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Brazilian Journal of Physics

Infinite Square Well, Self-Adjointness, and the Dirac Delta Function

Marcos Amaku, Francisco Antônio Bezerra Coutinho, E. Massad, L. N. Oliveira

Published Mar 19, 2026• 0 citations
Publisher

Abstract

Abstract One hundred years ago, Dirac presented a groundbreaking method for quantizing classical Hamiltonians. Simplicity is one of the virtues of the procedure, but certain classes of problems require special care. Here, we discuss one such class, the quantization of confined systems, with emphasis on the one-dimensional square well. We examine two presentations of the problem. The first presentation restricts the analysis to the interval $$0\le x\le L$$ . The second presentation considers the entire x axis, but an infinite potential traps the particle in the region $$0\le x\le L$$ . In contrast with the typical treatment in quantum mechanics textbooks, the first presentation imposes nontrivial boundary conditions. Proper treatment of the problem in the second presentation demands the use of delta functions or distributions.

DOI
10.1007/s13538-026-02041-7

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We present some simple arguments to show that quantum mechanics operators are required to be self-adjoint. We emphasize that the very definition of a self-adjoint operator includes the prescription of a certain domain of the operator. We then use these concepts to revisit the solutions of the time-dependent Schroedinger equation of some well-known simple problems - the infinite square well, the finite square well, and the harmonic oscillator. We show that these elementary illustrations can be enriched by using more general boundary conditions, which are still compatible with self-adjointness. In particular, we show that a puzzling problem associated with the Hydrogen atom in one dimension can be clarified by applying the correct requirements of self-adjointness. We then come to Stone\'s theorem, which is the main topic of this paper, and which is shown to relate the usual definitions of a self-adjoint operator to the possibility of constructing well-defined solutions of the time-dependent Schrödinger equation.
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For a quantum confinement model, the wave function of a particle is zero outside the confined region. Due to this, the negative energy states are, in fact, square integrable. As negative energy states are not physical, we need to impose some boundary conditions in order to avoid these states. For the case of the infinite square well model, we show the possible boundary conditions that avoid the existence of the negative energy states. The well-known boundary condition requiring the wave function be continuous on both sides of the well is one of the boundary conditions that avoids negative energy states. However, there are other types of boundary condition that can also avoid the negative energy states. This shows that there are the different branch of physical systems that can be established on the infinite square well model.