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

• URL: http://arxiv.org/abs/2403.13422v1
• Date: Wed, 20 Mar 2024 09:08:56 GMT
• Title: Acceptable solutions of the Schrodinger radial equation for a particle in a two-dimensional central potential
• Authors: Jesus Etxebarria,
• Abstract summary: 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. We show that if R is singular, the complete wave function psi = Phi R fails to satisfy the full Schrodinger equation.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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