix of the hydrogen atom written as a path integral, is calculated by means of the exact summation of a ``modified'' perturbation expansion (W is expanded as a power series in β with β = 1/kT). Performing this summation is equivalent to solving a problem of moments with infinite moments. For a wide class of potentials the perturbation expansion for W converges faster than the power-series expansion for the exponential function (for the Coulomb potential the convergence rate of both expansions is the same). It is shown how the complete energy spectrum can be obtained by this method. It is also illustrated how the wavefunctions might be obtained by transforming W.
A particle in a one-dimensional \ensuremath{\delta}-function potential possesses both discrete and continuum solutions. The configuration-space Green's function and propagator for this problem are derived by explicit summation over the spectrum of eigenstates. The momentum-space Green's function is also obtained. The propagator does not contain the classical action function in any simple way, in contrast to the usual structure in Feynman's path-integral formalism. Various analogies between the \ensuremath{\delta}-function and Coulomb problems are discussed.
A motion of a Gaussian wave packet of a single electron in a harmonic potential is obtained by solving the time dependent Schrödinger equation analytically. The analytic formula of the phase of the wave function which depends on both time and space is also presented explicitly. The probability density of the wave packet moves changing its width and central position periodically. The period of the change of the width is half of that of the central position. The interpretation of these periodic changes in terms of the Green's function which can be derived by the path integration formulation is also discussed.