# Energy levels of a quantum particle on a cylindrical surface with non-circular cross-section in electric and magnetic fields

## Abstract

We calculate the energy levels of a quantum particle on a cylindrical surface with non-circular cross-section in uniform electric and magnetic fields. Using separation of variables method and a change of independent variable, we show that the problem can be reduced to a one-dimensional Schrödinger equation for a periodic potential. The effects of varying the shape of the cross-section while keeping the same perimeter and the strengths of the electric and magnetic fields are investigated for elliptical, corrugated, and nearly-rectangular tubes with radial dimensions of the order of a nanometer. The geometric potential has minima at the angular positions where there is a significant amount of curvature. For the elliptical and corrugated tubes, it is shown that as the tube departs from the circular shape of cross-section the double-degeneracy between the energy levels is lifted. For the nearly-rectangular tube, it is shown that energy level crossings occur as the horizontal dimension of the tube is varied while keeping the same perimeter and radius of circular corners. The interplay between the curvature and the strength of the electric and magnetic fields determines the overall behavior of the energy levels. As the strength of the electric field increases, the overall potential getsmore »

- Authors:

- Publication Date:

- OSTI Identifier:
- 22617494

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Annals of Physics; Journal Volume: 379; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOUND STATE; CROSS SECTIONS; ELECTRIC FIELDS; EXCITED STATES; GROUND STATES; MAGNETIC FIELDS; ONE-DIMENSIONAL CALCULATIONS; QUANTIZATION; SCHROEDINGER EQUATION

### Citation Formats

```
Cruz, Philip Christopher S., E-mail: pscruz1@up.edu.ph, Bernardo, Reginald Christian S., E-mail: rcbernardo@nip.upd.edu.ph, and Esguerra, Jose Perico H., E-mail: jesguerra@nip.upd.edu.ph.
```*Energy levels of a quantum particle on a cylindrical surface with non-circular cross-section in electric and magnetic fields*. United States: N. p., 2017.
Web. doi:10.1016/J.AOP.2017.02.004.

```
Cruz, Philip Christopher S., E-mail: pscruz1@up.edu.ph, Bernardo, Reginald Christian S., E-mail: rcbernardo@nip.upd.edu.ph, & Esguerra, Jose Perico H., E-mail: jesguerra@nip.upd.edu.ph.
```*Energy levels of a quantum particle on a cylindrical surface with non-circular cross-section in electric and magnetic fields*. United States. doi:10.1016/J.AOP.2017.02.004.

```
Cruz, Philip Christopher S., E-mail: pscruz1@up.edu.ph, Bernardo, Reginald Christian S., E-mail: rcbernardo@nip.upd.edu.ph, and Esguerra, Jose Perico H., E-mail: jesguerra@nip.upd.edu.ph. Sat .
"Energy levels of a quantum particle on a cylindrical surface with non-circular cross-section in electric and magnetic fields". United States.
doi:10.1016/J.AOP.2017.02.004.
```

```
@article{osti_22617494,
```

title = {Energy levels of a quantum particle on a cylindrical surface with non-circular cross-section in electric and magnetic fields},

author = {Cruz, Philip Christopher S., E-mail: pscruz1@up.edu.ph and Bernardo, Reginald Christian S., E-mail: rcbernardo@nip.upd.edu.ph and Esguerra, Jose Perico H., E-mail: jesguerra@nip.upd.edu.ph},

abstractNote = {We calculate the energy levels of a quantum particle on a cylindrical surface with non-circular cross-section in uniform electric and magnetic fields. Using separation of variables method and a change of independent variable, we show that the problem can be reduced to a one-dimensional Schrödinger equation for a periodic potential. The effects of varying the shape of the cross-section while keeping the same perimeter and the strengths of the electric and magnetic fields are investigated for elliptical, corrugated, and nearly-rectangular tubes with radial dimensions of the order of a nanometer. The geometric potential has minima at the angular positions where there is a significant amount of curvature. For the elliptical and corrugated tubes, it is shown that as the tube departs from the circular shape of cross-section the double-degeneracy between the energy levels is lifted. For the nearly-rectangular tube, it is shown that energy level crossings occur as the horizontal dimension of the tube is varied while keeping the same perimeter and radius of circular corners. The interplay between the curvature and the strength of the electric and magnetic fields determines the overall behavior of the energy levels. As the strength of the electric field increases, the overall potential gets skewed creating a potential well on the side corresponding to the more negative electric potential. The energy levels of the first few excited states approach more positive values while the ground state energy level approaches a more negative value. For large electric fields, all bound state energy levels tend to more negative values. The contribution of weak magnetic fields to the overall potential behaves in the same way as the electric field contribution but with its sign depending on the direction of the component of the momentum parallel to the cylindrical axis. Large magnetic fields lead to pairing of energy levels reminiscent of 2D Landau levels for the elliptical and nearly-rectangular tubes.},

doi = {10.1016/J.AOP.2017.02.004},

journal = {Annals of Physics},

number = ,

volume = 379,

place = {United States},

year = {Sat Apr 15 00:00:00 EDT 2017},

month = {Sat Apr 15 00:00:00 EDT 2017}

}