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Revista Brasileira de Ensino de Física

Did you really miss a Delta function here?

Marcos Amaku, Francisco Antônio Bezerra Coutinho, P.C.C. dos Santos, Eduardo Massad

Published Jan 1, 2024• 0 citations
Publisher

Abstract

This is the third article in a series dedicated to examining the use of the delta function in the physics literature. Here we discuss the appearance of a delta function in the second derivative of the wave function if its derivative is discontinuous. When this happens it is often claimed in the literature that the proposed wave function must be eigenfunction of an operator that contains a delta function and, in some cases, it is rejected. We study in this paper what we believe are alternatives ways to deal with this situation.

DOI
10.1590/1806-9126-rbef-2023-0316

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This is the third article in a series dedicated to examining the use of the delta function in the physics literature. Here we discuss the appearance of a delta function in the second derivative of the wave function if its derivative is discontinuous. When this happens it is often claimed in the literature that the proposed wave function must be eigenfunction of an operator that contains a delta function and, in some cases, it is rejected. We study in this paper what we believe are alternatives ways to deal with this situation.
The stationary states of a particle in a central potential are usually taken as a product of an angular part Phi and a radial part R. The function R satisfies the so-called radial equation and is usually solved by demanding R to be finite at the origin. In this work we examine the reason for this requirement in the case of a two-dimensional (2D) central force problem. In contrast to some claims commonly accepted, the reason for discarding solutions with divergent R(0) is not the need to have a normalizable wave function. In fact some wave functions can be normalized even if R is singular at the origin. Instead, here we show that if R is singular, the complete wave function psi = Phi R fails to satisfy the full Schrodinger equation, but follows a equation similar to Schrodinger's but with an additional term containing the 2D Dirac delta function or its derivatives. Thus, psi is not a true eigenfunction of the Hamiltonian. In contrast, there are no additional terms in the equation for wave functions psi built from solutions R that remain finite at the origin. A similar situation also occurs for 3D central potentials as has been shown recently. A comparison between the 2D and 3D cases is carried out.
There is much discussion in the mathematical physics literature as well as in quantum mechanics textbooks on spherically symmetric potentials. Nevertheless, there is no consensus about the behavior of the radial function at the origin, particularly for singular potentials. A careful derivation of the radial Schrödinger equation leads to the appearance of a delta function term when the Laplace operator is written in spherical coordinates. As a result, regardless of the behavior of the potential, an additional constraint is imposed on the radial wave function in the form of a vanishing boundary condition at the origin.
Doubts and misconceptions about the origin and justification of a physical theory\'s most fundamental concept are bound to spill over into doubts and misconceptions about the origin and foundation of the theory as a whole: such is the case, in point of fact, of the concept of quantum discontinuity of orthodox quantum mechanics. This work, which comprises three chapters, each of which is an independent paper, is a comprehensive investigation of the origin and nature of the concept of quantum discontinuity. The first paper, entitled \"Rewriting the Quantum \'Revolution\',\" gives a novel account of the origin of quantum theory. It refutes, by direct counter-example, Thomas Kuhn\'s well-known thesis that revolutions in science do not happen on the basis of theory and experiment alone; the quantum revolution happened exactly in this way. The second paper, \"On the Lawfulness of Quanta,\" is a sequel to \"Rewriting the Quantum \'Revolution\'.\" It evinces that and how the existence of the energy quanta is a principled result; it refutes the traditional viewthat the existence of the energy quantawas originally introduced in physics as an ad hoc hypothesis in a self-contradictory argument by showing that it is instead a necessary consequence of the laws of thermodynamics of black-bodies. This result affects our understanding of what is actually involved in the classical-to-quantum paradigm-shift, of scientific method, and of the specific nature of the quantum hypothesis which was carried over into contemporary physics in the course of the quantum revolution. The third paper, \"The Dual Dynamical Foundation of Orthodox Quantum Mechanics,\" was written in collaboration with the mathematical physicist Dr. Ricardo Correa da Silva. It is a detailed critical study of the dynamical foundations of orthodox quantum mechanics; it contains several original results of mathematical, historical, and theoretical interest. The most important of them is the discovery of a dual action functional from which both of the dynamical postulates of von Neumann\'s quantum mechanics, process 1 and process 2, necessarily follow. It also contains the following results that have interest on their own: (i) a derivation of the canonical commutation relations from Matrix Mechanics; (ii) an analysis of Schrödinger\'s first derivation of the wave equation that improves on the understanding of it that is now current in the literature; (iii) a demonstration of how the canonical commutation relations follow fromWave Mechanics; (iv) a discussion, that improves on F. A. Muller\'s work, about the theoretical equivalence of Matrix andWave Mechanics; (v) a revaluation of von Neumann\'s statement of the measurement problem. Broadly speaking, the three papers, taken together, drive home the lesson that the concept of quantum discontinuity stands on a more solid foundation than the scholarship on the foundations of quantum mechanics has hitherto acknowledged