Simple Nature - Light and Matter

Here’s how the now-legendary student was presumably reasoning.We start with the equation v = fλ, which is valid for any sinewave, whether it’s quantum or classical. Let’s assume we alreadyknow E = hf, and are trying to derive the relationship betweenwavelength and momentum:λ = v f= vhE= vh12 mv2= 2hmv= 2h p.The reasoning seems valid, but the result does contradict theaccepted one, which is after all solidly based on experiment.The mistaken assumption is that we can figure everything out interms of pure sine waves. Mathematically, the only wave that hasa perfectly well defined wavelength and frequency is a sine wave,and not just any sine wave but an infinitely long sine wave, c. Theunphysical thing about such a wave is that it has no leading ortrailing edge, so it can never be said to enter or leave any particularregion of space. Our derivation made use of the velocity, v, and ifvelocity is to be a meaningful concept, it must tell us how quicklystuff (mass, energy, momentum, . . . ) is transported from one regionof space to another. Since an infinitely long sine wave doesn’t removeany stuff from one region and take it to another, the “velocity of itsstuff” is not a well defined concept.Of course the individual wave peaks do travel through space, andone might think that it would make sense to associate their speedwith the “speed of stuff,” but as we will see, the two velocities arein general unequal when a wave’s velocity depends on wavelength.Such a wave is called a dispersive wave, because a wave pulse consistingof a superposition of waves of different wavelengths will separate(disperse) into its separate wavelengths as the waves move throughspace at different speeds. Nearly all the waves we have encounteredhave been nondispersive. For instance, sound waves and light waves(in a vacuum) have speeds independent of wavelength. A water waveis one good example of a dispersive wave. Long-wavelength waterwaves travel faster, so a ship at sea that encounters a storm typicallysees the long-wavelength parts of the wave first. When dealingwith dispersive waves, we need symbols and words to distinguishthe two speeds. The speed at which wave peaks move is called thephase velocity, v p , and the speed at which “stuff” moves is calledthe group velocity, v g .c / Part of an infinite sine wave.d / A finite-length sine wave.e / A beat pattern created bysuperimposing two sine waveswith slightly different wavelengths.Section 13.3 Matter As a Wave 857