Simple Nature - Light and Matter

An infinite sine wave can only tell us about the phase velocity,not the group velocity, which is really what we would be talkingabout when we refer to the speed of an electron. If an infinitesine wave is the simplest possible wave, what’s the next best thing?We might think the runner up in simplicity would be a wave trainconsisting of a chopped-off segment of a sine wave, d. However, thiskind of wave has kinks in it at the end. A simple wave should beone that we can build by superposing a small number of infinitesine waves, but a kink can never be produced by superposing anynumber of infinitely long sine waves.Actually the simplest wave that transports stuff from place toplace is the pattern shown in figure e. Called a beat pattern, it isformed by superposing two sine waves whose wavelengths are similarbut not quite the same. If you have ever heard the pulsating howlingsound of musicians in the process of tuning their instruments to eachother, you have heard a beat pattern. The beat pattern gets strongerand weaker as the two sine waves go in and out of phase with eachother. The beat pattern has more “stuff” (energy, for example)in the areas where constructive interference occurs, and less in theregions of cancellation. As the whole pattern moves through space,stuff is transported from some regions and into other ones.If the frequency of the two sine waves differs by 10%, for instance,then ten periods will be occur between times when they arein phase. Another way of saying it is that the sinusoidal “envelope”(the dashed lines in figure e) has a frequency equal to the differencein frequency between the two waves. For instance, if the waves hadfrequencies of 100 Hz and 110 Hz, the frequency of the envelopewould be 10 Hz.To apply similar reasoning to the wavelength, we must define aquantity z = 1/λ that relates to wavelength in the same way thatfrequency relates to period. In terms of this new variable, the z ofthe envelope equals the difference between the z ′ s of the two sinewaves.The group velocity is the speed at which the envelope movesthrough space. Let ∆f and ∆z be the differences between thefrequencies and z ′ s of the two sine waves, which means that theyequal the frequency and z of the envelope. The group velocity isv g = f envelope λ envelope = ∆f/∆z. If ∆f and ∆z are sufficientlysmall, we can approximate this expression as a derivative,v g = dfdzThis expression is usually taken as the definition of the group velocityfor wave patterns that consist of a superposition of sine waveshaving a narrow range of frequencies and wavelengths. In quantummechanics, with f = E/h and z = p/h, we have v g = dE/ dp.In the case of a nonrelativistic electron the relationship between.858 Chapter 13 Quantum Physics