# Quantum States in Twisted Tubes with Linear Cross-Section Variation

## TL;DR

This paper investigates how linear geometric transformations of twisted quantum waveguides affect particle states, revealing effects like gauge fields and energy splitting, and highlighting the robustness of circular cross sections.

## Contribution

It derives an effective Hamiltonian for particles in twisted tubes with linear cross section variations, including explicit forms for rotation, scaling, and shearing transformations.

## Key findings

- Rotation introduces a gauge field coupled to angular momentum.
- Scaling and shearing produce geometric potentials affecting energy levels.
- Circular cross sections maintain degeneracy under transformations.

## Abstract

We study the quantum dynamics of a particle confined in a twisted tube with a linearly varying cross section. By relating a general linear transformation matrix to the system's Hamiltonian, we use an extended thin-layer method to derive an effective Hamiltonian for tangential motion under mild and general linear transformations. Explicit forms are provided for three fundamental transformations: rotation, scaling, and shearing. Rotation introduces a gauge field coupled to angular momentum, while scaling and shearing produce geometric potentials that lift degeneracies in non-circular cross sections. In square cross sections, these transformations cause energy splittings among formerly degenerate states, whereas circular cross sections retain degeneracy. Through an example combining rotation and squeezing, we analyze state evolution and compute the quantum geometric tensor to quantify geometric response. Our results demonstrate how geometric transformations can tailor quantum states and suggest that circular waveguides are more robust against mode mixing.