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

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 electrons in polyenes, we follow two approaches.

Rotovibrational states of the water molecule on the sun.

The infrared spectrum of water observed in sunspots is complex and dense, with bands separated by approximately 0.01 cm. For top asymmetrical molecules, there is no theoretical approach that allows for the calculation of rotovibrational energy with such precision.