  Context: 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.
  Method: To characterize the $\pi$ electrons in polyenes, we follow two
approaches. First, we utilize a weakened Coulomb potential associated with a
spiral curve. The solution to the Schr\"{o}dinger 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 $\pi$-$\pi^*$ transitions of
polyenes were calculated in good experimental agreement with both approaches,
although with different wave functions.
