On particle diffraction and its relation to the statistical interpretation of the wave function

Question

Particles can be diffracted due to their quantum nature and that is understood by their wave-like behavior. Clearly seen in e.g. plane wave solutions of the Schrodinger equation or a superposition of states which can be seen as a wave packet.

Statistical interpretation tells that those wave functions are merely to describe probability amplitudes, so one can argue they are not ontic/ physical waves.

On the basis of these two statements, why/how can de Broglie wavelength be used as in the classical wave approach to determine if a quantum object will be diffracted by a slit, lattice, nucleus etc. ?

Answer 1

IMO you would likely be ale to apply the de Broglie wavelength into a Feynman path integral, allowed paths would be "n" (n is integer) times the de Brogle length. Diffraction is a 2 step process 1) the particle interacts with the EM field of the slit edge material 2) the particle must interact with the screen in a quantum fashion .... this quantum fashion is the Feynman path which will produce the "interference " pattern. Areas of the screen that are dark have no particle impacts, bright areas have all the particles representing the most probable paths. All interactions are governed by the EM field and since the particle travels as a wave in this field it must interact with the screen according to Feynman/QM.

Answer 2

In a section of Heisenberg's The Physical Principles of Quantum Theory; he emphasizes Duane's perspective on corpuscular picture of diffraction/reflection and points out for example for a grating, if it is known that the particle will hit a certain $$\Delta x $$ of the grating then the momentum will become uncertain by an amount proportional to $$\frac{h}{\Delta x}$$ The reflection direction will therefore be uncertain according to the relation above. The value of this uncertainty can be calculated from the resolving power of a grating of

$$ \frac{\Delta x}{d} $$ lines; d being a plane ruled grating's constant. When, $$ \Delta x < < d $$ the interference maxima are no longer visible and the trajectory can be compared to that of a classical particle.