Use the de Brogue relation between the wavelength and the linear momentum of a non-relativistic particle, and the Planck relation between frequency and quantum energy, to obtain the phase and group velocities of the wave motion associated with a particle with velocity n << c. A de Brogue wave travelling in the z-direction passes through a narrow slit, of width a in the x-direction and long in the y-direction. Derive a relationship between the width of the slit and the angular spread of the emerging beam in the xz-plane. and show that it is consistent with the uncertainty principle.