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A free particle is a linear superposition of de Broglie waves and satisfies the Shroedinger equation with zero potential.

Do the component de Broglies waves have physical meaning or are they mathematical formalisms?

Now I turn on a potential and the particle is no longer free. Where did the de Broglie waves go? Do they tranform according to some physical process? Is there an analogue of de Broglie wave that describes the components of the wave function in the presence of this potential? or is the de Broglie wave just a fiction used to motivate the Shroedinger equation?