2024/03/20 by J. Etxebarría, Etxebarria, Jesus
Physics and Astronomy · #Crystallography and Radiation Phenomena #FOS: Physical sciences #Quantum Mechanics and Non-Hermitian Physics #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.2403.13422
openalex publication_date 2024/03/20 · openalex created_date 2024/03/22 · openalex updated_date 2026/07/28
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.