I have personally come across this famour Heisenberg’s Uncertainty Principle many times in my lifetime – when I was a college student, when I started to learn about what is short-time FT, when I prepared my courses on wavelet. I must admit it is a difficult concept to grasp and I often adopt the strategy of avoiding the elaboration of this principle in my teaching. Recently after reading Feynman’s lecture notes on physics, I have finally reached the peace with this mind-bothering concept. I am writing this blog to share my thought experience – not only as a reassurance for myself but also as an exemplar situation where the subtle role of mathematics in understanding nature can be illustrated. My lesson is: to understand some profound concept such as uncertainty principle, good intuition and logic are more important than fancy mathematical languages.
The reasoning starts from a popular double-slit experiment designed to explain the particle-wave nature of electrons. In short, you can run this experiment for particles (e.g., idealistic bullets) or waves and conclude how interfence is associated with wave (the interference term – determined by the distance between two holes – is needed to properly account for the recorded experimental results).
Now consider the same experiment for electrons. The most interesting observation is that you are going to get different mathematical results depending on whether you “look at” the electrons or not (so mathematics merely serves as a tool of describing the phenomenon though it does provide useful hints to the underlying law of nature). In Feynman’s word, “If the electrons are not seen, we have interference”. Therefore, one could strive harder to reduce the disturbance to the experiment by reducing the frequency of light source (the device which helps us “see” electrons). However, as the wavelength becomes comparable to the distance between two holes, “visual inspection” of electrons is impossible – what you will see is a big fuzzy flash. It is from this perspective that one reaches the conclusion that one cannot simultaneously determine the location (which hole did the electron go) and the frequency (to eliminate the interference) simultaneously.
Heisenberg’s Uncertainty Principle is the foundation of quantum mechanics. Even though wavelets are closely related to this principle, I tend to argue that wavelets, just like Fourier transform, is just a tool invented for us to probe into the fundamental property of nature. We might obtain seemingly-conflicting mathematical results by twisting the experiment slightly. What is more important is how the deeper result such as uncertainty principle could be obtained by reconcling the conflicts at the surface.
