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Home , Dispersion relation, Group velocity, Phase velocity

Phase velocity and

This note is part of the supporting online material for the textbook “Solid State Physics: An Introduction” by Philip Hofmann, Wiley-VCH, 2008. You can find more information on www.philiphofmann.net. The distinction between the and the group velocity of a is a concept of general significance for many different in physics: electromagnetic waves, particle waves, elastic waves and so on. It is explained and illustrated in this document, using little animations in the pdf file. You have to click on the images to start the animations. You need to have Quicktime installed for this to work. We start by considering a general one-dimensional wave

i(kx−ω(k)t) A(x, t) = A0e , (1) where A0 is the amplitude, k the wave number, ω(k) the angular , and t the time. The important thing to note is that ω depends on the wave number k or the λ = 2π/k. This phenomenon is called and it might be familiar from , where the in a material (or the index of ) depends on the wavelength. Starting from the dispersion ω(k), we define the phase velocity as ω v = (2) p k and the group velocity as ∂ω v = (3) g ∂k Obviously, these two are the same for a linear dispersion with ω(k) = ak. We are familiar with different kinds of dispersion relations and these are illustrated in Figure 1. For light in vacuum we have ω(k) = kc, (4) where c is the (vacuum) speed of light. This dispersion is shown in Figure 1(a). In general matter, n(ω) is greater than 1 and also depends on ω. This leads to a reduced and -dependent speed of light in matter kc ω(k) = , (5) n(ω) with n(ω) being the index of refraction. For a free, non-relativistic quantum mechanical particle of mass m, we have 2k2 E(k) = ω(k) = ~ , (6) ~ 2m n o h ehnclwv ersnigtevbain naoedmninlchain one-dimensional a in vibrations the have representing wave we mechanical the for and akg n codn oHiebr’ netit rnil hsipisalarge a implies range this small principle a uncertainty not only do Heisenberg’s use we will to Indeed, extension we according localized. because well in and be particularly respectively. shown package to not , case it is and three “particle” expect electrons the our even For , that see be space. will could in We “particles” localized these “particle” 1, a Figure as interpreted a around be centered will can waves we of such case, values of each velocity different package In a phase with form waves and dispersions. of velocity different propagation group these the of with study difference associated the situations respectively. at few a look (c), for shall and we 1(b) following, Figure the in In illustrated are cases These that so rpgtswt h aevlct steprilwvs h eoiyo h wave the of velocity The waves. “” partial velocity the the group that the as see is We velocity propagation around package “photon”. same peaked our the strongly to with is corresponds package propagates peak The this propagate and waves. waves position partial partial one these all are by waves that formed partial see package all velocity We the of phase respectively. and same (c), dot) the (blue and with package 2(b) the waves Figure of partial in of center package given the a at form wave we partial vacuum, the in light for different the g. with e. dispersion, linear a For ) b re o-eaiitcqatmmcaia atce(qain6,()teaosi branch acoustic the 8). (c) (equation 6), crystal (equation particle a mechanical in quantum vibrations non-relativistic of free, a (b) 4), 1: Figure

ω (arb. units) 0 0 0 a (b) (a) ∆ iprinrelations Dispersion x nspace. in k (arb.units) k ausidctdb h osi iue2a.Tepoaain of propagations The 2(a). Figure in dots the by indicated values ω ω ( k ( v ) k p o ieetpyia iutos a ih nvcu (equation vacuum in light (a) situations: physical different for 2 = ) = ω a ( ω/k steoedmninlltieconstant. lattice one-dimensional the is k r = ) k (arb.units) M = γ ~ v 2 c | g k m iue2d hw h rpgto of propagation the shows 2(d) Figure . sin 2 = , ∂ω/∂k ka k 2 pito neet uhapackage a Such interest. of -point | . (= k 0 (c) adtu of thus (and c ) nti atclrcase, particular this In . ∆ k ofr u wave our form to π k /a ω .Te we Then ). (7) (8) it is of course the same as the phase velocity. We also see that that the moving wave package does not change its shape. This is also a consequence of the linear dispersion that makes all partial waves travel with the same velocity.

(a) (arb. units) (arb. ω

0 0 k (arb. units)

Figure 2: (click on images to activate animations) Propagation of partial waves and wave package for a linear dispersion. (a) Dispersion with markers corresponding to the k values chosen for forming a wave package. (b) Propagation of the partial wave in the center of the package (the one with the blue dot). (c) Propagation of all seven partial waves. (d) Propagation of the package formed from all partial waves.

We now apply the same analysis to the quantum mechanical particle described by equation (6). The result is shown in Figure 3. Now the partial wave do not all move with the same velocity because of the quadratic dispersion. A very important consequence of this is that our initial wave package broadens out with time because the partial waves forming it gradually move out of phase with each other. So even if we start with a fairly localized “particle”, it will soon loose this localization. We can calculate the group velocity for this dispersion as ∂ω(k) k v = = ~ , (9) g ∂k m and this is perfectly consistent with the movement of a semiclassical particle for which the is p = ~k and the group velocity thus p/m = vg. We now study the dispersion of the lattice vibrations in equation 8 and Figure 1(c). Here, we have three interesting situations. For the first, we can choose a wave package around a small k value where the dispersion is linear. This is the aktde nedmv ntengtv ieto,ee huhalprilwaves partial all wave though our the even and past direction, linear) go (but negative negative the we obviously in as is move waves velocity indeed acoustic group of does the of packet at velocity Now centre dispersion phase zone 5. the and Brillouin Figure for group next the arises the near different be phase maximum, how The can image. of “catch the particle so: ones of illustration is faster a middle the this good the that all why particularly so in always While see ones A is slower can it not find: the but we does different with we 4(c), is “” up” what waves Figure localized partial precisely inspecting all the of is velocity, When velocity phase This all. positive at wave. a move have standing still a waves maximum velocity expect the partial group to would close the we sufficiently expect are would is, we we around If dispersion, waves 4. the Figure partial at of in from curve illustrated formed is dispersion sound. This package of the do boundary. wave waves speed do of a the the we maximum is that with so is the case but 2, difference light interesting Figure important of second in only speed that The The dispersion the see linear again. with We the here propagate for it not solid. as treat the to same have in the not waves) exactly (sound is waves situation the acoustic to corresponding regime hsnfrfrigawv akg.()Poaaino h ata aei h etro the of center Propagation the (d) waves. in partial wave waves. seven partial partial all all of the from Propagation of formed (c) the Propagation package dot). to the (b) blue corresponding of the markers with package. one with wave (the Dispersion package a (a) forming for dispersion. chosen quadratic a for package 3: Figure

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