On the quantum mechanics of a particle constrained to generalized cylinders with application to Möbius strips

Abstract

The intrinsic geometric approach to quantum mechanics has been applied to study nanostructures, as well as aromaticity and other electronic properties of molecules. Usually strong hypotheses on the geometry are made in order to allow for analytical solutions. We undertake the problem of stability, with respect to smooth deformations, of the spectra obtained under such simplifying hypotheses. In this article we show that symmetries can sometimes be removed by solving the problem of a particle constrained to a right cylinder with an arbitrary smooth cross section considering the effects of intrinsic geometry. The topology of the cross section, rather than its geometry, is shown to play a key role in this context. Our solution is applied to a generalized Möbius strip when the median curve is nearly a circle of diameter sufficiently large compared to the width of the strip. We also provide a systematic method for ordering the quantum states of generalized right cylinders according to their energies and apply this method to order the spectra of Möbius strips whose diameters are much larger than their widths. Finally, for the sake of comparison, our results are used to calculate the $\uppi$ electron energy spectra of five aromatic molecules (benzene, pyrazine, pyridine, 1,3 diazyne and 1,3,5 triazyne), showing a fair agreement with experimental data and quantum chemistry calculations.