Please help me in solving a question stuck in my head about symmetry and conservation laws? Can wave-particle duality be considered an atomic-particle symmetry? And if so, what is the conservation law behind this symmetry?
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Wave-particle duality is not a symmetry, although based on other places the word "duality" is used in physics, I could see why you might think it is.
If wave-particle duality were a symmetry, you would expect something like the following to be true. You can do a calculation with "wave" variables and get an answer. Then, in the original problem, you transform the "wave" variables to "particle" variables and repeat the calculation. At the end, you find the answer is exactly the same, regardless of whether the original problem was phrased in "wave" or "particle" variables. This is not a good description of wave-particle duality.
Instead, a "quantum thing" is neither a wave nor a particle. It is simply a different, quantum thing, that obeys its own rules. In certain limits, a "quantum thing" behaves like a wave. In other limits, a "quantum thing" behaves like a particle. There is no limit where a "quantum thing" simultaneously behaves like a wave and a particle. The fact that the "quantum thing" can sometimes behave like a wave, and sometimes like a particle, but never like both things, is the origin of the name "wave-particle duality."
A more advanced way of expressing this, is to talk about the operators used to represent a quantum field. You can express quantum fields in terms of the number operator, which counts how many particles (or "quanta") are present in a given state. States which are eigenstates of the number operator, have a clear particle interpretation. You can also express quantum fields in terms of the phase operator$^\star$, which tells you the phase of the field -- where the peaks and troughs are. States with a definite phase have a clear wave interpretation. However, the number and phase operators do not commute. Therefore, it is typically not possible for a system to be simultaneously in a state that both has a definite particle number, and a definite phase. Here we see explicitly that the quantum field is the "quantum thing", and if we use the quantum field to describe the physics, we always get the right answer (even though the right answer may be very counterintuitive). Meanwhile, we see in some special cases (if the quantum state is as a particle number eigenstate of a phase eigenstate) that we can give a particle or wave interpretation to the state, but there are no states where both interpretations are simultaneously possible. (With the exception of the vacuum state, where there are no particles and also no wave)
$^\star$ I'm simplifying a bit to keep this answer short. Really the "phase operator" is not Hermitian, and in a rigorous analysis one should break the field into operators associated sine and cosine components (or quadratures), as described for instance in https://journals.aps.org/ppf/pdf/10.1103/PhysicsPhysiqueFizika.1.49. However, the net result is that there is an uncertainty relationship between phase and number, which is the main point of this paragraph.
Andrew
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