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Co-Existence Effects of Square Weil and Square Barrier Potentials

Chao-Li R. Chen

Published Jan 1, 1982• 0 citations
OpenAlex

Abstract

This thesis is concerned with an elementary problem of quantum mechanics. We discuss the change in the behavior of the wave function when the potential is changed. The MAT ( maximum -amplitude-trajectory) method is used to present the results in a visually appealing form. We compare the results for three different potentials: (1) a simple square barrier, (2) a square barrier with free space inside, and (3) a square barrier wit h a square well inside. By the comparison of these three results, we are able to understand the effects of co-existence of square barrier and square well. The results will be presented in the following order: (1)Introduction, (2) The Significance of MAT, (3)The Maximum-Amplitude-Trajectory for Simple Square Barrier, (4)The Maximum-Amplitude-Trajectory for Square Barrier with Free space inside, (5)The Maximum-Amplitude-Trajectory for Co-existing Square Well and Square Barrier, (6)Discussion, and (7)Conclusion.

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Impenetrable barriers in quantum mechanics

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The time evolution of quantum mechanical states in a square well of infinite depth with a Dirac δ function at its center has been examined for cases where the initial state was localized in one of the wells. Let well A denote the well in which the initial state is localized, and let P(t) denote the integrated probability density Ψ*Ψ in well A. For opaque barriers the time-dependent system is adequately described by a two-state model in which only the pair of stationary states of even and odd parity are considered whose wave functions in well A, apart from an arbitrary phase factor, are largely identical with the wave function of the initial state. For P(t) a harmonic oscillation is observed whose frequency νt is well approximated by the well-known formula for the tunneling frequency νt≂ΔE/h, where ΔE represents the energy separation among the pair of states in the model. For the present model of a symmetric double well it has been shown that for highly transparent barriers a three-state model can describe the time evolution adequately. The three stationary states involved in this model are a state of odd parity whose wave function in well A is largely the same as the wave function of the initial state and a pair of stationary states of even parity which on the energy scale are immediately above and below the first state. In this three-state model the function P(t) is a superposition of two sinusoidal functions with nearly identical amplitudes and frequencies plus a constant. As a consequence, the amplitude of P(t) changes harmonically. In the present model a δ function has been used as a barrier in order to minimize the mathematical detail involved in the time-dependent treatment. It is to be expected that the beating in P(t) can be observed also in the time evolution of a state localized in one of the wells of a symmetric double-minimum potential with a more realistic ‘‘low’’ barrier if the density of energy levels near the energies of the levels to be considered in the corresponding three-state model varies slowly with energy.

American Journal of Physics•Mar 1, 1992•6 citations

Continuous limit of the square well problem in quantum mechanics

Matheus D. Moro, Thiago T. Tsutsui, Antonio S. M. de Castro +1 more

The free-particle and square-well potentials are two of the most emblematic problems in quantum mechanics, illustrating essential concepts such as matter waves, energy quantization, and bound states. It is therefore natural to consider how the free-particle solutions emerge from the square well as the width approaches infinity. In this work, we present a systematic procedure to demonstrate this transition by applying a Fourier transform to the wave equation.

quant-pharXiv:2607.13872•Jul 16, 2026•0 citations
This thesis is concerned with an elementary problem of quantum mechanics. We discuss the change in the behavior of the wave function when the potential is changed. The MAT ( maximum -amplitude-trajectory) method is used to present the results in a visually appealing form. We compare the results for three different potentials: (1) a simple square barrier, (2) a square barrier with free space inside, and (3) a square barrier wit h a square well inside. By the comparison of these three results, we are able to understand the effects of co-existence of square barrier and square well. The results will be presented in the following order: (1)Introduction, (2) The Significance of MAT, (3)The Maximum-Amplitude-Trajectory for Simple Square Barrier, (4)The Maximum-Amplitude-Trajectory for Square Barrier with Free space inside, (5)The Maximum-Amplitude-Trajectory for Co-existing Square Well and Square Barrier, (6)Discussion, and (7)Conclusion.
This thesis is concerned with an elementary problem of quantum mechanics, and its purpose is to expand the use of the maximum amplitude trajectories. There are two motivations of this thesis: First, we want to investigate the change in scattering caused by a potential when its shape gradually changes. In particular, we are interested in understanding how a resonance phenomenon originates and in determining what is required of a potential to produce a sharp resonance. We start from a square repulsive potential which never produces a resonance. Subsequently, we carve a pocket in the inner region. As the pocket becomes deeper, we observe that a resonance is created and that it becomes ' sharper as the depth of the pocket increases. Secondly, we deal with this elementary quantum mechanical problem based on an advanced mathematical technique, i.e. the analytically continuable pertubation theory. That theory suggests that the method of the maximum-amplitude-trajectory (MAT) is useful. A substantial number of case studies are presented in this thesis. The solutions of the Schroedinger equation have been surveyed in a wide range of parameters with the help of the Vax-ll computer. The behavior of the phase shift as well as the cross section has also been studied for particular cases as a function of wave number. We compare the MAT determined for fifteen different sets of parameters of the potential. By the comparison of these fifteen results, we are able to understand systematically how the ratio and width parameters influence scattering. The results will be presented in the following order: 1) Introduction, 2) Review of scattering theory, 3) Review of Maximum Amplitude Trajectory, 4) Maximum Amplitude Trajectory for model potentials, 5) Results and discussions, 6) Conclusions.
Chapter 5 familiarizes the student with some of the most basic features of quantum systems, by applying the Schrödinger equation to simple, one-dimensional problems. It shows that it is possible to obtain qualitatively correct conclusions about the behavior of quantum particles in the presence of more or less arbitrary potentials by replacing them with piecewise-constant potentials, which greatly simplifies the mathematics. Replacing the real potential with a simpler one that retains its basic characteristics has the great advantage of allowing us to study the essence of the physical situation, as illustrated by the examples of the one-dimensional potential step, the square well, the square potential barrier and the symmetric double well. These examples are used to discuss specific features such as quantum tunneling, and applications such as masers and molecular clocks.
This thesis is concerned with a research project that aims to investigate the minimum barrier height that produces a sharp resonance by combining a square well potential with a square barrier potential. This combination is a simple quantum mechanical system. Yet, it deserves a thorough investigation. We deal with this elementary problem based on an advanced quantum mechanical technique i. e, analytically continuable perturbation theory. Before the systematic application of the analytically continuable perturbation theory, however, we need to orient ourselves to the entire project, determining the range of parameters in which thorough investigations must be carried out. A sufficient number of case studies are presented in this thesis. The solutions of the Schrodinger equation have been surveyed in a Wide range of ~meters with the help of the Sigma-9 computer. The behavior of phase shift and cross section as a function of energy has been studied for different combinations of the well and barrier. Finally by studying the shape of resonance peak, with three different values of barrier parameters, we have determined what kind of square well to be combined with each barrier in order to produce a typical case of sharp resonance.
"We derive the expression V (x) u(x) = c δ(x − a) + v(x) u(x) (where V (x) is the potential, u(x) the wave function, c a constant and v(x) a finite potential function for x ≤ a), which is present in the one-dimensional Schr ̈ dinger equation on the whole real line when we have o an impenetrable barrier at x ≥ a, that is, an infinite step potential there. By studying the solution of this equation, we identify, connect and discuss three different Hamiltonian operators that describe the barrier. We extend these results by constructing an infinite square-well potential from two impenetrable barriers."
The time evolution of quantum mechanical states in a square well of infinite depth with a Dirac δ function at its center has been examined for cases where the initial state was localized in one of the wells. Let well A denote the well in which the initial state is localized, and let P(t) denote the integrated probability density Ψ*Ψ in well A. For opaque barriers the time-dependent system is adequately described by a two-state model in which only the pair of stationary states of even and odd parity are considered whose wave functions in well A, apart from an arbitrary phase factor, are largely identical with the wave function of the initial state. For P(t) a harmonic oscillation is observed whose frequency νt is well approximated by the well-known formula for the tunneling frequency νt≂ΔE/h, where ΔE represents the energy separation among the pair of states in the model. For the present model of a symmetric double well it has been shown that for highly transparent barriers a three-state model can describe the time evolution adequately. The three stationary states involved in this model are a state of odd parity whose wave function in well A is largely the same as the wave function of the initial state and a pair of stationary states of even parity which on the energy scale are immediately above and below the first state. In this three-state model the function P(t) is a superposition of two sinusoidal functions with nearly identical amplitudes and frequencies plus a constant. As a consequence, the amplitude of P(t) changes harmonically. In the present model a δ function has been used as a barrier in order to minimize the mathematical detail involved in the time-dependent treatment. It is to be expected that the beating in P(t) can be observed also in the time evolution of a state localized in one of the wells of a symmetric double-minimum potential with a more realistic ‘‘low’’ barrier if the density of energy levels near the energies of the levels to be considered in the corresponding three-state model varies slowly with energy.