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CRC Press eBooks

Particle in a Box

James W. Rohlf

Published Oct 18, 2023• 0 citations
Publisher

Abstract

The particle in a box, also referred to as “infinite square well” is an important example because it illustrates the many features of the solution to the Schrödinger equation in an example requiring minimal math. The problem will be worked first in one dimension. The box is defined by a potential that is zero in some region (0
DOI
10.1201/9781003395515-6

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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.
The finite square well The infinite square well potential energy rigorously restricts the associated wavefunction to an exact region of space: it is infinitely “hard.” Potential energies encountered in more realistic physical scenarios are “softer” in that they permit wavefunctions to spread throughout less well-defined regions. An important toy example of the latter is the finite square well. In this problem, the potential energy function is U(x) = 0, if 0 < x < L, and U0 otherwise.
In a recent paper diffused in the literature, the quantum problem of a particle in a infinite square well potential was claimed to be solved via Laplace transform. Surprisingly, the correct solutions were found for an eigenvalue problem with variable defined on a finite range. As a matter of fact, the authors have misused the Laplace transform of the time-independent Schrödinger equation, and they have also used incorrectly the inversion process. In the present paper it is pointed out carefully and clearly those mistakes that might occur with the use of inappropriate methods for a given eigenvalue problem.
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.