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Physical Review A

Quantum solutions for the harmonic-parabola potential system

Kyu Hwang Yeon, Shou Zhang, Yong Dae Kim, Chung‐In Um, Thomas F. George

Published Mar 16, 2000• 8 citations
Publisher

Abstract

The quantum harmonic-parabola system, whose potential energy is the negative harmonic potential, is analyzed and applied to the cases of a quantum well, barrier, and periodic lattices. The eigenstate of the quantum Hamiltonian of the harmonic-parabola system is obtained. It is shown that any function may be expanded in terms of the eigenfunctions of the system in a finite interval, and the propagator of the system can be obtained from the eigenfunctions. The energy eigenvalues, uncertainty, and probability density for the first few states are treated for the infinite harmonic-parabola well. The energy eigenvalues and their enumeration, the energy band, and the probability density for the first few states are obtained for a well composed of the parabola and constant potentials. The transmission coefficients and each potential interval dependence are determined for this well and the barrier of the parabola-constant potential structure. Periodic lattices composed of the parabola potential and parabola-constant potentials are constructed, and their dispersion relations, energy states, and bands are obtained.

DOI
10.1103/physreva.61.042103

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The quantum harmonic-parabola system, whose potential energy is the negative harmonic potential, is analyzed and applied to the cases of a quantum well, barrier, and periodic lattices. The eigenstate of the quantum Hamiltonian of the harmonic-parabola system is obtained. It is shown that any function may be expanded in terms of the eigenfunctions of the system in a finite interval, and the propagator of the system can be obtained from the eigenfunctions. The energy eigenvalues, uncertainty, and probability density for the first few states are treated for the infinite harmonic-parabola well. The energy eigenvalues and their enumeration, the energy band, and the probability density for the first few states are obtained for a well composed of the parabola and constant potentials. The transmission coefficients and each potential interval dependence are determined for this well and the barrier of the parabola-constant potential structure. Periodic lattices composed of the parabola potential and parabola-constant potentials are constructed, and their dispersion relations, energy states, and bands are obtained.
An application of the quantum wave impedance method for the analytical study of quantum-mechanical systems containing a set of delta-wells and delta-barriers is considered in this study.The algorithm for determining both transmission and reflection coefficients for a set of delta-barriers was proposed.Similarly, the scheme for analytically deriving the expressions for energy levels and wave functions of bound states for a set of delta-wells was developed.In both cases, the solution for the quantum wave impedance function in the region with a constant potential [1] and the matching conditions at the origins of deltafunctions were used.As an illustration of the application of the proposed approach, the quantum-mechanical systems with symmetric double-barrier potential and double-well potential were investigated.In the case of a single delta-well, the eigenenergy value as well as the expression for the eigenfunction were obtained directly using the non-linear differential equation for the quantum wave impedance function.The results demonstrate the effectiveness of the quantum wave impedance method in handling quantum-mechanical systems with deltalike potentials, offering a robust framework for further studies in this area.
The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.
We analyze the behavior of a quantum system described by a one-dimensional asymmetric potential consisting of a step plus a harmonic barrier. We solve the eigenvalue equation by the integral representation method, which allows us to classify the independent solutions as equivalence classes of homotopic paths in the complex plane. We then consider the propagation of a wave packet reflected by the harmonic barrier and obtain an expression for the interaction time as a function of the peak energy. For high energies we recover the classical half-period limit.
The harmonic oscillator is a quantum mechanical system that represents one of the most basic potentials. In order to understand the behavior of a particle within this system, the time-independent Schrödinger equation was solved; in other words, its eigenfunctions and eigenvalues were found. The first goal of this study was to construct a family of single parameter potentials and corresponding eigenfunctions with a spectrum similar to that of the harmonic oscillator. This task was achieved by means of supersymmetric quantum mechanics, which utilizes an intertwining operator that relates a known Hamiltonian with another whose potential is to be built. Secondly, a generalization of the technique was used to work with the time-dependent Schrödinger equation to construct new potentials and corresponding solutions.
A consistent scheme is proposed for quantizing the potential amplitude in the one-dimensional Schrödinger equation in the case of negative energies (lying in the discrete-spectrum domain). The properties of the eigenfunctions ϕ n ( x ) and eigenvalues α n corresponding to zero, small, and large absolute values of energy E < 0 are analyzed. Expansion in the set ϕ n ( x ) is used to develop a regular perturbation theory (for E < 0), and a general expression is found for the Green function associated with the time-independent Schrödinger equation. A similar method is used to solve several physical problems: the polarizability of a weakly bound quantum-mechanical system, the two-center problem, and the tunneling of slow particles through a potential barrier (or over a potential well). In particular, it is shown that the transmission coefficient for slow particles is anomalously large (on the order of unity) in the case of an attractive potential is characterized by certain critical values of well depth. The proposed approach is advantageous in that it does not require the use of continuum states.
A general solution of the Schrödinger equation in the potential representation has been obtained in the form of integral equations. In this representation, the wave function for positive and negative energies or bound states can be expressed as a product of the unperturbed solution for model potential and the function which depends on the additional potential or potential perturbation. Here we have proved that this method is equivalent to the method of variation of constants for negative energies. The linearly independent solutions of Schrödinger equation for harmonic oscillator potential have been obtained for derivation of integral equations, which are used for finding eigenfunctions and eigenvalues for Woods–Saxon potential. Eigenvalues obtained by numerical iterations of these integral equations are in good agreement with results obtained by the discretization method. The kernels of the obtained integral equations are proportional to the perturbation or difference of Woods–Saxon and harmonic oscillator potentials.