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Journal of Mathematical Physics

Square-well representations for potentials in quantum mechanics

Richard L. Hall

Published Oct 1, 1992• 6 citations
Publisher

Abstract

A single particle moves in one dimension and obeys nonrelativistic quantum mechanics. If the potential shape f(x) is symmetric and vanishes at infinity, then, without loss of generality, it can be represented in the form of a mixture f(x)=∫0∞A(t)w(x/t)dt of square wells w(x). It is shown that A(t)=f′(t) and that the discrete eigenvalues of the Schrödinger operator H=−D2+vf(x) are given approximately by the general formula En≂EAn=mins≳0[s+v∫0∞f′(t)w̄n(st2)dt], where {w̄n(s)}n=0∞ are the known ‘‘kinetic potentials’’ associated with the simple square-well problem. For n=0 or 1, the approximations EnA are lower bounds to the corresponding exact energies En. The general formula is applied to the sech-squared potential and to the truncated quartic oscillator.

DOI
10.1063/1.529896

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A single particle moves in one dimension and obeys nonrelativistic quantum mechanics. If the potential shape f(x) is symmetric and vanishes at infinity, then, without loss of generality, it can be represented in the form of a mixture f(x)=∫0∞A(t)w(x/t)dt of square wells w(x). It is shown that A(t)=f′(t) and that the discrete eigenvalues of the Schrödinger operator H=−D2+vf(x) are given approximately by the general formula En≂EAn=mins≳0[s+v∫0∞f′(t)w̄n(st2)dt], where {w̄n(s)}n=0∞ are the known ‘‘kinetic potentials’’ associated with the simple square-well problem. For n=0 or 1, the approximations EnA are lower bounds to the corresponding exact energies En. The general formula is applied to the sech-squared potential and to the truncated quartic oscillator.
A simple procedure has been found for the general solution of the time-independent Schrödinger equation (SE) with the help of quantization of potential area in one dimension without making any approximation. Energy values are not dependent on wave functions. So, to find the energy values, it is enough to find the classic turning points of the potential function. Two different solutions were obtained, namely, symmetric and antisymmetric in bound states. These normalized wave functions are always periodic. It is enough to take the integral of the square root of the potential energy function to find the normalized wave functions. If these calculations cannot be made analytically, they should then be performed by numerical methods. The SE has been solved for a particle in many one-dimension and the spherical symmetric central potential wells as examples. Their energies and normalized wave functions were found as examples. These solutions were also applied to the theories of scattering and alpha decay. The results obtained with the experimental values were compared with the calculated values. One was seen to be very fit.
The ground-state energies and wave functions of a particle moving in one-dimensional potentials of the form V(x)=x2p, p=1,2,..., are obtained nearly exactly by iterative solution of the Schrödinger equation. These solutions are used as the ‘‘experimental benchmark’’ against which approximate solutions for the general case are tested. Both finite depth wells and infinitely deep wells are studied, as is the three-dimensional version (spherical symmetry, angular momentum quantum number l=0). The approach is aimed at student exploration of the physics of the family of potential wells, with emphasis upon connecting the familiar extreme cases, V(x)=x2 (simple harmonic oscillator) and the limiting case xk, k→∞ (square well).
It is well known that quantum mechanics allows the penetration into classically forbidden regions (tunneling). Less well known seems to be the fact that in some sense the converse is true also. Potentials with classically allowed regions where a particle can move freely to infinity can nevertheless lead to bound states in quantum mechanics due to the stringent requirements of the boundary conditions, thus forbidding an escape to infinity. This effect is demonstrated by using an obvious generalization of the well-known one-dimensional (D=1) square well potential to arbitrary space dimensions.
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.
A simple procedure has been found for the general solution of the time-independent Schrödinger Equation (SE) with the help of quantization of potential area in one dimension without making any approximation.Energy values are not dependent on wave functions. So, to find the energy values, it is enough to find the classic turning points of the potential function. Two different solutions were obtained, namely, symmetric and anti symmetric in bound states. These normalized wave functions are always periodic. It is enough to take the integral of the square root of the potential energy function to find the normalized wave functions. If these calculations cannot be made analytically, they should then be performed by numerical methods. The SE has been solved for a particle in many one-dimension and the spherical symmetric central potential well, the relativistic theory of Dirac as examples.
This chapter solves the eigen-problems for a few simple one-dimensional systems to acquire experience in solving the Schrodinger eigen-equation. One-dimensional problems are physically interesting not only because they are simple, but also because a number of more complicated higher-dimensional problems can be reduced to the solutions of equations in one dimension. For a symmetric potential, an eigenfunction of the Schrödinger equation can be always chosen to have a definite parity. The probability density of the plane wave solution is a constant, independent of the spatial coordinate of the particle. The eigensolution of a particle in an infinite square well consists of an infinite number of discrete energy levels. In classical mechanics, a particle cannot penetrate a potential barrier whose height is higher than the energy of the particle.