How Big Is a Photon and the Heisenberg Uncertainty Principle - Absolutely Small: How Quantum Theory Explains Our Everyday World - Michael D. Fayer 

Absolutely Small: How Quantum Theory Explains Our Everyday World - Michael D. Fayer (2010)

Chapter 6. How Big Is a Photon and the Heisenberg Uncertainty Principle

IN CHAPTER 5, WE LEARNED that a photon in an interferometer interferes with itself. In some sense, a photon can be in more than one place at a time. The photon location is described as a probability amplitude wave. This is not like a water wave, a sound wave, or even a classical electromagnetic wave. The wave associated with a photon (or other particles like electrons) describes the probability of finding the particle in some region of space. In the interferometer problem (Figures 3.4 and 5.1), a single photon was in leg 1 and leg 2 simultaneously, with equal probability of finding the photon in either of these regions of space. To understand and describe the location of a photon in more detail, it is necessary to discuss more aspects of waves. We need to know about the nature of the probability amplitude waves, particularly how they combine and what happens when a measurement is made.

The simplest problem to discuss is a free particle, which was introduced in Chapter 2. A free particle could be a photon, an electron, or a baseball. It is a free particle if no forces are acting on it. That is, there is no gravity, no electric or magnetic fields, no photons hitting an electron, no baseball bats hitting the baseball, no air resistance, and so on. With no forces acting on a particle, it has a perfectly defined and unchanging momentum. Thus, if it is moving in a particular direction, it will just keep going in that direction. We can call that direction anything we want, so let’s call it the x direction. Think of a graph with the horizontal axis x. We will just pick the direction of the x axis to be along the direction the particle is moving. In connection with Figure 2.5, we talked about a classical particle moving along x with a classical momentum p. Here we want to discuss the nature of a quantum particle with momentum p.

PARTICLES HAVE WAVELENGTHS

For a photon, the momentum is given as p = h/λ, where h is Planck’s constant and λ is the wavelength of the light. Therefore, the momentum is related to the wavelength (the color) of the light. Prince Louis-Victor Pierre Raymond de Broglie (1892-1987) won the Nobel Prize in Physics in 1929 “for his discovery of the wave nature of electrons.” De Broglie showed theoretically that particles, such as electrons or baseballs, also have a wave description. As discussed below, the wave description of electrons—or any type of particle—is in terms of the same types of waves as a photon, probability amplitude waves, as introduced in Chapter 5.

The wavelength associated with a particle is λ = h/p. This is a simple rearrangement of the formula for the photon momentum given above. If both sides of the photon momentum formula are multiplied by λ and divided by p, then the expression for the wavelength associated with a particle is obtained. De Broglie’s important result is that the relationship between the momentum and the wavelength is the same for photons (light) as it is for material particles, such as electrons and baseballs. Therefore, the properties of photons are described in fundamentally the same way as the properties of electrons, as well as baseballs. The wavelength associated with a particle is called the de Broglie wavelength. (We will see with physical examples in the next chapter why baseballs don’t seem to have wavelike properties, but photons and electrons do.)

WHAT A FREE PARTICLE WAVEFUNCTION LOOKS LIKE

For a free particle with some particular value of its momentum, p, what does the wavefunction look like? Recall that the wavefunction is related to the probability of finding the particle someplace in space. Figure 6.1 shows a graph of the wavefunction for a free particle with the momentum, p. As discussed above, the wavelength of the wavefunction associated with the particle is λ = h/p. As can be seen in the figure, the wavefunction for a free particle is represented by two waves called the real and the imaginary parts of the wavefunction. These components are equivalent. The term imaginary is a mathematical term. It does not imply that in some sense the imaginary component is less important than the part referred to as real. It is just jargon to identify the two components, although they do have differences in the way they are represented mathematically. The real and the imaginary components of the wavefunction have the same wavelength, but are shifted by one-fourth of the wavelength. That means that one wave is shifted in phase relative to the other by 90°. The two components of the wavefunction do not interfere with each other, either constructively or destructively, because in a mathematical sense and in essence they are perpendicular to each other.

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FIGURE 6.1. The wavefunction for a free particle with momentum p, which has wavelength, λ = h/p. A quantum mechanical wavefunction can have two parts, called real and imaginary. Both waves have the same wavelength. They are just shifted by one-fourth of a wavelength, which is the same as a 90° shift in the phase. These two components are separate from each other. They do not interfere either constructively or destructively. For a free particle with the well-defined value of the momentum, p, the wave function extends from positive infinity to negative infinity, +∞ to - ∞.

A PARTICLE WITH WELL-DEFINED MOMENTUM IS SPREAD OVER ALL SPACE

The important feature of the wavefunction shown in Figure 6.1 is that it extends from positive infinity to negative infinity, that is, from +∞ to -∞. In Figure 6.1, only a small section of the wavefunction in a small region of space is shown because we cannot plot +∞ to -∞ on a finite piece of paper. The wave shown in the figure just keeps going to the right and to the left. It is uniform across all space. This means that for a quantum mechanical particle with a definite value of the momentum, p, we are equally likely to find the particle anywhere along the x axis, the horizontal axis in the graph. The vertical axis tells the probability amplitude of finding the particle somewhere. Both the real (dashed) and imaginary (solid) components oscillate positive and negative. Both have places where they are zero.

The fact that the wavefunction oscillates positive and negative doesn’t matter. In explaining photon interference quantum mechanically following Figure 5.1, the Born interpretation of the wavefunction was introduced. In the Born interpretation, the absolute value squared of the wavefunction in a certain region of space gives the probability of finding a particle in that region of space. When the wavefunction is squared, it becomes only positive in the same way that 22 = 4 and (-2)2 = 4 because minus times a minus is a plus. Note that in Figure 6.1, whenever one wave is zero, the other wave is either at a positive or negative maximum. Where one wave is small, the other wave is big. When the wavefunction is analyzed mathematically, and as can be seen from the graph, at all locations the absolute value squared of the wavefunction is uniform along the x axis.

The absolute value squared of the wavefunction for a free particle is uniform along the x axis from +∞ to -∞. Therefore, the probability of finding the particle anywhere in space is uniform. The particle has equal probability of being found at x = 10, or x = -1,000,000, or anywhere else. Imagine that you are a tiny creature, frequently referred to as Maxwell’s Demon. You are standing next to the particle-wave shown in Figure 6.1. You make a grab for the particle. There is some probability that you will find it in your hand. If you start over and do this again and again, depending on the size of your hand, you may eventually come up with the particle. Each time you need to start fresh in your attempt to grab the particle. If you move somewhere else along the wave and do the same thing, the chance that you will come up with the particle is no different. This is what it means to say that there is equal probability for finding the particle anywhere. There is no best spot for Maxwell’s Demon to stand to try to grab the particle. All locations are equally good.

This picture of a free particle described by a wavefunction that represents equal probability of finding the particle anywhere doesn’t go along very well with our classical concept of a particle. In Figure 2.5, we described a classical particle as having a particular momentum and position at a given time. In discussing the photoelectric effect (Figure 4.3), Einstein described light as photons, which are quanta of light. One photon “hits” one electron, and the electron flies out of the piece of metal. This description almost sounds like both the photon and the electron are particles in the classical mechanics sense of particles. However, in discussing the interference of photons in conjunction with Figure 5.1, it was necessary to use the Born interpretation and describe photons as probability amplitude waves, with half of the probability going into each leg of the interferometer. In Figure 6.1, the plot of a free particle wavefunction is completely delocalized, spread out over all space. The description is the same for a photon or an electron.

INTERFERENCE OF WAVES WITH DIFFERENT WAVELENGTHS

So what are photons and electrons and rocks and anything else? Are they particles or waves? To see that there is no contradiction in the quantum mechanical description of the nature of things, we need to discuss waves and the interference of waves further. In connection with Figures 3.2 and 3.3, we discussed that waves could interfere constructively to give a bigger wave or destructively to give a smaller wave or no wave at all. In the examples in Figures 3.2 and 3.3, the waves have the same wavelengths. When they added constructively (Figure 3.2), all of the positive peaks lined up with the positive peaks and the negative peaks lined up with the negative peaks to give increased amplitude. When the waves added destructively (Figure 3.3), the positive peaks lined up with the negative peaks and vice versa, to give cancellation. However, waves of different wavelengths can also interfere.

Figure 6.2 shows a plot of five waves with different wavelengths. The units of length do not matter. What is important is that the five waves have wavelengths, λ = 1.2, 1.1, 1.0, 0.9, and 0.8. The phase of the waves are adjusted so that they all match at the point x = 0, where x is the horizontal axis. The waves match at x = 0 in the sense that each wave has a positive going peak at x = 0. However, because the waves have different wavelengths, the peaks do not necessarily match at other points along the x axis. For example, at about x = 10 or - 10, the dark gray wave has a maximum but the dashed light gray wave has a minimum. In addition, at approximately 10, one wave has a negative value and another has a positive value. At approximately x = 16 or - 16, two waves have maxima, but another wave is at a minimum. The important point is that for waves with different wavelengths, at one point (x = 0 in the example), all of the waves can be matched, but in general, at other points, some of the waves will be positive and some of the waves will be negative.

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FIGURE 6.2. Five waves are shown that have different wavelengths. The wavelengths are λ = 1.2, 1.1, 1.0, 0.9, and 0.8. The phases are adjusted so all of the peaks of the waves match at 0 on the horizontal axis. However, because the waves have different wavelengths, they do not match up at other positions, in contrast to Figure 3.2. Note that at a position of approximately 10 or - 10, the dark gray wave has a positive peak, but the dashed light gray wave has a negative peak.

Figure 6.3 shows the result of superimposing (adding up) the five waves in Figure 6.2. At x = 0 (horizontal axis) in Figure 6.2, all of the waves are exactly in phase. The superposition (adding the waves together) shown in Figure 6.3 yields a maximum. In Figure 6.2, the waves are all exactly in phase only at x = 0. Near x = 0, the difference in the wavelengths has not produced a large shift in the peaks of one wave relative to another, so the waves are still pretty much in phase. There is another set of maxima at about x = 6 and -6. However, these maxima are not as large as the one at x = 0, because the peaks of the waves are not all right on top of each other, as can be seen in Figure 6.2. Beyond x = ± 10, the amplitude of the superposition is getting small. At any point, some waves are positive and some waves are negative, and they tend to destructively interfere. Because there are only five waves, the destructive interference is only partial.

Figure 6.4 shows the superposition of 250 waves with different wavelengths. The waves have equal-sized steps in wavelength in the range of wavelengths from 0 to 4. As for the five waves and their superposition shown in Figures 6.2 and 6.3, each wave has the same amplitude. The phases of the 250 waves are adjusted to match at x = 0 (x is the horizontal axis). Because there are many more waves over a wider range of wavelengths than in the superposition shown in Figure 6.3, the peak around x = 0 is much narrower, and the rest of the superposition dies out much more rapidly. The little oscillations come from the fact that all of the waves in the superposition have the same amplitude. If the amplitude of the wave at the middle of the spread of wavelengths has the biggest amplitude and the amplitudes of the other waves get smaller and smaller for wavelengths further and further from the center wavelength, it is possible to create a superposition that decays smoothly to zero without the set of deceasing amplitude oscillations. This type of superposition will be discussed below.

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FIGURE 6.3. The superposition of the five waves shown in Figure 6.2. At x = 0 (horizontal axis), all of the waves in figure 6.2 are in phase, so they add constructively. Near x = 0, the waves are still pretty much in phase, but the next set of maxima at about x = 6 and -6 are not as large as the maximum at x = 0. In the regions between 10 and 20 and - 10 and -20, the difference in wavelengths makes some of the waves positive, where others are negative. There is significant cancellation.

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FIGURE 6.4. The superimposition of 250 waves with equally spaced wavelengths spanning the wavelength range 0 to 4. Compared to Figure 6.3, which is the superposition of five waves, this superposition has a much larger peak at x = 0, the region of maximum constructive interference, and destructive interference reduces the other regions more. The amplitude of the superposition is dying out going toward +20.

THE SUPERPOSITION PRINCIPLE

In Chapter 5, the interference experiment was analyzed in terms of the superposition of two photon translation states, T1 and T2. A photon in the interferometer is described as being in the 50/50 superposition state, T = T1 + T2. The idea of superposition is central to the quantum theoretical description of nature; it is called the Superposition Principle and assumes that “Whenever a system is in one state, it can always be considered to be partly in each of two or more states.”

An original state can be regarded as a superposition of two or more states, as in the interference problem in which the translation state T of the photon was described as a superposition of T1 and T2. Conversely, two or more states can be superimposed to make a new state. It is this second statement that we will now use to understand the fundamental nature of particles. The fact that a photon can act like a particle in the photoelectric effect but act like a wave to give rise to the interference effect follows from the superposition principle and leads to the Heisenberg Uncertainty Principle.

Eigenstates

In connection with Figure 6.1, it was stated that a free particle with perfectly defined momentum p is a delocalized probability amplitude wave spread out over all space. If a particle exists in such a state, it is said to be in a momentum eigenstate. In discussing the interference problem, we called T1 and T2 pure states, but the correct name for them is eigenstates. Eigen is German for characteristic, so an eigenstate is a characteristic state. An eigenstate for a particular observable property, such as momentum, is a state with a perfectly defined value of that property. The momentum eigenstates of a free particle are delocalized over all space. One such eigenstate exists for each of the infinite number of possible values of the momentum. Each of these momentum eigenstates is associated with an exact value of momentum of the particle. The particle’s location is uniform over all space because the wavefunction associated with the eigenstate is spread out over all space. However, the Superposition Principle tells us that we can superimpose any number of momentum eigenstates to make a new state.

Superposition of Momentum Eigenstate Probability Amplitude Waves

To understand the nature of real particles, photons, electrons, and so on, we will superimpose a range of momentum eigenstate probability amplitude waves, such as the one shown in Figure 6.1. For each momentum p, the wave has a different wavelength, λ = h/p. In Figures 6.3 and 6.4, we saw that adding together waves of different wavelengths concentrated the amplitude of the wave in a particular region. As mentioned in both examples above, the amplitude of each wave in the superposition was the same. Now we will superimpose momentum probability amplitude waves with different amplitudes. There is one wave (a particular value of p) with the largest amplitude. The other waves with different wavelengths have amplitudes that decrease as the wavelength becomes greater than or less than the wavelength of the wave with the maximum amplitude. So, we have a distribution of wavelengths centered around the wavelength of the maximum amplitude wave. The wavelength with maximum amplitude is at the center of the distribution. By a distribution, we just mean that there is a range of wavelengths, in the same way that if you have a room full of people there will be a distribution of ages. There will be some people of average age, the center of the distribution, and some people older than the average and some people younger. Here, we have a wave at the center of the distribution with other waves having shorter wavelengths and still others having longer wavelengths.

Figure 6.5 illustrates a distribution of momentum probability amplitude waves. p0 is the momentum of the wave at the center of the distribution of waves. It has a wavelength λ = h/p0. It is the wave with the biggest amplitude, that is, the biggest probability of finding it in the distribution. As the momentum is increased or decreased (λ is smaller or bigger) away from p0, the amount of a particular wave in the superposition (its probability) decreases. Δp is a measure of the width of the distribution. If Δp is large, then there is a large spread in p, and therefore, a large spread in wavelengths in the distribution. If Δp is small, the spread in wavelengths is small.

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FIGURE 6.5. A plot of the probability of finding a particle in a particular momentum eigenstate with momentum p given that it is in a superposition of momentum probability amplitude waves. p0is the middle wave with the biggest amplitude in the distribution. Δp is a measure of the width of the distribution of eigenstates.

Momentum of a Free Particle in a Superposition State

What is the momentum of a free particle that is in a superposition of momentum eigenstates such as that shown in Figure 6.5? A superposition of momentum eigenstates just means that we add together (superimpose) a bunch of waves (probability amplitude waves) with each of the waves having a specific value of the momentum associated with it (an eigenstate). In any measurement of a property of a system, a particular value of that property will be measured. If we make a measurement of the momentum of a particle, we will measure a single value of the momentum. The nature of the disturbance that accompanies the measurement of an absolutely small object is to collapse the superposition state into a single eigenstate. Making a measurement changes a system by taking it from its initial superposition state to one particular eigenstate. This is what is meant by collapse.

In discussing the interference problem, we said that if we tried to find out if the photon was in the state T1 by placing a detector in leg 1 of the interferometer, we would destroy the superposition state necessary for the interference. The superposition state T would jump into either T1 or T2. Since the state T is a 50/50 superposition of T1 and T2, half the time a measurement will result in finding the system in T1 and half the time in T2. On any single measurement, it is impossible to know ahead of time which result will occur. Many measurements show that the superposition is 50/50 because half the time we find that the photon is in leg 1 of the apparatus (state T1) and half the time we find the system is in leg 2 of the apparatus (state T2).

The superposition of momentum eigenstates shown in Figure 6.5 is composed of a vast (infinite) number of states spread over a range of momenta characterized by the width of the distribution, Δp. Therefore, there is a wide range of momentum values that can be measured on any single measurement. If we make a single measurement, we will measure one of the many values. Let’s say we make a measurement and find a momentum a little bigger than p0. Call it p1 because it is our first measurement. In the process of making the measurement, we made a nonnegligible disturbance of the system. It was changed from being in the superposition state to a single eigenstate with momentum p1. So to make another measurement, we need to start over again and prepare the particle (the system) in the same way we did originally to get the same distribution of momenta. We make a second measurement. This time we measure a value that is quite a bit smaller than p0. Call this value p2. We prepare the system again, and make another measurement. We measure p3. Each time we make a measurement on an identically prepared system, we will measure a particular value of the momentum. In advance, we don’t know what the value will be. If we make many, many measurements, we can plot the probability of measuring particular values of p. Such a plot gives the distribution like that shown in Figure 6.5. We can’t say what value we will obtain on a single measurement. However, we do know something. It is unlikely that we will measure a value of p that is much much greater or much much smaller than p0 because the distribution has very low amplitude in the wings (extremes) of the distribution. We are most likely to measure a value of p that is near p0 because this is the portion of the distribution where the amplitude is large.

Momentum Not Perfectly Defined for a Particle in a Superposition of States

A particle in a superposition of momentum eigenstates, such as that shown in Figure 6.5, does not have a perfectly well-defined value of the momentum. In a single measurement, we cannot say what value of the momentum will be measured. We can say we are most likely to measure a value near p0. With many measurements, we can determine the probability distribution. A classical particle, like the one illustrated in Figure 2.5, has a perfectly well-defined momentum. We can measure it without changing it. If it is a free particle, we can make many measurements of the momentum at different times, and we will always measure the same value of p. This is not the case for an absolutely small quantum particle in a momentum superposition state. We will measure a single value of p on a single measurement, but the act of making the measurement fundamentally changes the nature of the particle. The particle goes from being in a superposition state to being in an eigenstate (a single wave with a single value of the momentum). It goes from being in a state in which there is a probability distribution of momenta to a single value of the momentum that is observed. To recover the distribution, the particle needs to be prepared again.

WHERE IS A PARTICLE WHEN IT IS IN A MOMENTUM SUPERPOSITION STATE?

In connection with Figure 6.1, we said that a particle in a single momentum eigenstate is delocalized over all space. This doesn’t go along well with the description of the photoelectric effect. Now the question is where is a particle that is in a momentum superposition state? We have already hinted at the answer with the discussions surrounding Figures 6.2 through 6.4. In Figures 6.3 and 6.4, we saw that a superposition of waves with different wavelengths produced a spatial distribution that was concentrated in a region of space. In Figure 6.3, the wavelengths went from 0.8 to 1.2, and the pattern was not as concentrated as the one in Figure 6.4, which was formed from wavelengths that went from 0 to 4. Figure 6.6 shows the spatial distribution associated with distribution of waves (momentum eigenstates) shown in Figure 6.5. There is a position where the value is maximum, which is also the average value. For larger and smaller values of x relative to x0, the amplitudes (probabilities) become smaller.

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FIGURE 6.6. A plot of the probability of finding the particle at a location x given that it is in the superposition of momentum eigenstates shown in Figure 6.5. x0is the middle position with the greatest probability. Δx is a measure of the width of the spatial distribution.

What does the probability distribution of positions (x values) mean? A particle with the momentum probability distribution shown in Figure 6.5 gives rise to the spatial probability distribution shown in Figure 6.6. A single measurement of the position will measure a particular value of the position. Call it x1. When the position measurement is made on the absolutely small quantum particle, it causes a nonnegligible disturbance that collapses the position probability distribution into a position eigenstate with a perfectly defined value of the position. To make another measurement, the system (a particle) must be prepared again in the identical manner so that it has the same momentum probability distribution and, therefore, the same spatial probability distribution. The second measurement of the particle’s position will give a value, x2, 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.

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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 you that the momentum is both 10 and 50 at the same time. In a single measurement, a single well-defined value will be measured. How do we get a single value when our wave packet has a spread in momenta? The wave packet is made up of a superposition of momentum eigenstates, that is, momentum probability amplitude waves with associated well-defined values of the momentum. When a measurement is made, the nonnegligible disturbance accompanying the measurement causes the system to “jump” from being in a superposition state to a particular eigenstate. The measurement gives the value of the momentum that goes with that eigenstate. Now, the measurement changed the system. To make another measurement, we need to start over and prepare the particle in the same way. By preparing it in the same way, a wave packet composed of the same superposition of momentum eigenstates will be generated. We now make the same measurement we did the first time. In general, we will measure a different value of the momentum because the wave packet is composed of many momentum waves each with a different observable value of the momentum associated with it. If we do this over and over again, each time preparing the wave packet in the same way and then making the measurement, we will get a spread in the measured values of the momentum. After a huge number of measurements, we might measure (neglecting units) the value 400 a thousand times, 390 eight hundred times, 410 eight hundred times, but 200 and 600 only twenty times. If we make a plot of all of these numbers we get a probability distribution like those shown for momentum on the left side of Figure 6.7. A probability distribution is an experimental determination of the composition of the wave packet. We now know how much (what is the probability) of each wave in the packet. The same description also applies to the position of our wave packet. On each measurement of the position of identically prepared wave packets, a single location for the particle will be found. After many measurements, a distribution of positions is determined like those illustrated on the right side of Figure 6.7.

THE HEISENBERG UNCERTAINTY PRINCIPLE

A very important point is that there is a relationship between the spread in momentum and the spread in position that is fundamental to the superposition state description of particles. When the spread in momentum (Δp) is large, there are many waves spread out along the x axis (see Figure 6.1) that combine to make the wave packet. These waves have different wavelengths (see Figure 6.2). When many waves with a wide range of wavelengths interfere, the region of constructive interference dies out very fast away from the maximum (see Figures 6.3 and 6.4). That means that the spread in position (Δx) is small. If there is only a small range of the momentum waves that make up the wave pack (Δp small), then the spatial region of constructive interference dies out slowly away from the maximum in the position distribution (see Figure 6.7). Therefore, the spread or uncertainty in position Δx is large. All of this occurs because of the probability amplitude wave nature of the wave functions that describe the momentum eigenstates. A wave packet is located, more or less, in the region of constructive interference, and there is little probability of finding the particle in the regions of substantial destructive interference.

The formal relationship between the spread in the momentum and the spread in the position, that is, between Δp and Δx, is called the Heisenberg Uncertainty Principle. Werner Karl Heisenberg (1901-1976) won the Nobel Prize in Physics in 1932 “for the creation of quantum mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen.” The Heisenberg Uncertainty Principle is stated through a simple mathematical relationship, ΔxΔp ≥ h/4π, where h is Planck’s constant and the Δx and Δp define the widths of the distributions for position and momentum, as shown in Figure 6.7. (≥ means greater than or equal to.) Whether the equal sign (=) or the greater than sign (>) applies depends on the shapes of the probability distributions. The equal sign applies for a shape called a Gaussian after the great mathematician, Karl Friedrich Gauss (1777-1855). The curves shown in Figures 6.5 through 6.7 are Gaussians. A Gaussian is the standard “bell-shaped curve” that describes the distributions of things, such as test scores, for a properly designed test with a large enough number of people taking the test. Gaussian-shaped curves show up in many places in the physical sciences. The greater than sign applies to other shapes. For any shape, which is made up from a specific distribution of waves, it is possible to determine what the product ΔxΔp will be, but it will always be >h/4π unless the shape is a Gaussian.

To understand the nature of the Uncertainty Principle, it is sufficient to consider Gaussian shapes like those in Figure 6.7. Then, ΔxΔp = h/4π. The equation shows what it is possible to know simultaneously about the position and momentum of a particle. h/4π is a constant. Therefore, ΔxΔp equals a constant. So if the uncertainty in the momentum, Δp, is large, then the uncertainty in the position, Δx, must be small, so that the product is h/4π. On the other hand, if Δp is small, then Δx is large. The connection between Δp and Δx is illustrated in Figure 6.7. The uncertainty principle says that you can know something about the momentum of a particle and something about the position of a particle, but you can’t know both the position and the momentum exactly at the same time. This uncertainty in the simultaneous knowledge of the position and the momentum is in sharp contrast to classical mechanics. It is fundamental to classical mechanics theory, as illustrated in Figure 2.5, that the position and the momentum of a particle can be precisely known (measured) simultaneously. Quantum theory states that it is impossible to know both the position and the momentum precisely at the same time. We can know both within some uncertainties, Δx and Δp.

Examining the Uncertainty Principle relationship, ΔxΔp = h/4π, consider what happens as we make Δp smaller and smaller. As Δp becomes smaller and smaller, Δx grows. Dividing both sides of the equation by Δp gives Δx = 028 As Δp becomes smaller, we are dividing by a smaller and smaller number, so Δx grows. As Δp gets closer and closer to zero, Δx gets closer and closer to infinity. In the limit that Δp goes to zero, Δx goes to infinity. This limit has an important meaning. If Δp is zero, the momentum is known precisely, but the position is totally unknown. With Δx = ∞, the particle can be found anywhere with equal probability. This result is in accord with the discussion surrounding Figure 6.1, which shows the wavefunction for a momentum eigenstate. When a particle is in a momentum eigenstate, it has a perfectly well-defined value of its momentum. However, its probability amplitude function, which describes the probability of finding the particle in some region of space, is spread out (delocalized) over all space. The probability of finding the particle anywhere is uniform; Δx = ∞. This is in contrast to the wave packets shown in Figure 6.7, where a superposition of momentum eigenstates produces a state in which there is no longer a perfectly well-defined momentum, but there is some knowledge of the position. We know the position and momentum within some ranges of uncertainty.

If we rearrange the uncertainty relation to give Δp = 029 we see that in the limit that Δx goes to zero (perfect knowledge of the position), Δp goes to infinity. If we know the position perfectly, the momentum can have any value. A wave packet composed of all of the momentum eigenstates (Δp = ∞) has a perfectly well-defined value of the position. It is possible to know p precisely but with no knowledge of x; it is possible to know x precisely, but with no knowledge of p. This is called complementarity. You can know x or p but not both at the same time. In classical mechanics, you can know x and p. In quantum mechanics, you can know x or p. Generally for quantum particles, absolutely small particles, you know something about p and something about x, but you can’t know both precisely simultaneously.