Quantum Regime of Particles in Solids On my midterm today, I read that when the deBroglie wavelength of a particle exceeds the spacing between the particles in a solid or liquid, the particles begin to behave quantum dynamically. Why is this? I thought a larger deBroglie wavelength implied a less quantum mechanical behavior.
 A: The de Broglie wavelength


where lambda is the wavelength and p the momentum of a particle, E its energy and f the frequency in a proposition that the particle may appear as a wave.

The relations allow estimating whether a quantum mechanical entity will behave as a classical particle ( billiard ball) or as a probability wave. Accurate solutions of quantum mechanical equations justify this view.
Lambda is in units of length, and  the probability of finding the particle  within this length is high. If lambda is larger than the spacing of the solid's structure, for example, one will not be able to use the particle nature, but will have to consider the quantum mechanical solutions that will encompass more than one atom of the structure. (Though you do not refer in what situation a single particle will be found in  a solid, an electron in metal? )
Think of the particle passing through a slit. If its deBroglie wavelength is smaller than the width of the slit, it will act as a classical particle, if it is larger then the probability of hitting the wall is high.
A: It is because of the interference of particles on the crystal lattice. Imagine diffraction of visible electromagnetic waves on the microwave gratings (in this case the wavelength of photons is much smaller than the grating period). Nothing special happens. One can observe diffraction when the wavelength is comparable to the period. In macroscopic solids (if it is not a nanostructure), even if wavelength of the particle is very large it can always find a proper set of crystallographic planes to interfere with.
