Infinite square well: A common mistake

Brian P. Anderson

Benjamin Bernard, L. C. Lew Yan Voon

In the course of one's studies, it may be desired to approximate a particle constrained to a curved surface, such as a sheet of graphene or a fullerene, but without modelling the lattice of Coulomb potentials. This can be done using the machinery of differential geometry to formulate a Schr?dinger equation for a particle constrained so by applying an infinitely narrow infinite square well potential normal to the surface. The purpose of this paper is to clarify two mistakes that might be made when attempting to formulate this equation. These mistakes are: (1) incorrectly assuming that the Laplacian of the surface is the same as the Laplacian of the space, with the constrained term removed; and (2) neglecting to include the effective potential energy due to the curvature of the surface. It is also noted that the constrained approach produces results that, if the resulting Schr?dinger equation is separable, will produce one constant of separation instead of the traditional two. The emergence of the second quantum number is demonstrated and, hence, the apparent inconsistency is resolved. The prolate spheroid is given as a non-trivial example because it highlights the issues presented.

Antonio S. de Castro

In a recent paper diffused in the literature, the quantum problem of a particle in a infinite square well potential was claimed to be solved via Laplace transform. Surprisingly, the correct solutions were found for an eigenvalue problem with variable defined on a finite range. As a matter of fact, the authors have misused the Laplace transform of the time-independent Schrödinger equation, and they have also used incorrectly the inversion process. In the present paper it is pointed out carefully and clearly those mistakes that might occur with the use of inappropriate methods for a given eigenvalue problem.

David Peak

Some 1D infinite well stuff

Pieter Naaijkens

Pratik Adarsh, Sabyasachi Ghosh

There are various types of infinite potential well problems occurring in elementary quantum mechanics formalism. The infinite square well (one dimensional), cubical box and, spherical well are quite common in textbooks. In this paper, we consider a rather uncommon potential well, an infinite cylindrical well, and try to find its energy eigenvalues and eigenfunctions using Schrödinger equation. We also plot some radial wavefunctions and density plots.