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Schrödinger, 3

David Peak

Published Aug 28, 2017• 0 citations
OpenAlex

Abstract

The 3D infinite square well: quantum dots, wells, and wires In the preceding discussion of the Schrödinger Equation the particle of interest was assumed to be “moving in the x -direction.” Of course, it is not possible for a particle to be moving in one spatial direction only. If that were true, according to the HUP it could be anywhere in the y - and z -directions and therefore be undetectable with finite volume detectors. Now, we consider the more realistic case of motion in all three spatial directions. For this purpose, we start with the 3D infinite square well. This model provides a useful conceptual framework for understanding a burgeoning “nano-industry,” namely the production and use of “quantum dots.” The 3D infinite square well is a “rigid rectangular solid box.” The potential energy of a particle trapped inside is given by U(x,y,z) = 0 , if 0 < x < L1, 0 < y < L2, 0 < z < L3, and ∞, otherwise. The particle’s momentum now has three components and its kinetic energy is K = (px2 + py2 + pz2) /2m.

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The 3D infinite square well: quantum dots, wells, and wires In the preceding discussion of the Schrödinger Equation the particle of interest was assumed to be “moving in the x -direction.” Of course, it is not possible for a particle to be moving in one spatial direction only. If that were true, according to the HUP it could be anywhere in the y - and z -directions and therefore be undetectable with finite volume detectors. Now, we consider the more realistic case of motion in all three spatial directions. For this purpose, we start with the 3D infinite square well. This model provides a useful conceptual framework for understanding a burgeoning “nano-industry,” namely the production and use of “quantum dots.” The 3D infinite square well is a “rigid rectangular solid box.” The potential energy of a particle trapped inside is given by U(x,y,z) = 0 , if 0 < x < L1, 0 < y < L2, 0 < z < L3, and ∞, otherwise. The particle’s momentum now has three components and its kinetic energy is K = (px2 + py2 + pz2) /2m.
This chapter discusses the concept of quantum wells from one to three dimensions to see how particles behave in a three-dimensional quantum world. It considers the properties of particles confined in two- and three-dimensional potential structures called quantum wires and quantum dots. Determining energies and corresponding wave functions require some careful analysis but are worth the trouble. A quantum wire is sometimes referred to as a one-dimensional system. The energy of the particle in a quantum dot is quantized in all three directions. Because the motion of the particle is now restricted in every direction, the particle has zero dimension of freedom. Quantum dots are also regarded as artificial atoms . One can notice that the results obtained for the three-dimensional case are similar to that obtained for the one-dimensional case. In the three-dimensional case, the wave function and the energy are specified by the trio of integers. The energy level is degenerate, and the number of degenerate states, degeneracy.
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).
If the valley encountered by the particle is infinitely deep, then the particle must be trapped inside. The best analogy is an infinitely deep well. Therefore, this potential is called the infinite square well , or the particle in a box (PIB). The TISE for the PIB is very easy to solve, again using the physical principles from Chapter 6 . The solutions are sine waves with an integral or half-integral number of wavelengths fitting the width of the well. This fact that only certain wavelengths are allowed is a perfect example of quantization of a trapped particle. From the allowed wavelengths, we can also derive the quantized momentum and kinetic energy of the particles. Since momentum is a vector quantity, care must be taken to consider the direction in which the particle might be moving. With this caveat, these simple sine wave solutions are ideal for illustrating principles and procedures such as eigenfunctions, normalization, orthogonality, orthonormality, and expectation values.
In the previous chapter we have analyzed the peculiarities of quantized electron motion in layered structures with one-dimensional potential wells. From the mathematical point of view, the solution of the Schrödinger equation for one-dimensional potential profiles is much simpler. However, many quantum objects, such as atoms, molecules, and quantum dots, are three-dimensional objects. Thus, in order to analyze electron motion in such objects we need to find solutions of the Schrödinger equation for three-dimensional potential profiles. The electron motion in spaces with dimensionality higher than one, especially for rectangular potential profiles with infinite potential barriers, is not so difficult to analyze. At the same time we have to keep in mind that such potential profiles frequently represent some approximation of the more complex, real potential profiles. Depending on the type of structure and on the form of the potential profile, the electron motion may be limited in two directions (two-dimensional quantization) or in three directions (three-dimensional quantization). In this chapter we will show that the existence of discrete energy levels in the electron spectrum is an intrinsic feature of electron motion in potential wells of any form and dimensionality. An electron in a rectangular potential well (quantum box) In the previous chapter we studied the electron motion in one-dimensional potential wells. An electron's motion along one direction was confined by the potential profile and the momentum in this direction was quantized.
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.
The wave nature of particles plays an important role in their physical properties that, for example, particles confined into a small bounded area can have only particular discrete energies. This chapter explains how to solve the time-independent Schrodinger equation. It considers four cases of a one-dimensional motion of particles confined in potentials rapidly changing with x : infinite potential quantum well; a potential step; square-well potential; and tunneling through a potential barrier. The chapter helps to learn the characteristic properties of solutions to this equation. It shows the differences between the predictions of quantum mechanics and classical physics. Quantum tunneling is important in the understanding of a number of physical phenomena such as thermonuclear reactions and conduction in metals and semiconductors. The most advanced application of quantum tunneling is the scanning tunneling microscope. The details of the calculations are left for the readers as a tutorial problem.