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.
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.
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.
In the previous chapter we saw that for a particle in a one-dimensional potential well, physically acceptable solutions to the one-dimensional Schro¨dinger equation are possible only for particular discrete values of the total energy. Moreover, in the case of an electron in a well of atomic dimensions, the spacings between these energy levels are in qualitative agreement with the separations experimentally observed in atoms; we also interpreted the square of the wave function as a probability distribution for the position of the particle. These ideas led us to phenomena, such as quantum-mechanical tunnelling, that have been observed experimentally. The real world, however, is three-dimensional and, although one-dimensional examples often provide useful insights and analogies, we shall have to extend our theory into three dimensions before we can make quantitative predictions of most experimental results. In the present chapter, therefore, we shall set up the three-dimensional Schro¨dinger equation and obtain its solutions in a number of cases, culminating in a discussion of the hydrogen atom where we shall find that theory and experiment agree to a remarkable degree of accuracy.
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.
In these lecture notes we will consider systems in which the motion of electrons is confined to one dimension (1D). In these so-called quantum wires electron-electron interaction effects play an important role because the restricted dimensions enhance the scattering between the electrons and completely destroy the quasi-particle picture. New density wave excitations appear that are described by bosonic operators. Here, we will develop this bosonic description, following a pedestrian approach which does not require any previous knowledge in field theory methods. These notes therefore serve as a detailed introduction into bosonization by carefully deriving the most fundamental formulas.
The Schrödinger equation assumes that a particle behaves as a wave and yields a solution in terms of wave function and the energy of the particle under consideration. Once the wave function is known, then everything about the particle can be deduced from the wave function. This chapter covers the development of the Schrödinger wave equation and its applications to one-dimensional systems. Time-dependent and time-independent Schrödinger equations, bound and scattering states, probability density, probability current and expectation values, the solution of the time-independent 1D-Schrödinger equation for a free particle, infinite as well as finite potential wells, potential steps, and the finite potential barrier are discussed in detail in the chapter. The phenomenon of tunneling, relevance and the importance of the free particle, potential wells and potential barriers to contemporary fields of physics and chemistry, periodic solids and their band structures, the Kronig-Penney model, confined states in quantum wells, and wires and dots are other topics presented in innovative manners in the chapter. The importance of the topics presented here is demonstrated through carefully chosen solved examples and unsolved exercises from contemporary fields of sciences.
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.