A Propagating Wave Packet The Group Velocity

Transcription

1 Lecture 7 A Propagating Wave Pacet The Group Velocity Phys 375 Overview and Motivation: Last time we looed at a solution to the Schrödinger equation (SE) with an initial condition (,) that corresponds to a particle initially localized near the origin We saw that (,t) broadens as a function of time, indicating that the particle becomes more delocalized with time, but with an average position that remains at the origin To etend that discussion of a localized wave (pacet) here we loo at a propagating wave pacet The two ey things that we will discuss are the velocity of the wave pacet (this lecture) and its spreading as a function of time (net lecture) As we shall see, both of these quantities are intimately related to the dispersion relation ω ( ) This discussion has applications whenever we have localized, propagating waves, including solutions to the SE and the wave equation (WE) Key Mathematics: Taylor series epansion of the dispersion relation ω ( ) will be central in understanding how the dispersion relation is related to the properties of a propagating wave pacet The Fourier transform is again ey because the localized wave pacet will be described as a linear combination of harmonic waves I A Propagating Schrödinger-Equation Wave Pacet In the last lecture we found the formal solution to the initial value problem for the free particle SE, which can be written as i[ ( ) t ] ( t) = d C( ) e ω,, () π where the coefficients C ( ) are the Fourier transform of the initial condition (,) C π ( ) = d (, ) e i and the dispersion relation (for the SE) is given by h m,, () ω ( ) = (3) The eample that we previously considered was for the initial condition (, ) = e (4) D M Riffe -- 3/3/9

2 Lecture 7 Phys 375 We saw that for increasing positive time (,t) becomes broader (vs ), but its average position remains at the origin (indicating that, on average, the particle is motionless) So you might as, what initial condition would describe a particle initially localized at the origin, but propagating with some average velocity? Well, here is one answer: (, ) e i = e (5) As we will demonstrate you may thin of as some average wave vector (or momentum h through debroglie's relation p = h ) associated with the state (,t) As we did in the last lecture, let's find an epression for (,t) We start by using Eq () to calculate C ( ), so we have i ( de e ) C = π (6) This almost loos lie the Fourier transform of a Gaussian, which we can calculate Indeed, we can mae it be the Fourier transform of a Gaussian if define the variable =, so that the rhs of Eq (6) becomes π d e e i (7) This equals the Gaussian (in the variable ) a 4 e, (8) and now reusing the relation = we can write C ( ) 4 ( ) = e (9) As we stated in the last lecture, the Fourier transform of the Gaussian 4 e e is another Gaussian D M Riffe -- 3/3/9

3 Lecture 7 Phys 375 This equation is telling us something very important It tells us that the coefficients C ( ) are again described by a Gaussian, but it is a Gaussian centered at the wave vector That is, the state (,t) has an average wave vector with a distribution of wave vectors described by the Gaussian in Eq (9) With Eq (9) we can now use Eq () to write ( ) i[ = ( ) t t d e ω, e ], () 4 π eeping in mind that the dispersion relation ω ( ) is given by Eq (3) This loos complicated, so let's loo at some graphs of (,t) to see what is going on with this solution The following figure, which contains snapshots of the video SE Wavepacet 3avi, illustrates (,t) as a function of time (for a positive value of ) Notice that the wave pacet moves in the + direction with a constant velocity Notice also that (,t) is not simply a translation in time of the function (,) That is, the solution is not of the form g( vt), where v is some velocity This can be seen in the video by noticing that the center of the wave pacet travels faster than any of the individual t = t = Re( ( t, )) Im( ( t, )) ( t, ) ( t, ) t = 4 t = Re( ( t, )) Im( ( t, )) ( t, ) ( t, ) D M Riffe -3-3/3/9

4 Lecture 7 oscillation peas Phys 375 II The Group Velocity We now want to determine the velocity of the propagating wave pacet described by Eq () Because this solution (,t) can be thought of as having an average wave vector, you might guess that the velocity is simply the phase velocity v ph = ω( ) evaluated at the average wave vector That is, you might thin that the pacet's velocity is simply the velocity of the normal-mode traveling-wave solution i [ ( ) t, t e ] ω =, () which propagates in the + direction at the phase velocity v ph = ω ( ) = h m However, this is not correct! To figure out the pacet's velocity we must carefully analyze the propagating-pulse solution described by Eq () This solution lends itself to some approimation because part of the integrand, 4 e, is peaed at =, and for values of >> this part of the integrand is nearly zero The importance of this is that we only need to now what ω ( ) is for less than a few times the width parameter a That is, we only need to now what ω ( ) is for values of close to If ω ( ) is does not vary too much for these values of, then it maes sense to approimate ω ( ) by the first few terms of a Taylor series epansion about the point So we write + ω ( ) = ω( ) + ω ( )( ) + ω ( )( ) () If we now approimate ω ( ) in Eq () by the first two terms of the series, ω ( ) ω( ) + ω ( )( ), then we obtain i {[ ω( ) ( ) ] t} ( ) i [ ( ) t, t e d e e ] ω 4 ω (3) π This loos rather messy, but the integral can be calculated eactly to yield [ [ ] ] [ ] ω t ω t, e i t e (4) D M Riffe -4-3/3/9

5 Lecture 7 Phys 375 We can now easily see what is going on This (approimate) solution is the product of i the normal-mode traveling wave solution [ ( ω( ) ) t ], t = e (at the wave vector ), which travels at a speed equal to the phase velocity v ph ω ( ) = (5) (evaluated at ) and a Gaussian "envelope" function [ ω ( ) ] t e, which travels at a speed equal to ω ( ) The derivative ω ( ) is nown as the group velocity v gr dω ( ) = (6) d and so the envelope function, which describes the position of the pacet, travels at the group velocity (evaluated at = ) Note that the group and phase velocities are not necessarily equal The group velocity is typically more important than the phase velocity because the average position of the particle is given by the pea of the envelope function With these definitions of phase and group velocities we can now write Eq (4) as i [ ] [ ] v ph t vgr t a, t e e, (7) where both v ph and v gr are evaluated at the wave vector III Application to the Schrödinger Equation You may have noticed that nothing in the last section was necessarily directly related to the SE That is, Eq (7) can be applied to any situation where general solutions to the problem at hand can be described as a linear combination of harmonic, travelingwave solutions that have a dispersion relation ω ( ) The SE happens to be one of those situations; so let's apply the results of the last section to the SE We simply need to now that for the SE the dispersion relation is h m ω ( ) = (8) From Eq (8) we obtain the phase and the group velocities v ph ( ) = h m and ( ) = m, respectively Notice that the group velocity is twice the phase velocity v gr h D M Riffe -5-3/3/9

6 Lecture 7 Phys 375 This eplains the behavior of the pulse in the video, where the center of the pulse (which travels at v gr ) travels faster than any of the oscillation peas (which travel at v ph ) With the interpretation that Eq () describes a particle with an average momentum p = h, we see that the group velocity corresponds to the result for a classical, nonrelativistic particle v gr = p m IV Application to the Wave Equation We can also apply the results of Sec II to the wave equation Because harmonic traveling waves can also be used as basis functions for solutions to the WE (see Lecture notes ), we can also create a wave-pacet solution to the WE of the form of Eq (7) Again, we simply need to now the dispersion relation ω ( ) = c (9) in order to calculate the phase and group velocities, which are thus v ph ( ) = ω ( ) = c and v gr ( ) = ω ( ) = c, respectively So in this case the two velocities are equal! Then the solution given by Eq (7) becomes t = t = Re( ( t, )) Im( ( t, )) ( t, ) ( t, ) t = t = Re( ( t, )) Im( ( t, )) ( t, ) ( t, ) D M Riffe -6-3/3/9

7 Lecture 7 Phys 375 i [ ] [ ] ct ct a, t = e e () Notice that Eq () is a function of ct, and as such is an eact solution to the wave equation, rather than an approimate solution (as it is for the SE) (Why is that?) The figure on the preceding page shows snapshots of the video WE Wavepacet avi In contrast to the SE solution both the center of the wave pacet and the oscillation peas travel at the same velocity, consistent with the solution being a function of ct Eercises *7 Show that Eq (3) follows from Eq () with the linear Taylor's-series approimation described in the notes *7 Equation () is the Taylor's-series epansion of the dispersion relation about the point = For the dispersion relation appropriate to the WE, find all terms in this epansion Then argue why Eq (), is an eact solution to the WE **73 SE Approimate Solution (a) Calculate the integral on the rhs of Eq (3) and show that Eq (3) simplifies to Eq (4) (Hint: Transform the integral to be the Fourier transform of a Gaussian, and then use the fact that the Fourier transform of e 4 is e ) (b) Show that Eq (4) consistent with Eq (5), the initial condition *74 EM Waves For electromagnetic waves traveling in a dielectric material such as glass the dispersion relation is ω ( ) = ( c n), where n is the inde of refraction, which is often assumed to be a constant (a) If n is indeed a constant, calculate the phase and group velocities for these waves (b) Often, however, the inde of refraction depends upon the wave vector Assuming that n( ) = n + n, find the phase and group velocities (c) For n ( ) given in (b) show that v = v n ( n + n ) [ ] gr ph *75 Calculate * for the approimate wave function given by Eq (7) and show that * travels at the group velocity v gr D M Riffe -7-3/3/9