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American Journal of Physics

General series solution for finite square-well energy levels for use in wave-packet studies

David L. Aronstein, Carlos R. Stroud

Published Oct 1, 2000• 40 citations
Publisher

Abstract

We develop a series solution for the bound-state energy levels of the quantum-mechanical one-dimensional finite square-well potential. We show that this general solution is useful for local approximations of the energy spectrum (which target a particular energy range of the potential well for high accuracy), for global approximations of the energy spectrum (which provide analytic expressions of reasonable accuracy for the entire range of bound states), and for numerical methods. This solution also provides an analytic description of dynamical phenomena; with it, we compute the time scales of classical motion, revivals, and super-revivals for wave-packet states excited in the well.

DOI
10.1119/1.1285868

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We develop a series solution for the bound-state energy levels of the quantum-mechanical one-dimensional finite square-well potential. We show that this general solution is useful for local approximations of the energy spectrum (which target a particular energy range of the potential well for high accuracy), for global approximations of the energy spectrum (which provide analytic expressions of reasonable accuracy for the entire range of bound states), and for numerical methods. This solution also provides an analytic description of dynamical phenomena; with it, we compute the time scales of classical motion, revivals, and super-revivals for wave-packet states excited in the well.
We present an analytical investigation of revival phenomena in the finite square-well potential. The classical motion, revival, and super-revival time scales are derived exactly for wave packets excited in the finite well. These time scales exhibit a richer dependence on wave-packet energy and on potential-well depth than has been found in other quantum systems: They explain, for example, the difficulties in exciting wave packets with strong classical features at the bottom of a finite well, or with clearly resolved super-revivals in a shallow well. In the proper regions of validity, the time scales predict the instances of wave-packet reformation extremely accurately. Revivals at the bottom of the well are explored as a ``universal'' limit of the general theory, which offers the clearest connection with the series of fractional and full revivals seen in the dynamics of the infinite square-well potential.
The quantum Finite Square Well (FSW) model is a well-known topic in most quantum mechanics (QM) books.A couple of equations can be derived from one dimensional Schrodinger equation for a finite potential well for describing the bound Eigen states within the well.Sometimes the FSW problem does not have an exact solution, yet, there are in fact exact solutions.In this work a computational techniques is adopted to find the exact solution for FSW.To achieve this computational solution, a computer program has been written in Basic language for calculating the Eigen state energy for a particle confined in finite potential well using the iterative method (IM).Six values of potential width have been studied with potential depth.The results showed that wider potential width led us to more bound states, while narrower potential width led us to less bound states.Six values of potential depth have been studied with potential width.The results showed that larger potential depth led us to more bound states, while smaller potential width led us to less bound states.A Comparison Between Finite and Infinite Potential Well has been also presented.The result of comparison showed that energy levels of an infinite well are much higher than that the corresponding energy levels for finite potential well.In general, the matching between the results of the iterative method and the graphical method (GM) proving that the iteration method can be regarded as a useful tool for describing the solutions of the 1-dimensional FSW problem.
The harmonic oscillator and the infinite square well are two of the simplest one-dimensional quantum systems which exhibit wave packet revivals (trivially so in the case of the oscillator.) These two potentials can be thought of as special cases of the general one-dimensional power-law potential given by V(k)(x)=V0|x/a|k (with k=2 and k→∞ for the oscillator and square well, respectively.) Using an autocorrelation function approach and the WKB approximation for the quantized energy levels in such potentials, we exhibit numerical evidence for wave packet revivals in the case of arbitrary k>0 which agree well with more analytic results. We derive expressions for the revival and collapse time scales in terms of the physical quantities of the system as well as the wave packet parameters. We find that both times scale with the power-law exponent k as |(k+2)/(k−2)|. In this way, we can explicitly exhibit the approach to the two familiar limiting cases. We also briefly consider the case of a “half” well where an infinite wall in added at the origin.
The computation of allowed energy levels for a particle bounded to a finite square well potential is ubiquitous in modern physics and introductory quantum mechanics textbooks. For stationary bound states, the matching conditions for the wave functions lead to a pair of transcendental equations whose roots correspond to the energy eigenvalues. However, the graphical solutions available do not make clear the dependence of the energies on the well potential parameters. In this note, I present a simpler graphical solution involving only one dimensionless parameter that determines a straight-line crossing identical sinusoidal curves. I then reduce this solution to a single cosine curve, and from a three-point interpolation, I derive an approximate formula for all energy levels valid for any square quantum well and that demands only a pocket calculator.
We have investigated quantum revivals of wave packets in a one-dimensional infinite square well potential containing a $\ensuremath{\delta}$ well in the middle. The time-dependent Schr\"odinger equation for this composite potential admits formally exact solutions. We present analytical results for revival properties in three physically motivated approximations: wave packets containing eigenstates with large numbers in the presence of an arbitrary $\ensuremath{\delta}$ well, ``shallow'' and ``deep'' $\ensuremath{\delta}$ wells. Analytical results in the case of a ``shallow'' $\ensuremath{\delta}$ well have been tested numerically.