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arXiv:1009.1918 · Canadian Journal of Physics

s-wave scattering and the zero-range limit of the finite square well in arbitrary dimensions

Aaron Farrell, Brandon P. van Zyl

Published Nov 1, 2010• 8 citations
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Abstract

We examine the zero-range limit of the finite square well in arbitrary dimensions through a systematic analysis of the reduced, s-wave, two-body, time-independent Schrödinger equation. A natural consequence of our investigation is the requirement of a delta function multiplied by a regularization operator to model the zero-range limit of the finite-square well when the dimensionality is greater than one. The case of two dimensions turns out to be surprisingly subtle, and needs to be treated separately from all other dimensions.

DOI
10.1139/p10-061

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We examine the zero-range limit of the finite square well in arbitrary dimensions through a systematic analysis of the reduced, s-wave, two-body, time-independent Schrödinger equation. A natural consequence of our investigation is the requirement of a delta function multiplied by a regularization operator to model the zero-range limit of the finite-square well when the dimensionality is greater than one. The case of two dimensions turns out to be surprisingly subtle, and needs to be treated separately from all other dimensions.
Abstract We discuss the procedure of solving the time-dependent Schrödinger equation via a separation ansatz that leads to the time-independent Schrödinger equation. The resulting equation is in the form of an eigenvalue problem for the Hamiltonian, whose solutions are the stationary states. We derive and discuss the bound states and corresponding energy levels for several one-dimensional potentials of interest, including the finite and infinite potential well, the Dirac-delta potential, and the square double-well modelling the spatial configuration of the ammonia molecule. We then turn to the scattered solutions and discuss reflection from and transmission through a finite potential barrier, thus encountering the tunnel effect, which we in turn formulate more generally using the Gamow factor. This leads us to the interesting phenomenon of tunnelling resonances
This paper presents several cases in which the effects of the addition of a delta function potential on bound states can be computed exactly. In the case of the one dimensional Schrödinger equation, a Green’s function technique is used to compute an exact implicit expression that gives the effect of the one−dimensional delta function potential on the bound states eigenvalues. It is also shown that the weak coupling limit of the implicit expression agrees with the perturbative treatment of the delta function. It is pointed out that this technique cannot be applied to obtain an exact treatment of the delta function potential δ3(r) in the three−dimensional Schrödinger equation. Instead, an alternate definition of the delta function in three dimensions as a limit of a sequence of square well potentials is used. Unlike the one dimensional case, the exact effect on the eigenvalue does not agree in the limit of weak coupling with the perturbative results. This indicates that the three−dimensional delta function potential should not be used to approximate strongly coupled short range potentials for which perturbation theory is inadequate.
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.