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Cambridge University Press eBooks

Additional examples of quantized motion

Vladimir Mitin, Dmitry I. Sementsov, Nizami Z. Vagidov

Published May 20, 2010• 0 citations

Abstract

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.

DOI: 10.1017/cbo9780511845161.005

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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.

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 this chapter we shall discuss some elementary examples of quantum dynamics. In Sec. 4.1 we shall go back to the problem of a particle in a box, this time with finite potential wells. In Sec. 4.2 we shall analyze the effects of a potential barrier on a moving particle. In Sec. 4.3 we shall consider another quantum effect which has no analogue in the classical domain: a quantum particle can tunnel in a classically forbidden region. In Sec. 4.4 perhaps the most important dynamical typology (with a wide range of applications) is considered: the harmonic oscillator. Finally, in Sec. 4.5 several types of elementary fields are considered. Finite potential wells In Sec. 3.4 we have considered what is perhaps the simplest example of quantum dynamics, that is a free particle moving in a box with infinite potential walls. Consider the motion of, say, a one-dimensional particle in a rectangular potential well with finite steps. In Fig. 4.1 we show two of such potentials, symmetric in (a) and asymmetric in (b). Let us consider the case pictured in Fig. 4.1(a) and indicate with V 0 the energy of the potential well. We may therefore distinguish three regions on the x -axis: region I ( x < 0), where the potential energy is equal to V 0 ; region II (0 ≤ x ≤ a ), where the particle is free; and region III ( x > a ), where the potential energy is again equal to V 0 .

Abstract The simple ‘particle in a box’ (Piab) is introduced in this chapter so that the reader can get familiar with applying the quantum recipe and atomic units. The PiaB is introduced in its one, two, and three dimensional variants, which demonstrates the use of the separation of variables technique as a strategy to solve the Schrodinger equation for a particle with two or three degrees of freedom. It is shown that the confinement of the particle causes the energy to be quantized. The one-dimensional PiaB is then applied to treat the electronic spectra of cyanine dyes and their absorption colors. The chapter then introduces more general setups with finite potential wells, in order to introduce the phenomenon of quantum tunnelling and to discuss more generally with the unintuitive ‘quantum behavior’ of particles such as electrons. Scanning tunnelling and atomic force microscopes are also discussed briefly.

This chapter evaluates some of the simplified Schrödinger equations obtained in the previous chapter, and presents explicit solutions for the energy levels and the corresponding wavefunctions. It begins by looking at the free particle and the particle in a box. The particle in a box is the simplest example of a confined particle, and enables us to understand the origin of many features of such systems, such as the appearance of discrete energy levels. It is a general feature of particles confined by a potential to some region of space that solutions to the Schrödinger equation can only be found at certain discrete energies. The chapter then considers the harmonic oscillator; angular momentum; rotations of a polyatomic molecule; and the hydrogen atom.

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.