Basic de Broglie relation question

Physics Asked by Anastassis Kapetanakis on May 9, 2021

First of all I am completely new to quantum mechanics and some equations arise while I am studying. Some of them are related with the de Broglie wave-particle duality of matter.

When de Borglie first postulated the wave-particle duality of matter, what did he mean by the wave-number of the particle? Did he mean that the position of the particle with respect to time was following a certain sinusoidal path or did he mean that the probability of finding the particle across space and time has a sinusoidal form?

As far as the de Broglie relation is concerned($λ=hp$), when the wavefunction is a superposition of waves with different wave-numbers, it has as a consequence the fact that different momenta arise for every wave-number. But I know that an object has only one momentum. How is that possible? Does that mean that momentum is quantized (as well as the energy does) and the particle has at the time of measurement only one momentum among them all, which has a specified probability associated with it?

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