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

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 \(\varvec{\pi }\) $\mathit{\pi }$ 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 \(\varvec{\pi }\) $\mathit{\pi }$ - \(\varvec{\pi ^{*}}\) ${\mathit{\pi }}^{\mathbf{\ast }}$ transitions of polyenes were calculated in good experimental agreement with both approaches, although with different wave functions.