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BEHAVIOUR OF THE WAVE FUNCTION NEAR THE ORIGIN IN THE RADIAL CASE

Emily C. Martin

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

We give one more proof in two and three space dimensions that the irregular solution of the Schrodinger equation, for zero angular momentum, is in fact the solution of an equation con- taining an extra delta function. We propose another criterium to eliminate the irregular solution which is to require the validity of the virial theorem of which we give a general proof in the classical and quantum cases.

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

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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
We give one more proof in two and three space dimensions that the irregular solution of the Schrodinger equation, for zero angular momentum, is in fact the solution of an equation con- taining an extra delta function. We propose another criterium to eliminate the irregular solution which is to require the validity of the virial theorem of which we give a general proof in the classical and quantum cases.
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.