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On Wave-Packets Dynamics On Wave-Packets Dynamics Learning notes Paul Durham Scientific Computing Department, STFC Daresbury Laboratory, Daresbury, Warrington WA4 4AD, UK 11 February 2020 Abstract These are working notes on wave-packets: their construction, behaviour and dynamics. Wave equations in classical and quantum physics are often linear. In such cases, wave-packets – linear combinations of solutions corresponding to different frequencies or energies – are themselves solutions of the wave equation, and may possess useful properties such as normalisability, localizability etc. They also tend to exhibit the motion occurring in quantum systems in a way that corresponds to classical concepts. These notes deal with the free motion and potential scattering of wave-packets, almost always in one dimension 1 . The basic theory here is (very) well known. Indeed, the quantum dynamics is essentially trivial, because we are considering only motion under a Hamiltonian that is constant in time. This means that we never have to solve the time-dependent Schrödinger equation (TDSE) directly; the solutions of the TDSE are simply linear combinations of energy eigenstate multiplied by the standard dynamical phase factor, which is what I mean by the term wave-packet. The fun part comes with a set of numerical calculations on simple models that demonstrate in great detail how wave-packet dynamics actually works. In these models, the wave-packets are always built from plane waves, because that fits the simple systems considered. But the basic theory applies to wave-packets constructed from energy eigenstates of any kind, depending on the system. C:\Blogs\Blog list\Post 3 - On wave-packet dynamics\Wave-Packet Dynamics.docx Contents 1 Classical wave-packets .................................................................................................................... 3 1.1 The wave equation ................................................................................................................................ 3 1.2 Properties of classical wave-packets ..................................................................................................... 3 1.2.1 At t=0 ................................................................................................................................................................. 4 1.2.2 For t>0 ............................................................................................................................................................... 5 2 Quantum wave-packets .................................................................................................................. 7 2.1 The time-dependent Schrödinger equation .......................................................................................... 7 2.2 Density and current ............................................................................................................................... 9 3 Wave-packet dynamics in free space ............................................................................................ 10 3.1 Motion ................................................................................................................................................. 10 3.2 Wave-packet spreading ....................................................................................................................... 11 4 Scattering of wave-packets in 1 ............................................................................................... 12 4.1 The time-independent scattering problem ......................................................................................... 13 4.2 Wave-packet dynamics ....................................................................................................................... 16 4.2.1 Total reflection ................................................................................................................................................. 17 4.2.2 Partial reflection-transmission ......................................................................................................................... 19 5 A general theory of wave-packet scattering ................................................................................. 21 References ............................................................................................................................................ 24 Appendix A Gaussian wave-packets ................................................................................................. 25 A.1. Normalisation Integral ........................................................................................................................ 25 A.2. Fourier Transforms .............................................................................................................................. 25 2 1 Classical wave-packets 1.1 The wave equation Consider the following partial differential equation governing the behaviour of a field A x, t - we work only in 1 here: 22AA1 (1.1) x2 c 2 t 2 This would be the wave equation of classical electromagnetic theory, for example. There are many other wave equations in classical physics [1]. The solutions are obviously waves: A x, t Nei kx t (1.2) where the frequency is related, through the wave equation, to the wavelength by k ck (1.3) This plane wave moves with speed ck / relative to the positive x-direction (in the sense that planes of equal phase – points of equal phase in this 1 case – move at this speed). One says that the phase velocity is / k . Because the wave equation is linear, we know that a linear combination of the plane wave solutions is also a solution of the wave equation. In the next section we construct and examine such a linear combination – a wave-packet. 1.2 Properties of classical wave-packets Working in 1 , take a wave-packet of general shape: 1/2 i kx k t A x,2 t dka k e (1.4) Remember that here the frequency k is a function of k because A x, t is a solution of some wave equation that relates and k . 2 2 The normalisation dx A 1(or, equivalently, dk a k 1) implies that A has the dimensions 1/2 1/2 length (and that a has the dimensions length ). We also want ak to be peaked at 1 kk 0 with a k-width of l , say. If we introduce a dimensionless function fq of a dimensionless variable q such that fq is peaked at q 0 and f q 11 when q , then we can write a k Cf k k0 l 1/2 where C is some overall amplitude scale with dimensions length that allows us to assume that fq is of order 1. If we take fq to be normalised in the sense that dqf 2 1 , then it is easy to 3 show that normalisation of A x, t implies that Cl 1/2 . Thus our canonical normalised form for the envelope function is 1/2 a k l f k k0 l (1.5) For example, if we take the standard normalised Gaussian form (see Appendix A ) for fq 2 f q 1/4 e q /2 (1.6) then 1/2 2 2 l k k0 l /2 a k 1/2 e (1.7) Thus (1.4) becomes 1/2 l i kx k t A x, t dkf k k0 l e 2 or, using the dimensionless variable q k k0 l , ikxkt 0 0 1/2 iqx/ l ikqlkt 0 / 0 Axte ,2 l dqfqee (1.8) We now look explicitly at the time evolution, ie the motion, of this wave-packet. 1.2.1 At t=0 From (1.8), we have 1/2 A x,0 eik0 x 2 l dqf q eiqx/ l (1.9) eik0 x l1/2 F x/ l where the (dimensionless) Fourier transform of fq is 1/2 F 2 dq f q eiq (1.10) Equation (1.9) says that at t 0the wave-packet is a plane wave eik0 x whose amplitude varies in space as F x/ l . Because we postulated the characteristics described above for fq , we can say that the amplitude function F will be peaked at 0 with width ~1 in the sense that F will be of order 1 and F 11 when (1.11) Moreover, if fq is normalised, so is F : 4 22 dq f q 1 d F (1.12) 2 For example, the Gaussian form (1.6) for fq gives Fe 1/4 /2 (see Appendix A ). The amplitude function F x/ l in (1.9) is thus peaked at x 0 and has width ~ l . Hence the standard picture: 1 0.5 -3 -2 -1 1 2 3 -0.5 -1 Note 1: It’s easy to show that for the following wave-packet (1.13) we get So this means that a wave-packet of the form (1.13) is localised at position at time . 1.2.2 For t>0 Now we have to know the function k . Note, from (1.8) that if is independent of k the wave- packet does not move at all; it just sits at x 0 oscillating up and down with frequency 1. Since we know that wave-vectors near k0 are important in the integral (1.8), we make a Taylor expansion of k around k0 : 1 Recall that in band theory a completely flat band means no hopping. But energy bands are a quantum, not classical, concept. The connection is just the wave equation in the two cases – see section 2.1. 5 2 k 0 v 0 k k 0 w 0 k k 0 ... (1.14) where 00 k dk v (1.15) 0 dk kk 0 1 dk2 w 0 2 dk2 kk 0 If we take only the linear term in (1.8) we get 1/2 A x, t ~ A1 x , t 2 l ei k0 x 0 t dq f q e iq x v 0 t/ l (1.16) 1/2 i k00 x t l e F x v0 t / l 1 i k x t Thus A x, t has a phase function e 00 representing a plane wave moving with speed 00/ k relative to the x direction and an amplitude function that is the same in shape as at t 0 but now centred at x v0 t . So the wave-packet moves with the group velocity v0 relative to the x- direction and maintains its width l . There’s no wave-packet spreading in this linear approximation. Now if we put in the quadratic term in (1.14) we get 2 1/2 i k x t iw q22 t/ l iq x v t/ l AxtAxt , ~ , 2 l e0 0 dqfqe 0 e 0 (1.17) 1/2 i k00 x t l e F x v0 t / l where F is the Fourier transform of the modified amplitude function 22 f q f q e iw0 q t/ l (1.18)
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