I would appreciate a quantum mechanic to answer this question. A colleague claimed that the wave vs. particle duality is analogous to the discrete vs. continuous duality between functions on the circle and functions on the set of integers (the Fourier coefficients). Is this an analogy or does it go deeper than that. Is the wave vs. particle duality an instance of either Fourier series or some other example of Pontrjagin duality?
Roughly speaking, a particle is a thing with a well-defined position, and a wave in quantum mechanics is a thing with a well-defined momentum. Since the position wavefunction is the Fourier transform of the momentum wavefunction (to within some constants), yes, you could say that the wave-particle duality is deeply connected to Fourier transforms.
Chronos, from what I’ve seen of your posts you’re obviously pretty knowledgeable in these subjects. Can you please explain what “wavefunctions” in general are and how they relate to the “momentum” and “position” in your post? Some mathematics is fine, but a general idea is what I’m grasping at at the moment.
In QM the wavefunction is simply the solution to Schrodinger’s equation for a specific potential.
In position space the amplitude squared gives you the probablity of finding the particle at a certain location, and in momentum space the probability of what its momentum is.
The two are Fourier transforms of each other.
That never occurred to me before. I understand Fourier transforms pretty well, but the wave-particle duality seemed utterly mysterious. Wonder why no one ever mentioned this before.