A Discussion of Particles in Triangular Potential Wells and the Quantum Harmonic Oscillator
By Varun Jain
My first real encounters with physics were in Year 9 (the United Kingdom equivalent of 8th grade). It was here that my passion for the subject was sparked, thanks to my teacher at the time: Mr. Andrew Brittain. His immense enthusiasm rubbed off on me. Two of his lessons, in particular, are imprinted in my memory. One was devoted entirely to particle physics and in the second, he discussed quantum tunnelling. Both these topics were well beyond our curriculum, but he thought that they would be of interest and serve as a valuable reminder that there was more to physics than studying changes of state or rolling a ball down a ramp. I remember being instantly captivated by the mystical world of quarks, and intrigued that it was, in fact, possible (though extremely unlikely) to run through a wall and appear on the other side! As a result of my journey into calculus, I was able to participate in the Senior Physics Challenge 2019, where I completed over 600 challenging questions in an eight-month period. This led to me being one of 41 students invited to a four-day residential at Cambridge University at the start of July. At the famous Cavendish Laboratory, Professor Mark Warner delivered a series of enthralling lectures in which he gave us a rigorous mathematical introduction to quantum mechanics. I was finally able to make sense of ideas I had run into previously like eigenvalues, eigenstates, the Schrodinger wave equation, infinite and finite potential wells, and forbidden regions. I also learnt about how the structure of atoms, and thus of the universe, is governed by the complex interplay of electrical attraction and kinetic energy of localisation. The residential prompted me to write this paper, returning to where it all began in Year 9 (when I got my first look at quantum physics) but with a more informed perspective and an in-depth calculus-based approach which lent clarity to this inherently unintuitive field. I truly relished writing this paper and hope you can draw some inspiration from it!