Approximating the finite square well with an infinite well: Energies and eigenfunctions

Abstract

Polynomial expansions are used to approximate the equations of the eigenvalues of the Schrödinger equation for a finite square potential well. The technique results in discrete, approximate eigenvalues which, it is shown, are identical to the corresponding eigenvalues of a wider, infinite well. The width of this infinite well is easy to calculate; indeed, the increase in width over that of the finite well is simply the original width divided by the well strength. The eigenfunctions of this wider, infinite well, which to first order has the same width for the ground state and all excited states, are also good approximations to the exact eigenfunctions of the finite well. These approximate eigenfunctions and eigenvalues are compared to accurate numeric calculations and to other approximations from the literature.

The static electric polarizability of a particle bound by a finite potential well

“…Due to this a variety of useful approximations had been developed to replace the conventional method. [11][12][13] In Ref.…”

Superrevivals in the quantum dynamics of a particle confined in a finite square-well potential

“…The formula of Barker et al is more accurate for tightly bound eigenstates (deeper wells) than for weakly bound states (shallow wells) [14] The revival features for the dynamics of any initial state can be understood by examining the absolute square of the autocorrelation function: Table I. Thus for short times the dynamics of the wavepacket in the finite well is similar to that in the infinite well with modified revival times which depend on the depth of the well.…”

“…The revival times, however, are in general longer than that of the infinite well and depend on the well strength (depth), ǫ, of the finite well. Barker et al [14] have shown via a first order Taylor series expansion of the transcendental equations (8) and (10) for the eigenvalues that the energy levels of a finite well of length L and well strength ǫ can be approximated as:…”

“…as the 'well-strength' [14]. A finite well of well strength ǫ would contain a finite number of bound states, N , where…”

Comment on “On the energy levels of a finite square-well potential” [J. Math. Phys. 41, 4551 (2000)]

“…The asymptotic limit of the energy spectrum, Eq. ͑4͒, and its interpretation in terms of the infinite square-well spectrum were addressed in detail by Barker et al 2 Calculations of the ''effective length'' or ''tunneling length'' of the finite square-well energy eigenstates beyond the well boundaries at xϭϮL/2 were given by Garrett 3 and Rokhsar. 4 To our knowledge, the work by Paul and Nkemzi is the first to derive the finite-well energy levels in an asymptotic expansion in powers of 1/p; however, power series expansions in the quantum number k have been developed by JOURNAL OF MATHEMATICAL PHYSICS VOLUME 41, NUMBER 12 DECEMBER 2000…”