Eigenvalues and Eigenfunctions for Angular Double-Well Potential

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This Demonstration presents the quantum-mechanical problem of a particle confined to a ring under the influence of a symmetric double quantum well (DQW) potential along the angular coordinate. The Schrödinger equation is solved for the allowed energy levels and the corresponding wavefunctions. One realization of this system might be a single-electron semiconductor quantum ring placed between two lateral gates, which we model as parallel negatively charged wires equidistant from the ring center and lying in the plane of the ring.


The DQW potential is governed by three parameters:

, the global potential strength in terms of the contribution from a particular gate

, the relative contribution from the other gate

, the ratio of the gate-ring distance

The potential can be approximated by the sum of two trigonometric functions, valid when , which leads to the single-particle Hamiltonian and its corresponding solutions. The gate-induced DQW potential and its trigonometric approximation are plotted.


Contributed by: Thomas P. Collier (June 2019)
Open content licensed under CC BY-NC-SA


We study the lowest single-electron energies and wavefunctions obeying the Schrödinger equation. The dimensionless external potential felt by an electron of charge and effective mass , confined between two in-plane gates is given by


where is the energy scale of the interlevel separation in the ring of radius , and is the electrostatic potential along the ring due to an individual negatively charged gate. We model the gates as charged wires, as in [1]. Thus the individual wire electrostatic potential is given by


Here labels the wires, is the linear charge density on a wire, and is its corresponding distance from the center of the ring. Zero potential is defined to be along the axis perpendicular to the reflection axis . For simplicity, equidistant gates are assumed. Making the approximation , we can expand up to second-order terms, which leads to the Hamiltonian



characterizes the global potential strength in terms of the contribution from gate . The relative contribution from gate is given by


We have accordingly shifted all calculated values by the arbitrary constant resulting from the expansion. Note that the approximated potential here is a corrected form of the potential used in [1–3] (which has a (harmless) error in the parameter in front of ). Solutions must be -periodic due to the ring periodic boundary conditions and are found as a sum of free particles on a ring basis function


The Demonstration plots the energy levels and wavefunctions for the Hamiltonian , as well as the approximate trigonometric potential for (red) and the full potential (dotted black), as functions of , and the ratio .

Snapshot 1: results for equivalent gate contributions

Snapshot 2: results for non-equivalent gate contributions and small global potential strength

Snapshot 3: results for deep global potential strength and non-equivalent gate contributions


[1] T. P. Collier, V. A. Saroka and M. E. Portnoi, "Tuning Terahertz Transitions in a Double-Gated Quantum Ring," Physical Review B, 96, 2017, 235430. doi:10.1103/PhysRevB.96.235430.

[2] T. P. Collier, V. A. Saroka and M. E. Portnoi, "Tuning THz Transitions in a Quantum Ring with Two Gates," Physics, Chemistry and Applications of Nanostructures (V. E. Borisenko, S. V. Gaponenko, V. S. Gurin and C. H. Kam, eds.), Singapore: World Scientific, 2017 pp. 172–175. doi:10.1142/9789813224537_0040.

[3] T. P. Collier, A. M. Alexeev, C. A. Downing, O. V. Kibis and M. E. Portnoi, "Terahertz Optoelectronics of Quantum Rings and Nanohelices," Semiconductors, 52(14), 2018 pp. 1813–1816. doi:10.1134/S1063782618140075.


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