Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

American Journal of Physics

Acceptable solutions of the radial Schrödinger equation for a particle in a central potential

J. Etxebarría

Published Sep 21, 2023• 0 citations
Publisher

Abstract

We revisit the discussion about the boundary condition at the origin in the Schrödinger radial equation for central potentials. We give a transparent and convincing reason for demanding the radial part R(r) of the wave function to be finite at r = 0, showing that if R(0) diverges the complete wave function ψ does not satisfy the full Schrödinger equation. If R(r) is singular, we show that the corresponding ψ follows an equation similar to Schrödinger's, but with an additional term involving the Dirac delta function or its derivatives at the origin. Although, in general, understanding some of our arguments requires certain knowledge of the theory of distributions, the important case of a behavior R ∝ 1/r near r = 0, which gives rise to a normalizable ψ, is especially simple: The origin of the Dirac delta term is clearly demonstrated by using a slight modification of the usual spherical coordinates. The argument can be easily followed by undergraduate physics students.

DOI
10.1119/5.0141536

Related papers

On the appearance of a Dirac delta term at the origin in the Schroedinger radial equation

J. Etxebarria

We revisit a recent discussion about the boundary condition at the origin in the Schroedinger radial equation for central potentials. Using a slight modification of the usual spherical coordinates, the origin of a previously reported Dirac delta function term at the origin is clearly shown up. As a consequence, a vanishing boundary condition at the origin must be imposed in solving the radial equation, regardless the kind of potential.

arXiv:1305.2782•May 13, 2013•0 citations

Acceptable solutions of the Schrodinger radial equation for a particle in a two-dimensional central potential

J. Etxebarría

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.

arXiv:2403.13422•Mar 20, 2024•0 citations

What is the boundary condition for the radial wave function of the Schrödinger equation?

Anzor Khelashvili, Teimuraz Nadareishvili

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.

arXiv:1009.2694•American Journal of Physics•May 13, 2011•31 citations

Unexpected Delta-Function Term in the Radial Schrodinger Equation

Anzor Khelashvili, Teimuraz Nadareishvili

Careful exploration of the idea that equation for radial wave function must be compatible with the full Schrodinger equation shows appearance of the delta-function while reduction of full Schrodinger equation in spherical coordinates. Elimination of this extra term produces a boundary condition for the radial wave function, which is the same both for regular and singular potentials.

arXiv:1002.1278•Feb 5, 2010•0 citations

Dirac’s reduced radial equations and the problem of additional solutions

Anzor Khelashvili, Teimuraz Nadareishvili

We show that additional solutions must be ignored (in differences of the Schrödinger and Klein–Gordon equations) in the Dirac equation, where usually the second-order radial equation is passed, called the reduced equation, instead of a system. Analogously to the Schrödinger equation, in this process, the Dirac’s delta function appears, which was unnoted during the full history of quantum mechanics. This unphysical term we remove by a boundary condition at the origin. However, the distribution theory imposes on the radial function strong restriction and by this reason practically for all potentials, even regular, use of these reduced equations is not permissible. At the end, we include consideration in the framework of two-dimensional Dirac equation. We show that even here the additional solution does not survive as a result of usual physical requirements.

arXiv:1610.07936•International Journal of Modern Physics E•Jul 1, 2017•12 citations

Delta function singularity in the Reduction of Radial Schrodinger Equation

Anzor Khelashvili, T. Nadareishvili

We obtain the extra delta-like singularity while reduction of the Laplace operator in spherical coordinates, elimination of which restricts the radial wave functions at the origin. This restriction has the form of boundary condition for the radial wave function.

arXiv:1009.3612•Sep 19, 2010•0 citations
We revisit the discussion about the boundary condition at the origin in the Schrödinger radial equation for central potentials. We give a transparent and convincing reason for demanding the radial part R(r) of the wave function to be finite at r = 0, showing that if R(0) diverges the complete wave function ψ does not satisfy the full Schrödinger equation. If R(r) is singular, we show that the corresponding ψ follows an equation similar to Schrödinger's, but with an additional term involving the Dirac delta function or its derivatives at the origin. Although, in general, understanding some of our arguments requires certain knowledge of the theory of distributions, the important case of a behavior R ∝ 1/r near r = 0, which gives rise to a normalizable ψ, is especially simple: The origin of the Dirac delta term is clearly demonstrated by using a slight modification of the usual spherical coordinates. The argument can be easily followed by undergraduate physics students.
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.
We show that additional solutions must be ignored (in differences of the Schrödinger and Klein–Gordon equations) in the Dirac equation, where usually the second-order radial equation is passed, called the reduced equation, instead of a system. Analogously to the Schrödinger equation, in this process, the Dirac’s delta function appears, which was unnoted during the full history of quantum mechanics. This unphysical term we remove by a boundary condition at the origin. However, the distribution theory imposes on the radial function strong restriction and by this reason practically for all potentials, even regular, use of these reduced equations is not permissible. At the end, we include consideration in the framework of two-dimensional Dirac equation. We show that even here the additional solution does not survive as a result of usual physical requirements.