An extension of the particle in a one dimensional box model
Cristiano C. Bastos, Gerson S. Paiva, Eduardo S. G. Leandro, and Antonio C. Pavão
Under the hypotheses commonly employed in textbooks, we calculated the spectrum of a particle in a one‐dimensional regular curve embedded in 2D with a general smooth parameterization x = g(t) and y = h(t), where t is the curve parameter. The solution of the one‐dimensional Schrödinger equation under the confined boundary conditions corresponding to our generalized particle in a box model shows that the linear box has the same energy spectrum as parabolas, cubics, exponentials, or any other open regular curves: E = h2n2/8mL2, whereas the circular box has the same energy spectrum as ellipses, ovals, or any other closed regular curves: E = h2n2/2mL2. There are many studies of quantum mechanics on curves using different theoretical approaches, however the present elementary approach and its conclusions are not found in the literature. We observe that the Schrödinger equation does not make sense when applied to spaces that are not manifolds.