The discussion clarifies that the wavefunction of the n-th excited state of a harmonic oscillator is expressed as the product of the n-th Hermite polynomial and the exponential function {\large e^{-\frac{x^{2}}{4l}}}, where l = \sqrt{\frac{\hbar}{2m\omega}}. It emphasizes that while Hermite polynomials are integral to the solutions of the Schrödinger equation for a harmonic oscillator, they alone cannot serve as solutions due to their non-normalizability over all x. Thus, the complete solutions must incorporate both Hermite polynomials and exponential functions.
PREREQUISITESStudents and professionals in physics, particularly those focused on quantum mechanics, as well as mathematicians interested in the applications of Hermite polynomials in physical systems.