Seminar Teaching and Learning in Mathematics

Abstract: An important milestone in teaching quantum mechanics is modeling the hydrogen atom. Usually, the path starts with solving the Schrödinger equation for the square well potential, placing the electron inside a one-dimensional box. It then moves up in complexity and closer to the hydrogen atom through the harmonic potential, the cubic well, and maybe even the quantum rotor. However, the most intuitive approximation to an atom, that of placing the electron inside a spherical box, is rarely included in textbooks, and important concepts, such as angular momentum, are lost. Here we introduce a graphical method of teaching the spherical well potential using an analogy between optics and quantum mechanics. It is backed up by recent experimental data on the optical resonances of an evaporating droplet. The explanation helps build a quantum mechanical intuition based on the shape of the potentials and the resulting resonances. As opposed to the square and harmonic potentials, it includes the concept of quantized angular momentum, while staying in one dimension, the radial one. The model can also highlight other quantum concepts such as tunneling, ground and excited states, and Fano resonances. This approach is akin to the analogy between a square well potential and a Fabry-Perot cavity and provides the missing link on the path to teaching the hydrogen atom.

This research seminar is arranged together with the research group PER at the Department of Physics.