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Springer eBooks

Particles in Boxes and Their Applications

James K. Freericks

Published Jan 1, 2026• 0 citations
Publisher

Abstract

In this chapter, we introduce the particle in a box. This problem has become a standard in most quantum texts, even though it does not accurately represent too many real physical systems. Instead, it is used to illustrate quantum principles because it is viewed as one of the simplest quantum systems to discuss. Within our approach, however, this problem is a bit more challenging to solve. Nevertheless, it does provide a number of interesting insights, and it also allows us to prove an important result about the existence of bound states in one, two, and three dimensions. Our work here also allows us to answer the question why are some eigenvalues in quantum mechanics quantized, while others are not.

DOI
10.1007/978-3-031-87845-9_12

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In this chapter, we introduce the particle in a box. This problem has become a standard in most quantum texts, even though it does not accurately represent too many real physical systems. Instead, it is used to illustrate quantum principles because it is viewed as one of the simplest quantum systems to discuss. Within our approach, however, this problem is a bit more challenging to solve. Nevertheless, it does provide a number of interesting insights, and it also allows us to prove an important result about the existence of bound states in one, two, and three dimensions. Our work here also allows us to answer the question why are some eigenvalues in quantum mechanics quantized, while others are not.
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.
In the previous chapter, the postulates helped us to use the wave function to find the expected results of experimental measurements by letting a dynamical quantity F (x, p, t) → F (xˆ, pˆ, t) and integrating over space. In the examples, however, the wave functions were typically given and not derived from first principles. We now endeavor to find the appropriate wave functions for a variety of physical systems in one dimension by solving the Schro¨dinger equation where for time-independent cases, the total energy is an eigenvalue. In subsequent chapters, we will extend the method to three dimensions and develop the appropriate operators, eigenvalues, and eigenfunctions for angular momentum. We will also develop methods for examining time-dependent systems. The latter chapters address special systems of interest that may be referred to as applications of quantum mechanics.
This chapter should be treated as an appendix. It is included for completeness and covers certain fundamentals of quantum mechanics that will aid the understanding of topics discussed in this textbook. The reader is urged to refer to many of the good introductory textbooks on the subject to learn or revisit the fundamental concepts [1, 2, 3, 4]. First, we briefly discuss three key ingredients of quantum mechanics, i.e., the blackbody radiation spectrum, the concept of the photon and de Broglie wavelength, and the related wave-particle duality concept. Next, we briefly restate the four major postulates of quantum mechanics used in the so-called orthodox or Copenhagen interpretation of quantum mechanics that is in vogue. We continue on to describe a generalized form of the Heisenberg Uncertainty Principle which is illustrated with a few familiar bound state problems (particle-in-a-box and one-dimensional harmonic oscillator) and a few spin related problems.
The exact formulation of the correspondence principle and in particular understanding the quantum‐to‐classical transition remains an open problem in quantum mechanics. In this paper we present our investigation into the quantum‐to‐classical transition of the most trivial of quantum systems — a particle in a box. Whilst it is perhaps surprising, even this example can produce new physical insight into these fundamental problems. With modern fabrication techniques of nano‐mechanical systems we will be able to experimentally investigate these results and directly observe the quantum‐to‐classical transition. This will enable us to build technologies that probe the fundamental questions of quantum mechanics, such as the maximum size of a quantum object.
Abstract Schrödinger developed an operator method for solving quantum mechanics. While this technique is overshadowed by his more familiar differential equation approach, it has found wide application as an illustration of supersymmetric quantum mechanics. One reason for reticence in its usage for conventional quantum instruction is that the approach for simple problems like the particle-in-a-box is much more complicated than the differential equation approach, making it appear to be less useful for pedagogy. We argue that the approach is still quite attractive because it employs only algebraic methods, and thereby has a much lower level of math background needed to use it. We show how Schrödinger’s operator method can be streamlined for these particle-in-a-box problems greatly reducing the complexity of the solution and making it much more accessible. As an application, we illustrate how this approach can be used to prove an important result, the existence of bound states for one- and two-dimensional attractive potentials, using only algebraic methods. The approach developed here can be employed in undergraduate classes and possibly even high school classes because it employs only algebra and requires essentially no calculus.
One of the classic first problems in quantum theory textbooks is that of the particle trapped in a box. Despite the apparent artificiality of the situation, it has quite a few interesting features that we can quickly apply to some more physically plausible situations. The particle in a box is literally an object, such as an electron, trapped inside a ‘box.’ In a realistic physical example, this ‘box’ would be a volume of space bound by some constraining force acting on the particle. For example, an atom is a ‘box’ as far as the orbiting electrons are concerned, with the electrostatic force between the positive protons in the nucleus and the negative electrons providing the constraint. The first move is to assume that the wave function for any particle in the box will have separate spatial and temporal dependencies. One of the key steps in obtaining our wave function was the quantization of the momentum.