In the realm of quantum physics, certain systems have become iconic due to their fundamental nature and the insights they provide. Among these are the particle in a box and the Coulomb potential. Now, a researcher from the University of British Columbia, Don MacMillen, has delved into the quantum solution for an electron confined between the grounded planes of an infinite capacitor, a problem that has intrigued physicists since at least 1929.
MacMillen’s work, published in the Journal of Physics A: Mathematical and Theoretical, explores the behavior of an electron in this unique setup. The potential in this system arises from the image charges that form in the grounded planes, with the added condition that the wavefunction must be zero at the planes’ boundaries. This setup effectively couples a hydrogen-like system to a particle-in-a-box (PIB) scenario, based on the distance between the planes, denoted as L.
The researcher provides a concise derivation for one of the limiting cases, yielding a compact expression for the potential. He demonstrates how the Kellogg infinite summation formula converges to this value. The potential in question is a symmetric double well potential, which means that many of its properties will be familiar to those versed in quantum physics.
Using this potential, MacMillen solves Schrödinger’s equation using a spectral technique. He finds that the limiting forms of a particle in a box for small L (and high energy, E) and that of a (degenerate) bound image charge at large L and small energy are recovered. Additionally, he discusses the tunneling level splitting that occurs in the transition from the large L to the small L regime.
For the energy sector, this research could have implications for understanding and manipulating quantum effects in electronic devices, particularly those involving quantum dots or other nanoscale structures. Quantum dots, for instance, are already used in some advanced photovoltaic cells and quantum computing applications. A deeper understanding of quantum behaviors in confined spaces could lead to more efficient and innovative energy technologies. However, the practical applications of this specific research are still in the theoretical stages and will require further exploration.
This article is based on research available at arXiv.