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European Journal of Physics

1D potential wells of the form V x < a = − V o 1 − x n a n

Abby Corrigan, Janine Shertzer

Published Dec 4, 2020• 1 citation
Publisher

Abstract

Abstract We consider the finite potential well defined by V x < a = − V o 1 − x n a n ,where n is an integer. The Schrödinger equation is recast into dimensionless form; the size of the well is characterized by a single parameter α = 2 m V o a / ℏ . We use three different mathematical techniques to solve for the energy eigenvalues and wave functions. Analytic solutions are given for the triangular well ( n = 1), the parabolic well ( n = 2), and the square well ( n → ∞). The finite element method is used to obtain accurate numerical results for 1 ⩽ n ⩽ ∞ over a full range of α . Finally, both the energy and wave function are expanded in a Taylor series in order to study the analytic behavior of the ground state energy (as a function of n ) as α → 0.

DOI
10.1088/1361-6404/abd0d9

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Abstract We consider the finite potential well defined by V x < a = − V o 1 − x n a n ,where n is an integer. The Schrödinger equation is recast into dimensionless form; the size of the well is characterized by a single parameter α = 2 m V o a / ℏ . We use three different mathematical techniques to solve for the energy eigenvalues and wave functions. Analytic solutions are given for the triangular well ( n = 1), the parabolic well ( n = 2), and the square well ( n → ∞). The finite element method is used to obtain accurate numerical results for 1 ⩽ n ⩽ ∞ over a full range of α . Finally, both the energy and wave function are expanded in a Taylor series in order to study the analytic behavior of the ground state energy (as a function of n ) as α → 0.
The objective of this study is to numerically solve and apply two approximate methods to investigate the Time Independent Schrödinger Equation (TISE) in one dimension for an infinite potential square well. These two numerical methods are the Numerov Method (NM) and Matrix Method (MM). As a simulation tool, MATLAB, a high-level programming language and an efficient simulation tool, is used for modeling and solving TISE in one dimension. Exact analytical solutions for these potential functions are obtained and compared with numerical solutions and computational techniques. The energy eigenvalues and Eigen functions of a particle (such as an electron) restricted to move inside this potential are discussed as an illustration. The numerically calculated energies of several states with increasing numbers of points were obtained from both methods and compared with the simulation results of the exact solution. As an exemplary case, the first five wave functions are accurately determined numerically where the discreteness is found since the wave function vanishes at the boundary. The obtained results show very good agreement and the similarity is clearly confirmed between the three cases. This agreement confirms that this approach was highly accurate and efficient. The accuracy and the convergence of the numerical obtained results were easily checked. The stability of these methods is due to the fact that there are no restrictions on the time steps to be taken. The merits of these numerical methods are to avoid a huge expense in time when solving Schrödinger equation.
Abstract This article examines a particle in a one-dimensional potential well V a (x) ∝ x a , where a ≥ 2 is an even integer. The system is referred to as a harmonic oscillator when a = 2 and as a particle in a box when a → ∞. The finite difference method is applied to solve the Schrödinger equation, determining the ground-state energies and wave functions. Giving accurate description in the cases of a = 2 and a → ∞, the numerical method allows us to explore the system for intermediate values of a, revealing how the energies and wave functions evolve between these two limits. The study includes discussions on the virial theorem, the uncertainty principle, excited states, and the limitation of the numerical method.
This paper investigates applying the approximate method; Graphical Solution Method (GSM), to theoretically solve the Time Independent Schrödinger Equation (TISE) in one dimension for a finite square well using MATLAB. With just few lines of MATLAB coding, calculating and plotting accurate eigenvalues (energy), eigenvectors (wave functions) and the bound eigenstates are possible for the finite square well of a negative potential (depth of the well) of -400 eV and a well width of 0.1 nm for an electron confined to this quantum well. These eigenvalues, eigenvectors and eigenstates are obtained and discussed. The found energy eigenvalues and states are discrete and yield physical acceptable solutions. The even and odd solutions of the TISE are also considered. The graphical solutions for the finite potential well are shown. The locations of discrete eigenvalues for even and odd solutions are also presented. These eigenvalues are tested confirming the correct eigenfunctions. The precision of these solutions depend on well width L and on the interval dx used to integrate the equation. Exact analytical solutions for this case are obtained and compared with results from the GSM. The accuracy and the convergence of the numerical results are easily checked. The results showed that the GSM can be considered as a suitable mean for determining the one dimensional solutions for the finite square well.
This chapter examines some explicitly one-dimensional systems, including the finite square well, the harmonic oscillator, and the application of harmonic oscillator theory to molecular vibrations. It analyzes the computer solutions of the Erwin Schrodinger equation. Although there are no real one-dimensional square wells in nature, many real systems have potentials that resemble the square well. Near the center of force the different wave functions must differ from one another or else they would not satisfy the Schrodinger equation for different values of energy. In contemporary research almost all the manipulation of the Schrodinger equation is done not analytically but rather by computer using numerical methods. A major advantage of the dimensionless form of the Schrodinger equation is that a single solution for one kind of system is trivially easy to extend to all similar systems. The dimensionless form of the Schrödinger equation can be solved numerically if it is approximated by a so-called difference equation.
For the quantum particle in a finite and semi-infinite well, we scale space and energy to consider a dimensionless Schrdinger equation that can be solved for the allowed energy levels. In contrast to traditional approaches that determine the energies from dimensional transcendental equations associated with the boundary conditions, we solve for the square root of the dimensionless energies, which are also solutions of dimensionless transcendental equations. Our ability to define classes of wells depends on the value of the square root of the dimensionless well depth. We are also able to determine the maximum number of energy levels quite easily, and, besides deriving the relevant transcendental equations, we obtain asimple expression for the corresponding wave functions, which can be plotted using the numerical values of the allowed energies.
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.