Gratings Measure Momentum of Particle Using Wave Interference
If a particle beam (with a definite momentum per particle) falls at right angles on the grating, one sees several "orders" of a very sharp patterns of scattered particles in definite directions because of constructive interference from the lines of the grating which are analogous to the slits in our previous example. The relative uncertainty in the particle quantum wavelength is 1/Nm where N is the total number of lines in the grating and m is the order of the diffraction pattern. L is the difference in distance between the total distance that the particle has to travel if it scatters from the top of the grating compared to it scattering from the bottom of the grating. If the wave packet of the incident particle is shorter than L the whole grating is not used and the scattered peaks widen. The wave packet must be long enough so that at some moment the entire wave is scattering simultaneously from all parts of the grating. Under these conditions the resolution in the wave number k is 2pi/L. Therefore the bigger the grating the smaller the uncertainty in the momentum which is p = hk/2pi. The length of the incoming wave packet is obviously the uncertainty in the particle position. We have seen that when this length is smaller than L the uncertainty in the wave number is bigger than 2pi/L. Put it all together and it is the Heisenberg uncertainty principle.
Reference: 2-2 Feynman Lectures Vol III and Ch 30 of Vol I.
