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The Infinite Well: Physical Aspects

Published Apr 13, 2006• 0 citations
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Abstract

Abstract The problem of a particle moving in an infinite well or one-dimensional box, defined by the potentialis one of the most familiar and easiest to solve of all problems in introductory quantum mechanics. It is the simplest case in which to study the phenomenon of quantized energy levels in bound states, one of the “smoking guns “ of wave mech- anics. In this chapter and the next, we use this problem (and some related ones) as a tool in understanding the formalism used in solving the time-independent Schrödinger equation for bound states in a potential, understanding their phys- ical interpretation, as well as some of the formal properties of its solutions. In Chapter 7, we also use the infinite well as a model system in which to examine the role that spin and the Pauli exclusion principle play in multiparticle quantum systems.

DOI
10.1093/oso/9780198530978.003.0005

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Abstract The problem of a particle moving in an infinite well or one-dimensional box, defined by the potentialis one of the most familiar and easiest to solve of all problems in introductory quantum mechanics. It is the simplest case in which to study the phenomenon of quantized energy levels in bound states, one of the “smoking guns “ of wave mech- anics. In this chapter and the next, we use this problem (and some related ones) as a tool in understanding the formalism used in solving the time-independent Schrödinger equation for bound states in a potential, understanding their phys- ical interpretation, as well as some of the formal properties of its solutions. In Chapter 7, we also use the infinite well as a model system in which to examine the role that spin and the Pauli exclusion principle play in multiparticle quantum systems.
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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.
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.
This chapter discusses several applications of the formalism of theory of quantum mechanics. The time-independent Schrodinger equation is applied to understand simple quantum systems such as a free particle, a particle in an infinite square well, step potentials, potential barrier and finite square well. The chapter deals with the simplest case, a free particle. It is the most elementary application of the formalism. A free particle means that no forces are acting on the particle, and hence, there is no interaction between the particle and its environment. According to classical mechanics, a free particle moves with a constant velocity along a straight line. The energy of the wave associated with the free quantum particle is same as that of the classical free particle. The chapter describes a quantum free particle as a wave packet traveling in space and time. It discusses a quantum particle under the influence of a force.
This chapter solves the eigen-problems for a few simple one-dimensional systems to acquire experience in solving the Schrodinger eigen-equation. One-dimensional problems are physically interesting not only because they are simple, but also because a number of more complicated higher-dimensional problems can be reduced to the solutions of equations in one dimension. For a symmetric potential, an eigenfunction of the Schrödinger equation can be always chosen to have a definite parity. The probability density of the plane wave solution is a constant, independent of the spatial coordinate of the particle. The eigensolution of a particle in an infinite square well consists of an infinite number of discrete energy levels. In classical mechanics, a particle cannot penetrate a potential barrier whose height is higher than the energy of the particle.