Making a decent semi-classical approximation even better

What is it about?

It is often difficult to formulate an analytical solution to a quantum mechanical problem. This reality has led to the development of a number of approximate techniques, many of which are semi-classical in nature. Amongst these, the so called WKB (Wentzel–Kramers–Brillouin) approximation is the most famous, historically leading to insights in tunnelling within nuclei and with the scanning tunnelling microscope (STM). The WKB is well-known for attaining accurate energies, however, it is generally not appreciated how inaccurate the accompanying wave functions are. In this work, not only do we expose this deficiency, but we also apply a lesser known semi-classical approximation, the Modified Airy Function (MAF) technique, to two important systems in quantum mechanics; the harmonic oscillator and double-well potentials. In these systems we find that the MAF wave function solutions are very accurate, in contrast to those of the WKB. WE believe that courses that teach the WKB approximation should include the MAF approximation as well.

Featured Image

Photo by ThisisEngineering on Unsplash

Photo by ThisisEngineering on Unsplash