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Bonding, Structure and Solid-State Chemistry$

Mark Ladd

Print publication date: 2016

Print ISBN-13: 9780198729945

Published to Oxford Scholarship Online: May 2016

DOI: 10.1093/acprof:oso/9780198729945.001.0001

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(p.463) A21 Wave Packets

(p.463) A21 Wave Packets

Source:
Bonding, Structure and Solid-State Chemistry
Author(s):

Mark Ladd

Publisher:
Oxford University Press

A21.1 Particle and wave

The de Broglie equation draws together the experimentally observed wave and particle properties of the electron through the well-known equation:

(A21.1)
p=h/λ,

whereby momentum, a particle property, and wavelength, a wave property, are reconciled through the Planck constant.

A21.2 Localized particle: superposition of waves

From the de Broglie equation (A21.1) that relates the particle and wave properties of an electron, it follows that:

(A21.2)
p=h/λ=mev(vee)and12mev2=E=hν(nu)

whence

(A21.3)
λν=h/mev×12mev2=12v

The speed λν of the wave appears to be one half the speed υ‎‎ of the electron, and must be considered further.

Consider an electron moving in vacuo. The Born interpretation of the wavefunction ψ‎‎ states that |ψ|2dτ is the probability of finding the electron in a volume element dτ,ordx in one dimension. If this probability is very high at any time t, then there is a high probability that there will be regions of zero electron density where the electron has left or not yet encountered. Given that the electron has energy E, it must possess finite values of momentum and wavelength (or wave number), so that the wave, in one dimension, has the spatially infinite form:

(A21.4)
ψ(x,t)=Asin(kxωt)

(p.464) where k=2π/λ and ω=2πν. In order to begin to represent a localized particle, two waves of the type Eq. (A21.4), differing by the small amounts ±Δk and ±Δν, must be compounded. By the usual trigonometric manipulation, the resultant wave expression is

(A21.5)
ψ(x,t)=2Asin(kxωt)cos(ΔkxΔωt)

(p.465) The first term on the right-hand side of Eq. (A21.4) is a wave oscillating at the average frequency, but modified by the following cosine term which oscillates by ±π/Δk=±Δλ/2. Thus, waves originally in phase at the origin become completely out of phase but subsequently come into phase again at the further distance of 2(±Δλ/2), and so on.

The beat frequencies decompose the continuous wave into a series of wave packets. A single travelling electron requires a single packet, which is obtained by a superposition of waves of the different orders represented by Δk. Since there is a multitude of wavelengths the out-of-phase waves can be in phase only at the origin.

The superposition of sine waves of the form Asin(πnX/a), where n represents frequency and X/a runs from −576 through zero to 576, is shown in Fig. A21.1, in which the curves represent a succession of 15 waves in (a) and 60 waves in (b). The diagrams show an increasing tendency to form a localized particle. However, it is not a true localized particle since only sine terms have been used and the plots are antisymmetric across the origin: the sine terms are those employed also in the electron-in-a-box equation (Section 2.10) and will be discussed further shortly.

A21 Wave PacketsA21 Wave Packets

Fig. A21.1 The superposition of sine waves of the form A sin (π‎‎nX/a), X = −576–576: (a) n = 1–15; (b) n = 1–60. The succession is moving towards the representation of a localized particle, but the functions are not totally harmonic.

A21.3 Phase and group velocities

Electron waves are unlike electromagnetic waves but resemble water waves in that they differ in phase and group velocities. Referring to Eq. (A21.5), the first term on the right-hand side represents waves with the speed ½υ‎‎ whereas the cosine term has the speed Δω/Δk.

From Eq. (A21.2):

(A21.6)
dE/dp=p/me=v

Since E=hν=hω/2π and p=h/λ=hk/2π, it follows that

(A21.7)
Δω/Δk=ΔE/Δp=v

Thus, the wave packet travels at the same speed υ‎‎ as the electron, whereas the waves within the packets travel at ½υ‎‎.

(p.466)

A21.4 Localized particle

The situations just discussed may be approached in another way. If a particle is situated at a specific location its representative wavefunction will be of large (p.467) magnitude, ideally infinite, at that site and zero elsewhere. This function can be created by the superposition of a large number of harmonic functions of the type Aexpikx, or its equivalent in terms of sine and cosine functions.

The wavefunction Aexpikx describes a particle travelling in the positive x direction with the precise momentum given by px=k. However, from the uncertainty principle (Section 2.7) the position of the particle is then (p.468) (p.469) indeterminate. Conversely, if the location of the particle is specified then its momentum cannot be known. The wavefunction of a localized particle can be created by the superposition of exponential functions or, equivalently, a combination of sine and cosine functions: Fig. A21.2 is Fig. A21.1b to which has been added the corresponding cosine functions. Thus, the linear combination of wavefunctions is beginning to form a localized wavefunction or wave packet.

A21 Wave PacketsA21 Wave PacketsA21 Wave PacketsA21 Wave Packets

Fig. A21.2 The superimposition of harmonic waves of the form A[sin(π‎‎nX/a) + cos sin(π‎‎nX/a)], X = −576−576: (a) n = 1–60, (b) n = 1–103, (c) n = 1–104, (d) n = 1–106, all of the same range of X as in Fig. A21.1. For the very large (albeit not infinite) number of waves in (d), the particle is completely located within limits of observation but its momentum now is indeterminate.

As the position becomes more and more localized by the continued addition of harmonics, so the particle is determined more and more exactly in position and less and less precise in momentum. In the limit, represented here by the superimposition of 106 waves, the visible result is a mathematical line of indeterminate height (Fig. A21.2d).

These results may be expressed through the Heisenberg uncertainty principle (Section 2.7) in the form:

(A21.8)
ΔpΔx/2

Each uncertainty in Eq. (A21.8) is defined as the rms deviation from its mean, that is, Δp=<p2><p>2 and Δx=<x2><x>2. If, as noted above, the momentum is known precisely, that is, Δp=0, then Eq. (A21.8) implies Δx=, which is an expression of indeterminacy of position. Conversely, if Δx=0, as implied by Fig. A21.2d, then Δp=, and the momentum is totally uncertain.

Although these results have been developed in one-dimensional space for convenience, they apply equally in two- or three-dimensional space.

Exercise A7

A proton travels at a speed of 0.5 Mm s−1. If the uncertainty in its momentum is to be reduced to 0.02%, calculate the uncertainty in its position. (Answer at end of Appendices.)

(p.470) Answers to Exercises

A1

b(100),e(010),d(001),

f(110),g(110),c(101),a(101),p(011),o(011),

r(111),m(111),q(111),n(11ˉ1ˉ).

A2

24.28.

A3

r=2nm,θ=135,ϕ=45or135(18045).

A4

(a)1.384×103Jm1(N)(b)1.384×105J.

A5

5033Vm1(NC1),awayfrom the fly.

A6

a=0.4514nm1,b=2.2710nm1,c=1.2726nm1,

α=88.66,β=84.82,γ=74.94.

V=0.04257nm3,V=23.490nm3.

A7

3.152×1010m(315.2pm).