Question

1. Suppose there exists an infinite one-dimensional system satisfying the dispersion relation w(k)ak2 where a is a constant. Suppose at t 0 the wave is composed entirely of Fourier components traveling -HA Rekoj where A and I are to the right (a "running" wave) and the wave function is ψ(x,0) constants and kol »1. 1+(x/e) (a) Show that at a later time (and in particular at a time which is not so much later that the shape of the pulse has had time to substantially disperse), that to good approximation wave function is Rle ko(-at)) e!" ), Show that this means that the pulse travels of a distance of order (kol) before the wave substan- tially disperses. (Note that this is qualitative so that factors of two don't matter.) Since kol ! (b) The dispersion of the envelope function is small for times which are small compared tou this means that the pulse travels many times its length before dispersing It turns out that dispersion relations of this type arise in quantum mechanics; however, in that context the wave functions are not real for running waves!

Answer 1

In above probleam gaiven w(k) =a K^2

here , k =n pi /a then we get A^2 a/2 =1 by useing noremalization condition

the tranmitted flux N (t) = N (i) /N (t)

= 4 k1^2 /k1^2 + K3^2 A^2

here x- 2k0at / l =0 ,we geat t

2 k0 a t / l =x

t = x l / 2 k0 a