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which in general will not be the same as x1. If the system is prepared again and again, and many measurements of the position are made, the position probability distribution shown in Figure 6.6 will be mapped out. Δx is a measure of the width of the spatial distribution. The spatial distribution shown in Figure 6.6, which is determined by many measurements on identically prepared systems, tells the likelihood of measuring any particular position. A measurement is more likely to find the particle somewhere in the vicinity of x0, but on any single measurement, it is impossible to say where the particle will be found. However, there is only a small probability of measuring a position that is far from x0.

Wave Packets

A particle in a superposition of momentum eigenstates, such as that shown in Figure 6.5, is called a wave packet. Its momentum is more or less known depending on how big Δp is. Because the momentum is the mass times the velocity and the mass of a particle is known, we more or less know the particle’s velocity. A bigger Δp (the wider the spread of momenta in the wave packet) results in the momentum being less well defined, which means that on any single measurement, one of a broader range of values of the momentum will be measured. The wave packet also has a spread in its position. The particle is not at a particular value of x as is a classical particle. There is a spread in positions given by the distribution like that in Figure 6.6, which can be quantified by the width Δx.

Spread in Momentum and Position

Figure 6.7 illustrates two wave packets. The top panels display a wave packet composed of a comparatively wide distribution of momentum eigenstates. The broad distribution of momentum eigenstates (large Δp) produces a spatial distribution that is relatively narrow (small Δx). The lower portion shows a wave packet composed of a relatively narrow distribution of momentum eigenstates (small Δp), which results in a relatively broad spatial distribution (large Δx).

The relationship between Δp and Δx illustrated in Figure 6.7 is general. A wave packet with a broad range of momenta (large uncertainty in momentum) will have a narrow spread of positions (small uncertainty in position). This relationship is produced by interference. A wave packet composed of a broad range of momentum eigenstates has a broad range of wavelengths because each momentum eigenstate has associated with it a probability amplitude wave with wavelength, λ = h/p. All of the probability amplitude waves in the packet can constructively interfere at some point in space. However, as shown in Figure 6.2, as the distance from this center point of constructive interference increases, destructive interference sets in. At any point far from the center, some waves will be positive while other waves are negative, as can be seen in Figure 6.2. When the spread in wavelengths is large, the vast differences in the wavelengths cause the onset of destructive interference very close to the center point of maximum constructive interference, and the packet is narrow (large Δp, small Δx). When the spread in wavelengths is small, the wavelengths are not very different from one to another. Therefore, it is necessary to move far from the center point of perfect constructive interference before an equal number of waves will be positive and negative at a given point. In this case, Δp is small so Δx is large.

FIGURE 6.7. The momentum (p) probability distributions and position (x) probability distributions for two wave packets. At the top, there is a large spread p (large Δp), which produces a small spread in x (small Δx). At the bottom, there is a small spread in p (small Δp), which gives rise to a large spread in x (large Δx).

Because the idea of a spread in momentum and a related spread in position is so important, let’s reprise the meaning of a spread. Everything is related to experiments. In a single experiment to measure the momentum of a particle, only one value can be measured. You have some instrument. It reads out a number. It can’t tell