WAVE PACKETS

4.5 WAVE PACKETS

In Section 4.3, we described measurements of the wavelength or frequency of a wave packet, which we consider to be a finite group of oscillations of a wave. That is, the wave amplitude is large over a finite region of space or time and is very small outside that region.

Before we begin our discussion, it is necessary to keep in mind that we are discussing traveling waves, which we imagine as moving in one direction with a uniform speed. (We’ll discuss the speed of the wave packet later.) As the wave packet moves, individual locations in space will oscillate with the frequency or wavelength that characterizes the wave packet. When we show a static picture of

a wave packet, it doesn’t matter that some points within the packet appear to have positive displacement, some have negative displacement, and some may even have zero displacement. As the wave travels, those locations are in the process of oscillating, and our drawings may “freeze” that oscillation. What is important is the locations in space where the overall wave packet has a large oscillation amplitude and where it has a very small amplitude. ∗

In this section we will examine how to build a wave packet by adding waves together. A pure sinusoidal wave is of no use in representing a particle—the wave

∗ By analogy, think of a radio wave traveling from the station to your receiver. At a particular instant of time, some points in space may have instantaneous electromagnetic field values of zero, but that

doesn’t affect your reception of the signal. What is important is the overall amplitude of the traveling wave.

120 Chapter 4 | The Wavelike Properties of Particles

extends from − ∞ to + ∞ , so the particle could be found anywhere. We would like the particle to be represented by a wave packet that describes how the particle is

localized to a particular region of space, such as an atom or a nucleus. The key to the process of building a wave packet involves adding together waves of different wavelength. We represent our waves as A cos kx, where k is the wave number (k = 2π/λ) and A is the amplitude. For example, let’s add together two waves:

y(x) = A 1 cos k 1 x+A 2 cos k 2 x=A 1 cos(2π x/λ 1 )+A 2 cos(2π x/λ 2 ) (4.17) This sum is illustrated in Figure 4.23a for the case A 1 =A 2 and λ 1 = 9, λ 2 = 11.

This combined wave shows the phenomenon known as beats in the case of sound waves. So far we don’t have a result that looks anything like the wave packet we are after, but you can see that by adding together two different waves we have reduced the amplitude of the wave packet at some locations. This pattern repeats endlessly from − ∞ to + ∞ , so the particle is still not localized.

Let’s try a more detailed sum. Figure 4.23b shows the result of adding 5 waves with wavelengths 9, 9.5, 10, 10.5, 11. Here we have been a bit more successful in restricting the amplitude of the wave packet in some regions. By adding even more waves with a larger range of wavelengths, we can obtain still narrower regions of large amplitude: Figure 4.23c shows the result of adding 9 waves with wavelengths 8, 8.5, 9, . . . , 12, and Figure 4.23d shows the result of adding

13 waves of wavelengths 7, 7.5, 8, . . . , 13. Unfortunately, all of these patterns (including the regions of large amplitude) repeat endlessly from − ∞ to + ∞ , so

even though we have obtained increasingly large regions where the wave packet has small amplitude, we haven’t yet created a wave packet that might represent

a particle localized to a particular region. If these wave packets did represent particles, then the particle would not be confined to any finite region.

FIGURE 4.23 (a) Adding two waves of wavelengths 9 and 11 gives beats. (b) Adding 5 waves with wavelengths ranging from 9 to 11. (c) Adding 9

waves with wavelengths ranging from 8 to 12. (d) Adding 13 waves with wavelengths from 7 to 13. All of the patterns repeat from − ∞ to + ∞ .

4.5 | Wave Packets 121

The regions of large amplitude in Figures 4.23b,c,d do show how adding more waves of a greater range of wavelengths helps to restrict the size of the wave packet. The region of large amplitude in Figure 4.23b ranges from about x = −40 to +40, while in Figure 4.23c it is from about x = −20 to +20 and in Figure 4.23d

increases from 2 to 4 to 6, the size of the “allowed” regions decreases from about

80 to 40 to 30. Once again we find that to restrict the size of the wave packet we must sacrifice the precise knowledge of the wavelength. Note that for all four of these wave patterns, the disturbance seems to have a wavelength of about 10, equal to the central wavelength of the range of values of the functions we constructed. We can therefore regard these functions as a cosine wave with a wavelength of 10 that is shaped or modulated by the other

cosine waves included in the function. For example, for the case of A 1 =A 2 = A,

Eq. 4.17 can be rewritten after a bit of trigonometric manipulation as

y(x) = 2A cos λ −

cos

If λ 1 and λ 2 2 −λ 1 ≪λ 1 ,λ 2 ), this can be

approximated as y(x) = 2A cos

where λ av = (λ 1 +λ 2 )/ 2≈λ 1 or λ 2 . The second cosine term represents a wave

with a wavelength of 10, and the first cosine term provides the shaping envelope 1 2A x sin that produces the beats. ∆lpx l 0.5 cos 0 2px 2 l

Any finite combination of waves with discrete wavelengths will produce

patterns that repeat between − ∞ to + ∞ , so this method of adding waves will

not work in constructing a finite wave packet. To construct a wave packet with a

finite width, we must replace the first cosine term in Eq. 4.19 with a function that

is large in the region where we want to confine the particle but that falls to zero

( a)

as x → ± ∞ . For example, the simplest function that has this property is 1/x, so we might imagine a wave packet whose mathematical form is 1

Ae –2(∆lpx/l 0 2 ) 2 0.5 cos 2 p x λ

2A o y(x) =

Here λ 0 represents the central wavelength, replacing λ av . (In going from Eq. 4.19 to Eq. 4.20, the cosine modulating term has been changed to a sine; otherwise the –1

( b)

function would blow up at x = 0.) This function is plotted in Figure 4.24a. It looks more like the kind of function we are seeking—it has large amplitude only in a FIGURE 4.24 (a) A wave packet

small region of space, and the amplitude drops rapidly to zero outside that region. in which the modulation envelope Another function that has this property is the Gaussian modulating function:

decreases in amplitude like 1/x. (b) A wave packet with a Gaussian

2 modulating function. Both curves are y(x) = Ae

0 ) 2 cos

λ 0 drawn for λ 0 which corresponds approximately to

which is shown in Figure 4.24b. Figure 4.23b.

122 Chapter 4 | The Wavelike Properties of Particles

Both of these functions show the characteristic inverse relationship between an that is used in constructing the wave packet. For example, consider the wave

packet shown in Figure 4.24a. Let’s arbitrarily define the width of the wave packet as the distance over which the amplitude of the central region falls by 1/2. That occurs roughly where the argument of the sine has the value ±π/2, which gives

0 , consistent with our classical uncertainty estimate. These wave packets can also be constructed by adding together waves of

differing amplitude and wavelength, but the wavelengths form a continuous rather than a discrete set. It is a bit easier to illustrate this if we work with wave number k = 2π/λ rather than wavelength. So far we have been adding waves in the form of A cos kx, so that

(4.22) where k i = 2π/λ i . The waves plotted in Figure 4.23 represent applications of the

y(x) =

A i cos k i x

general formula of Eq. 4.22 carried out over different numbers of discrete waves. If we have a continuous set of wave numbers, the sum in Eq. 4.22 becomes an integral:

(4.23) where the integral is carried out over whatever range of wave numbers is permitted

y(x) =

A(k) cos kx dk

(possibly infinite).

For example, suppose we have a range of wave numbers from k 0

at k 0 . If all of the waves have the same amplitude A 0 , then from Eq. 4.23 the form of the wave packet can be shown to be (see Problem 24 at the end of the chapter)

2A 0

y(x) =

This is identical with Eq. 4.20 with k

0 = 2π/λ 0 0 . This to obtain Eq. 4.6. With k = 2π/λ, taking differentials gives dk = −(2π/λ 2 ) dλ. Replacing the differentials with differences and ignoring the minus sign gives the

A better approximation of the shape of the wave packet can be found by letting ) e 2 −(k−k 0 /

A(k) vary according to a Gaussian distribution A(k) = A 2

0 . This gives a range of wave numbers that has its largest contribution at the central

wave number k 0 and falls off to zero for larger or smaller wave numbers with a

− ∞ to + ∞ gives (see Problem 25)

2 / y(x) = A 2

0 k 2π e

cos k 0 x (4.25)

which shows how the form of Eq. 4.21 originates.

By specifying the distribution of wavelengths, we can construct a wave packet of any desired shape. A wave packet that restricts the particle to a region in space

4.6 | The Motion of a Wave Packet 123

distribution. The mathematics of this process gives a result that is consistent with the uncertainty relationship for classical waves (Eq. 4.4).