# Quantum Eigenstates of Curved and Varying Cross-Sectional Waveguides

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## Abstract

**:**

## 1. Introduction

## 2. The General Equations

#### 2.1. Constant Orientation

#### 2.2. Variable Orientation

#### 2.3. Non Arc Length Parametrisation

#### 2.4. 2D Problems

#### 2.5. Schrödinger Equation

## 3. Transmission Studies through a Straight Waveguide with a Varying Thickness

## 4. 3D Eigenstate Problems

#### 4.1. Line with Circular Cross Section

#### 4.2. Elliptic Helix with Circular Cross Section

#### 4.3. Closed Ellipse with Circular Cross Section

#### 4.4. Tennis Ball Curve with a Square Cross Section

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Validation

#### Appendix A.1. Logarithmic Spiral

**Figure A1.**The six different cases in Table A1.

**Table A1.**The first five eigenvalues for the case $h=\frac{\alpha {e}^{\alpha u}}{{e}^{\alpha}-1}$, $\u03f5=0.05$. Calculated using both 1D and 2D equations.

$\mathit{\alpha}$ | ${\mathit{\lambda}}_{1}$ | ${\mathit{\lambda}}_{2}$ | ${\mathit{\lambda}}_{3}$ | ${\mathit{\lambda}}_{4}$ | ${\mathit{\lambda}}_{5}$ | |
---|---|---|---|---|---|---|

0.0 | 1D | 996.71 | 1026.32 | 1075.66 | 1144.75 | 1233.58 |

2D | 996.72 | 1026.34 | 1075.71 | 1144.82 | 1233.69 | |

0.2 | 1D | 925.10 | 1015.99 | 1094.11 | 1169.73 | 1255.79 |

2D | 925.12 | 1016.02 | 1094.15 | 1169.78 | 1255.88 | |

0.4 | 1D | 834.43 | 969.69 | 1088.59 | 1199.90 | 1307.49 |

2D | 834.45 | 969.72 | 1088.64 | 1199.95 | 1307.56 | |

0.6 | 1D | 752.84 | 918.86 | 1068.12 | 1210.19 | 1348.52 |

2D | 752.86 | 918.89 | 1068.16 | 1210.24 | 1348.57 | |

0.8 | 1D | 682.39 | 871.21 | 1044.48 | 1211.89 | 1376.83 |

2D | 682.39 | 871.22 | 1044.49 | 1211.89 | 1376.82 | |

1.0 | 1D | 622.31 | 828.74 | 1021.79 | 1210.85 | 1399.10 |

2D | 622.31 | 828.72 | 1021.74 | 1210.77 | 1398.99 |

**Figure A2.**The relative difference between 1D and 2D calculations as a function of $\alpha $. (

**left**) The first five eigenvalues in the case $\u03f5=0.05$. (

**right**) The first eigenvalue for different values of $\u03f5$.

**Figure A4.**The two cases $R=1$, $\mu =0.45\pi $, $\u03f5=0.3$, and $\alpha =\pm 0.8/L$, in Table A2. The colours represent the fifth eigenfunction calculated based on the 2D equation.

$\mathit{\alpha}\mathit{L}$ | ${\mathit{\lambda}}_{1}$ | ${\mathit{\lambda}}_{2}$ | ${\mathit{\lambda}}_{3}$ | ${\mathit{\lambda}}_{4}$ | ${\mathit{\lambda}}_{5}$ | |
---|---|---|---|---|---|---|

−0.8 | 1D | 15.99 | 17.64 | 19.09 | 20.44 | 21.75 |

2D | 15.88 | 17.58 | 19.06 | 20.43 | 21.75 | |

0.8 | 1D | 16.00 | 17.65 | 19.09 | 20.45 | 21.75 |

2D | 15.99 | 17.64 | 19.08 | 20.44 | 21.75 |

## Appendix B. Significance of the Different Terms

Case | ${\mathit{\lambda}}_{1}$ | ${\mathit{\lambda}}_{2}$ | ${\mathit{\lambda}}_{3}$ | ${\mathit{\lambda}}_{4}$ | ${\mathit{\lambda}}_{5}$ |
---|---|---|---|---|---|

(A1) | 74.13 | 103.74 | 153.08 | 222.17 | 311.00 |

(A2) | 81.25 | 112.16 | 157.22 | 228.03 | 316.90 |

(A3) | 57.48 | 110.62 | 167.66 | 236.54 | 324.39 |

(A4) | 63.80 | 120.12 | 173.19 | 241.69 | 330.02 |

Case | ${\mathbf{\lambda}}_{\mathbf{1}}$ | ${\mathbf{\lambda}}_{\mathbf{2}}$ | ${\mathbf{\lambda}}_{\mathbf{3}}$ | ${\mathbf{\lambda}}_{\mathbf{4}}$ | ${\mathbf{\lambda}}_{\mathbf{5}}$ |

(A1) | 74.13 | 103.74 | 153.08 | 222.17 | 311.00 |

(A2) | 101.52 | 105.88 | 210.91 | 230.51 | 385.76 |

(A3) | 72.45 | 110.92 | 153.34 | 228.77 | 311.94 |

(A4) | 108.18 | 113.17 | 215.24 | 237.46 | 386.54 |

**Figure A6.**The probability distributions for the full equation (case A4) for $\beta =20$ and $\beta =1000$.

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**Figure 1.**The physical domain. We consider a neighbourhood of a curve $\mathbf{r}$ (in black). The tangent vectors $\mathbf{t}$ (in red) are orthogonal to the normal planes (in pink). They intersect in this case the neighbourhood in disks of varying radius $\u03f5h$ centered at the curve. The vectors $\mathbf{p}$ and $\mathbf{q}$, in green and blue respectively, give an orthonormal basis for the normal plane.

**Figure 2.**The case ${h}_{0}=1\mathrm{n}\mathrm{m}$, $b=2\mathrm{\AA}$, ${L}_{m}=10\mathrm{n}\mathrm{m}$, and $n=1$, $\u03f5=1$.

**Figure 3.**The case ${h}_{0}=1\mathrm{n}\mathrm{m}$, $b=-2\mathrm{\AA}$, ${L}_{m}=10\mathrm{n}\mathrm{m}$, and $n=1$, $\u03f5=1$.

**Figure 4.**The case ${h}_{0}=1\mathrm{n}\mathrm{m}$, $b=2\mathrm{\AA}$, ${L}_{m}=10\mathrm{n}\mathrm{m}$, and $n=2$, $\u03f5=1$.

**Figure 5.**Energies for a tube around a straight line of length ${L}_{0}=10\mathrm{n}\mathrm{m}$, average width $1\mathrm{n}\mathrm{m}$ ($\u03f5=0.5\mathrm{n}\mathrm{m}$), and with trigonometric variation of the width, $h\left(s\right)=1+\frac{\alpha}{2}sin(\beta s-{\psi}_{0})$ and $\alpha =0.5$.

**Figure 6.**Energies for the elliptic helix with length $10\mathrm{n}\mathrm{m}$ and constant width $1\mathrm{n}\mathrm{m}$.

**Figure 7.**Eigenvalues for the closed ellipse with length $10\mathrm{n}\mathrm{m}$ and constant with $1\mathrm{n}\mathrm{m}$.

**Figure 8.**Three square tubes with length $25\mathrm{n}\mathrm{m}$ and constant thickness $1\mathrm{n}\mathrm{m}$ around a tennis ball curve with $b=a/20$. In the middle plot, the case with one quarter turn is shown. To the right with one half turn.

**Table 1.**The first five eigenvalues for the tennis ball curve with $b=a/20$, length $25\mathrm{n}\mathrm{m}$, average width $1\mathrm{n}\mathrm{m}$ ($\u03f5=0.5\mathrm{n}\mathrm{m}$), no turn, and trigonometric variation of width $h=1+\frac{\alpha}{2}sin(\beta s-{\psi}_{0})$, $\alpha =0.5$, ${\psi}_{0}=0$.

$\mathit{\beta}\mathit{L}$ | ${\mathit{E}}_{1}$ [eV] | ${\mathit{E}}_{2}$ [eV] | ${\mathit{E}}_{3}$ [eV] | ${\mathit{E}}_{4}$ [eV] | ${\mathit{E}}_{5}$ [eV] |
---|---|---|---|---|---|

0 | 2.10 | 2.21 | 2.21 | 2.52 | 2.53 |

$\pi $ | 1.54 | 1.90 | 2.27 | 2.64 | 3.00 |

$2\pi $ | 1.80 | 1.80 | 2.54 | 2.56 | 3.17 |

$3\pi $ | 2.10 | 2.11 | 2.12 | 3.13 | 3.13 |

$4\pi $ | 2.41 | 2.46 | 2.46 | 2.53 | 3.48 |

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Gravesen, J.; Willatzen, M.
Quantum Eigenstates of Curved and Varying Cross-Sectional Waveguides. *Appl. Sci.* **2020**, *10*, 7240.
https://doi.org/10.3390/app10207240

**AMA Style**

Gravesen J, Willatzen M.
Quantum Eigenstates of Curved and Varying Cross-Sectional Waveguides. *Applied Sciences*. 2020; 10(20):7240.
https://doi.org/10.3390/app10207240

**Chicago/Turabian Style**

Gravesen, Jens, and Morten Willatzen.
2020. "Quantum Eigenstates of Curved and Varying Cross-Sectional Waveguides" *Applied Sciences* 10, no. 20: 7240.
https://doi.org/10.3390/app10207240