Wavefunctions in Position and Momentum A Fourier Transform Exploration

In this video, Dr. Hudis solves a fascinating problem in quantum mechanics by exploring the ground state of the quantum harmonic oscillator (QHO) potential. He calculates the uncertainty in position (x) and momentum (p) and demonstrates how these uncertainties relate through the Heisenberg Uncertainty Principle. Using wavefunctions in both the position basis and momentum basis, Dr. Hudis illustrates how to use a Fourier transform to switch between these two representations, showcasing the beauty of basis transformations in quantum mechanics.
Throughout the video, Dr. Hudis explains critical concepts such as the ground state energy of the harmonic oscillator, the importance of the uncertainty relation, and how the wavefunctions change when observed in different bases. The video features detailed plots of the wavefunctions in both bases, providing a clear visual of how quantum states transform.
Whether you're studying the quantum harmonic oscillator, trying to grasp the Heisenberg Uncertainty Principle, or learning about the role of the Fourier transform in quantum mechanics, this video provides a step-by-step explanation designed to deepen your understanding of these fundamental topics.