Quantum mechanics of particles constrained to spiral curves with application to polyene chains

TL;DR

This paper explores how quantum particles behave on spiral curves, applying the findings to understand electron behavior in polyene chains.

Contribution

The paper introduces a new method for modeling π electrons in polyenes using a weakened Coulomb potential on spiral curves.

Findings

The Schrödinger equation with Dirichlet boundary conditions yields Bessel function solutions.

Both the weakened Coulomb potential and particle-in-a-box models agree with experimental π-π* transition data.

Effective mass corrections are incorporated in the particle-in-a-box approach.

Abstract

Due to advances in synthesizing lower-dimensional materials, there is the challenge of finding the wave equation that effectively describes quantum particles moving on 1D and 2D domains. Jensen and Koppe and Da Costa independently introduced a confining potential formalism showing that the effective constrained dynamics is subjected to a scalar geometry-induced potential; for the confinement to a curve, the potential depends on the curve’s curvature function.

To characterize the \documentclass[12pt]{minimal}
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\begin{document}$$\varvec{\pi }$$\end{document}π electrons in polyenes, we follow two approaches. First, we utilize a weakened Coulomb potential associated with a spiral curve. The solution to the Schrödinger equation with Dirichlet boundary conditions yields Bessel functions, and the spectrum is obtained analytically. We employ the particle-in-a-box model in the second approach, incorporating effective mass corrections. The \documentclass[12pt]{minimal}
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\begin{document}$$\varvec{\pi }$$\end{document}π-\documentclass[12pt]{minimal}
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\begin{document}$$\varvec{\pi ^{*}}$$\end{document}π∗ transitions of polyenes were calculated in good experimental agreement with both approaches, although with different wave functions.

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