Equation (6-21) is an infinite sum of quantum-mechanical plane waves. Inserting the proper...

Equation (6-21) is an infinite sum of quantum-mechanical plane waves. Inserting the proper A(k) and carrying out the integral gives a wave function with distinct phase and group velocities. The basic ideas can be illustrated, however, by replacing the integral with a sum of a small number of cosine functions and plotting the results as a function of x at a series of time values. Our simplified function is

Note that parentheses after A and ω indicate function arguments, not multiplication. Where equation (6-21) has a k, this sum has (1 + n/10), and its seven terms thus correspond to values of k between 0.7 and 1.3. Two functions still need to be specified. Define A(k) as e-25(k-1) This says that the cosine function for which k = 1 has an amplitude of 1, and the amplitudes of the other cosines fall off. The farther k is from central value 1, the smaller the amplitude. Define ω(k) as simply k. (a) Plot f(x,0) from x = -10 to x = 40. Given that the central value of k is 1.0, does your plot appear to have the correct approximate wavelength? (b) Make a series of plots from x = -10 to x = 40, with each successive plot incrementing the value of t by 0.5, ending with t = 20. Given that you have defined ω(k) as simply k, what should be the phase and group velocities, and do the crests and envelope indeed move at these speeds? (Animating the series of plots is helpful.) (c) Define ω(k) as k2/2. Make another series of plots with the same x and f values, and, remembering that your A(k) still makes k = 1 the central value, answer the questions in part (b) again, (d) Define ω(k) as 2k1/2. Make another series of plots with the same x and t values, and once again answer the questions in part (b).