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arXiv:1609.07380

Direct manifestation of Ehrenfest's theorem in the infinite square well model

Chyi-Lung Lin

Published Sep 23, 2016• 1 citation
arXivHTMLPDF

Abstract

Ehrenfest's theorem in the infinite square well is up to now only manifested indirectly. The manifestation of this theorem is first done in the finite square well, and then consider the infinite square well as the limit of the finite well. For a direct manifestation, we need a more precise formula to describe the degree of infiniteness of the divergent potential energy. We show that the potential energy term term, which is the product of the potential energy and the energy eigenfunction, is a well defined function which can be expressed in terms of Dirac delta functions. This means that the infinity in this model is not that vague but has obtained a specification. This results that expectation values can be calculated precisely and Ehrenfest's thereom can be confirmed directly.

DOI
10.48550/arxiv.1609.07380

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Ehrenfest's theorem in the infinite square well is up to now only manifested indirectly. The manifestation of this theorem is first done in the finite square well, and then consider the infinite square well as the limit of the finite well. For a direct manifestation, we need a more precise formula to describe the degree of infiniteness of the divergent potential energy. We show that the potential energy term term, which is the product of the potential energy and the energy eigenfunction, is a well defined function which can be expressed in terms of Dirac delta functions. This means that the infinity in this model is not that vague but has obtained a specification. This results that expectation values can be calculated precisely and Ehrenfest's thereom can be confirmed directly.
The original model of the infinite square well contains a vague notation infinity and therefore results some ambiguities. We investigate to obtain a functional form for the potential energy V(x). This is done by substituting back the original energy eigenstates and eigenvalues into the Schrodinger equation. We then obtain a precise functional form of the V(x). From this reformed model, we show that energy eigenstates and eigenvalues can directly be obtained without the need of imposing boundary condition, Ehrenfest's theorem can directly be confirmed, and ambiguities in the original model can be resolved.
Abstract We address an interesting problem in elementary quantum mechanics, namely how can one obtain the momentum state eigenfunctions for the infinite square well potential as a direct solution of the time-independent Schrödinger equation in momentum space. The conventional method for obtaining the momentum state eigenfunctions for the infinite square well potential is to take the Fourier transform of the coordinate space eigenfunctions. Unfortunately, the Schrödinger equation in momentum space for the infinite well potential is not well-defined. As such we must use a potential whose Fourier transform is well-defined and then take a limit in which the potential approaches that of the infinite well potential. We present three approaches to the problem: one based on dimensional analysis, one in which the infinite well potential is viewed as the limit of a finite well whose depth approaches infinity, and one in which the infinite well potential is viewed as the limit of a power law potential in the limit that the power approaches infinity.
Abstract: This work considered the infinite square well in one dimension with a contact potential. The Dirac delta derivative function potential λδ^' where λ is a coupling constant was used to represent the contact potential. Using Green’s function technique, exact implicit expressions of the energy eigenvalues and eigenfunctions were obtained. The energy eigenvalues were expressed using a transcendental equation. The energy eigenfunctions satisfy the Schrödinger equation and the infinite square well boundary conditions. Also, the eigenfunctions and their first derivative were shown to be discontinuous. The values of these discontinuity jumps agreed with the required conditions for a self-adjoint extension Hamiltonian. In the weak coupling region, the energy eigenvalues are close to that of the even parity solution before adding the contact potential. The energy eigenvalues in the strong coupling regime reveal the energy eigenvalues of the odd parity solution. Keywords: Point interactions, Infinite square well, Green’s function technique. PACS numbers: 03.65.-w, 03.65.Db, 03.65.Ge
Polynomial expansions are used to approximate the equations of the eigenvalues of the Schrödinger equation for a finite square potential well. The technique results in discrete, approximate eigenvalues which, it is shown, are identical to the corresponding eigenvalues of a wider, infinite well. The width of this infinite well is easy to calculate; indeed, the increase in width over that of the finite well is simply the original width divided by the well strength. The eigenfunctions of this wider, infinite well, which to first order has the same width for the ground state and all excited states, are also good approximations to the exact eigenfunctions of the finite well. These approximate eigenfunctions and eigenvalues are compared to accurate numeric calculations and to other approximations from the literature.
The goal of Quantum Mechanics I, A Problem Text, is to enable students to solve problems appropriate to their first course in quantum mechanics. Significantly detailed solutions to traditionally posed problems dominate. Theory and rationale are addressed in prescript and postscript narratives and/or within the solved problems to guide students; and clarify vocabulary, symbology, and terminology. Applications of the postulates are presented initially using arguments from linear algebra appropriate to discrete systems with low-dimensional vectors and matrix operators, introducing orthogonality, orthonormality, Hermiticity, eigenvectors, eigenvalues, probability, expectation value, and degeneracy. These arguments blend into infinite-dimensional vectors and operators, and calculus-based arguments for continuous systems where delta functions, theta functions, and Fourier transforms are initially discussed. Dirac notation is fully developed within these first five chapters. Chapter six discusses Ehrenfest’s theorem, the Heisenberg uncertainty relations, and Gaussian wave functions and wave packets. Increasingly sophisticated potential energy functions are then treated to include the free particle, the infinite square well, the one-dimensional scattering state, the simple harmonic oscillator, the finite square well, orbital angular momentum, and the hydrogen atom. Ladder operators are featured in the developments of the simple harmonic oscillator and orbital angular momentum. The diversity of narratives and detailed solutions intend