Quantum single-particle properties in a one-dimensional curved space

TL;DR

This paper explores quantum properties of a single particle constrained to a curved one-dimensional wire, comparing different Hamiltonian formulations and their resulting eigenvalues and eigenfunctions.

Contribution

It introduces multiple Hamiltonian models for a particle on a curved wire and analyzes their spectral differences, highlighting the limitations of geometric potential approximations.

Findings

Different quantization methods yield varying eigenvalues and eigenfunctions.

The JWKB approximation reproduces the square well spectrum without potential.

Geometric potential validity is limited to small curvature regimes.

Abstract

We consider one particle confined to a deformed one-dimensional wire. The quantum mechanical equivalent of the classical problem is not uniquely defined. We describe several possible hamiltonians and corresponding solutions for a finite wire with fixed endpoints and non-vanishing curvature. We compute and compare the disparate eigenvalues and eigenfunctions obtained from different quantization prescriptions. The JWKB approximation without potential leads precisely to the square well spectrum and the coordinate dependent stretched or compressed box related eigenfunctions. The geometric potential arising from an adiabatic expansion in terms of curvature is at best only valid for very small curvature.