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

Chapter 3. Some Things About Waves

TO ADDRESS THE NATURE of the inherent disturbance that accompanies a measurement and to understand what can and cannot be measured about an absolutely small quantum mechanical system, first it is necessary to spend some time discussing classical waves and the classical description of light. At the beginning of the twentieth century, a variety of experiments produced results that could not be explained with classical mechanics. The earliest of these involved light. Therefore, we will first discuss an experiment that seemed to show that classical ideas work perfectly. Then, in Chapter 4, we will present one of the experiments that demonstrated that the classical mechanics description could not be correct and, furthermore, that a classical reanalysis of the experiment seemed to work, but actually didn’t. Finally, the correct analysis of the experiment involving light will be given using quantum ideas, which will bring us back to Schrödinger’s Cat.

WHAT ARE WAVES?

There are many types of classical waves, water waves, sound waves, and light waves (electromagnetic waves). All waves have certain common properties, including amplitude, wavelength, speed, and direction of propagation (the direction in which a wave is traveling). Figure 3.1 shows a wave traveling in the x direction. The amplitude of the wave is the “distance” between its positive and negative peaks, the up-to-down distance. The wavelength is the distance along the direction of propagation between two positive or negative peaks. This is the distance over which the wave repeats itself. If you are riding on the wave and you move any integer number of wavelengths forward or backward along the wave, everything looks the same. The wave is traveling with some velocity, V.

WAVES HAVE VELOCITIES AND FREQUENCIES

The velocity depends on the type of wave, and the velocity of a wave needs a little discussion. Imagine you are standing beside the wave in Figure 3.1, but the wave is so long that you cannot see its beginning or end. Still, you can determine its velocity using a timing device. Start timing when a positive peak just reaches you and stop timing when the next positive peak reaches you. You now have enough information to determine the wave’s velocity. The wave has traveled d, a distance, of one wavelength, in time t. The distance equals the velocity multiplied by the time, d = Vt. (If you are in a car going at velocity, V = 60 miles per hour, and you travel for a time, t = 1 hour, then you have traveled a distance, d = 60 miles.) If we take the distance of one wavelength and divide it by the time it took to travel one wavelength, then we know the velocity, V = d/t. Watching the wave go by is like watching a very long train go by. You see boxcar after boxcar pass you. If you know the length of one boxcar and how long it takes that one boxcar to pass by, then you can determine the velocity of the train.

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FIGURE 3.1. A wave traveling in the x direction. The black line represents zero amplitude of the wave. The wave undergoes positive and negative oscillations about zero. The distance between the peaks is the wavelength. The wave is traveling along x with a velocity V.

Another important property of waves that is related to their velocity and wavelength is the frequency. Scientists love using Greek letters to represent things because we tend to use up all of the Roman letters early on. There is no reason the velocity has to be V or distance d or time, t, but these are usually used, and many of the letters of the Roman alphabet have common usages. Therefore, we turn to the Greek alphabet. It is common to call a wave’s wavelength λ (lambda) and a wave’s frequency ν (nu). To see what the frequency is, again consider the train of box cars passing by. If you count how many boxcars go by in a certain amount of time, you have found the box car frequency. If 10 boxcars go by in a minute, the frequency is 10 per minute, which would usually be written as 10/minute. The frequency of a wave is determined by how many cycles (peaks) go by a point in a second. If 1000 cycles pass by a point in a second, the frequency is ν = 1000/s = 1000 Hz. Lowercase s is used for the units of seconds. Per second has its own unit, Hz, for Hertz, which is in honor of Gustav Ludwig Hertz (1887-1975), who shared the Nobel Prize in Physics in 1925 with James Franck “for their discovery of the laws governing the impact of an electron upon an atom.” The wavelength, velocity, and frequency of a wave are related through the equation, λν = V.

OCEAN WAVES

Waves in the deep ocean travel with the crest above the average sea level and the troughs below sea level. A typical ocean wave has a wavelength λ = 160 m (520 ft) and travels with a velocity of 60 km/hr (60 kilometers per hour, or 38 miles per hour). The period, which is the time between wave crests, is 10 s, so the frequency ν = 0.1 Hz. The amplitude is just the distance between a crest and a trough. Therefore, it is relatively straightforward to visualize the amplitude. (Waves break at the beach because the troughs drag on the ocean bottom in shallow water, which slows them down. The crests move faster than the troughs and fold over to produce the breaking waves we see at the beach. Waves traveling in the ocean do not break.)

SOUND WAVES

Sound waves are density waves in air. A standard tuning fork A above middle C is 440 Hz. When you strike the tuning fork, the tines vibrate at 440 Hz. The vibration produces sound waves. The tines moving back and forth “push” the air back and forth at 440 Hz, producing a wave with frequency, ν = 440 Hz. At 70°F, the speed of sound is V = 770 miles per hour, which is 345 m/s. Because λν = V, the wavelength of the 440 Hz sound wave is λ = 0.78 m (2.55 ft). The sound wave consists of air density going above the average density and then below the average density, more air and then less air. The density is the weight of air in a unit of volume, for example the number of grams in a cubic centimeter (g/cm3). Increased density can be associated with increased pressure. So you could also think of the sound wave as a pressure wave in which the air pressure goes up and down at 440 Hz. When the sound wave enters your ear, the up-and-down oscillation of the pressure causes your eardrum to move in and out at the frequency of the sound wave, in this case, 440 Hz. The motion of the eardrum transfers the sound into the interior of the ear and tiny hairs are wiggled depending on the frequency of the sound. The motion of these hairs stimulates nerves, and the brain decodes the nerve impulses into what we perceive as sounds.

The amplitude of a sound wave is the difference between the maximum and minimum density (maximum and minimum pressure). In contrast to an ocean wave, you cannot see the amplitude of a sound wave, but you can certainly hear the differences in the amplitudes of sound waves. It is relatively simple to obtain electrical signals from sound waves, which is what a microphone does. Once an electrical signal is produced from a sound wave, its amplitude can be measured by measuring the size of the electrical signal. Like all classical waves, sound waves propagate in a direction and have an amplitude, a wavelength, and a velocity.

CLASSICAL LIGHT WAVES

The discussion of ocean waves and sound waves sets the stage for the classical description of light as light waves. In the classical description of light, explicated in great detail with Maxwell’s Equations (James Clerk Maxwell, 1831-1879), light is described as an electromagnetic wave. The wave has an electric field and a magnetic field, both of which oscillate at the same frequency. You have experienced electric and magnetic fields. If you have seen a magnet pull a small object to it, then you have seen the effect of a magnetic field. The magnetic field from a magnet is static, not oscillatory as in light. You may have also seen the effects of electric fields. If you have combed your hair on a very dry day with a plastic comb, you may have noticed that your hair is attracted to it. After combing, very small bits of paper may jump to the comb as the comb is brought close to them. These effects are caused by a static electric field. An electromagnetic wave has both electric and magnetic fields that oscillate.

Unlike ocean waves, which travel in water, and sound waves, which travel in air, light waves can travel in a vacuum. In a vacuum, the velocity of light is given the symbol c, and c = 3 × 108 m/s. The speed of light is about a million times faster than the speed of sound. This is the reason why you see distant lightning long before you hear it. Sound takes about 5 seconds to travel a mile. Light takes about 0.000005 s or 5 μs (microseconds) to travel a mile. The velocity of light is slower when it is not traveling in a vacuum. In air it is almost the same as in a vacuum, but in glass it travels at about two-thirds of c.

What is an electromagnetic wave, which is the classical description of light? In a water wave, we have the height of the water above and below sea level oscillating. In a sound wave, the air density or pressure oscillates above and below the normal values. If you take a small volume, the amount of air (number of molecules that make up air, mostly oxygen and nitrogen) goes above and below the average amount of air in the volume. In an electromagnetic wave, two things actually oscillate, an electric field and a magnetic field. We usually talk about the electric field because it is easier to measure than the magnetic field. The oscillating electric field is an electric wave. When you listen to the radio, the radio antenna is a piece of wire that detects the radio waves. Radio waves are just low frequency electromagnetic waves. They are the same as light waves, but much lower in frequency. The electric field in an electromagnetic wave oscillates positive and negative from a maximum positive amplitude value to the same negative value. The metal in a radio antenna has many electrons that can be moved by an electric field. (Electrons will be discussed in detail further on, and electrical conduction will be discussed in Chapter 19.) The oscillating electric field of a radio wave causes the electrons in the antenna to oscillate back and forth. The electronics in the radio amplify the oscillations of the electrons in the antenna and convert these oscillations into an electrical signal that drives the speakers to make the sound waves that you hear. So we can think of light classically as an oscillating electric field and an oscillating magnetic field. Both oscillate at the same frequency and travel together at the same speed in the same direction. This is why they are called electromagnetic waves.

VISIBLE LIGHT

For light in a vacuum, λν = c. The visible wavelengths, that is, the wavelengths we can see with our eyes, range from 700 nm (red) to 400 nm (blue). (A nm is a nanometer, which is 10-9 meters or 0.000000001 meters.) The visible wavelengths of light are very small; the velocity of light is very high. Therefore, the frequencies of visible light waves are very high. Red light has ν = 4.3 × 1014 Hz, and blue light has ν = 7.5 × 1014 Hz. 1014 is 100 trillion. Contrast light frequencies to a sound wave frequency (440 Hz) or an ocean wave frequency (0.1 Hz). Unlike an ocean wave or a sound wave, there is a complication in measuring the amplitude of a light wave. The frequency of light is so high that even the most modern electronics cannot see the oscillations. Rather than measuring the amplitude of the wave, defined as the amplitude of the oscillating electric field, the intensity of light is measured. The intensity, I, is proportional to the absolute valued squared of the electric field E, which is written asI∝|E|2. The absolute value, the two vertical lines | |, just means, for example, if there is a sign, positive or negative, we ignore it and just make everything positive. A photodetector, like the CCD in a digital camera (a CCD, or charge coupled device, makes an electrical signal when light strikes it), measures the amount of light, the intensity, rather than the amplitude of a light wave. Your eye does not directly measure the frequency of light waves in contrast to your ear, which measures the frequency of sound waves.

ADDING WAVES TOGETHER—INTERFERENCE

Waves of any kind, including light waves, can be added together to give new waves. Figure 3.2 shows on the left two identical waves (same wavelength, same amplitude, propagating in the same direction) that are in phase. (The waves are actually on top of each other, but they have been displaced so that we can see them individually.) “In phase” means that the positive peaks of one wave line up exactly with the positive peaks of the other wave, and therefore, the negative peaks also line up. The vertical dashed line in Figure 3.2 shows that the peaks line up. When waves are in phase, we say that the phase difference is 0° (zero degrees). One cycle of a wave spans a phase of 360°. Starting at any point on a wave, if you go along the wave for 360°, you are in an equivalent position, like going 360° around a circle. When two identical waves are added in phase, the resultant wave has twice the amplitude. This is called constructive interference, as shown on the right side of Figure 3.2.

Waves that are 180° out of phase can also be added together. As shown on the left side of Figure 3.3, waves that are 180° out of phase have the positive peaks of the top wave exactly lined up with the negative peaks of the bottom wave and vice versa. (Again, for interference to occur the waves need to actually be on top of each other, but they have been displaced so that we can see them clearly.) The dashed vertical line in Figure 3.3 shows that the positive peak of one wave is exactly lined up with the negative peak of the other wave. When two identical waves that are 180° out of phase are added, the positive peaks and the negative peaks exactly cancel. For example, take the maximum positive value to be +1 and the maximum negative value to be -1. Adding +1 and -1 gives zero. In Figure 3.3 each point on the top wave that is positive lines up perfectly with a point on the bottom wave that is the same amount negative, and each point of the top wave that is negative lines up with an equivalent point on the bottom wave that is the same amount positive. Therefore, the waves exactly cancel to give zero amplitude as shown on the right side of the figure. This cancellation is called destructive interference.

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FIGURE 3.2. Two identical waves that are in phase. The waves undergo positive and negative oscillations about zero (horizontal line). The positive peaks line up, and the negative peaks line up. They undergo constructive interference (are added together) to form a wave with twice the amplitude.

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FIGURE 3.3. Two identical waves that are 180° out of phase. The waves undergo positive and negative oscillations about zero (horizontal line). The positive peaks of the top wave line up exactly with the negative peaks of the bottom wave, and the negative peaks of the top wave line up exactly with the positive peaks of the bottom wave. The two waves undergo destructive interference when they are added together to produce zero amplitude.

INTERFERENCE PATTERNS AND THE OPTICAL INTERFEROMETER

Waves do not have to be right on top of each other and going in the same direction to interfere. They just have to overlap in some region of space, and interference can occur in that region. When Davies Symphony Hall in San Francisco was opened in 1980, it had acoustic problems. While the problems were very complicated, it is easy to see how they developed. Imagine that you are sitting in the audience pretty far back from the orchestra. When a 440 Hz A is played, the acoustic wave comes directly at you but it also bounces off of the walls on either side of you. If there is a reflection from the wall to your right and a reflection from the wall to you left so that the reflected acoustic waves (sound waves) from each wall comes to your row of seats at, for example, a 30° angle, an interference pattern will be produced along your row of seats. There will be places where reflected waves constructively interfere and make the sound louder and places where the waves destructively interfere and make the sound softer. The spacing between a peak and a null of the interference pattern is 2.4 ft (see below for the spacing formula). So depending on your seat, the 440 Hz A will be louder or softer. Of course, there are many frequency acoustic waves coming at you from many directions. The combined interference effects distorted the sound that should have been coming straight at you from the orchestra. The problem in Davies Hall was fixed in 1992 by the installation of 88 carefully designed panels hanging from the ceiling along the two side walls. No two panels are identical. They are filled with sand and weigh as much as 8500 pounds. These panels prevented the reflections from the walls from going into the audience.

Light can also undergo interference phenomena. The classical view of optical interference patterns can reproduce experimental results, as we are about to see. However, as discussed in Chapters 4 and 5, ultimately the classical description fails when other experiments are considered. The correct description will introduce the quantum mechanical superposition principle and bring us back to Schrödinger’s Cats.

Figure 3.4 shows a diagram of an interferometer used by Michelson (Albert Abraham Michelson, 1853-1931) in his studies of the nature of light waves. Michelson won the Nobel Prize in Physics in 1907 “for his optical precision instruments and the spectroscopic and metrological investigations carried out with their aid.” Michelson and Morley, who was a coworker of Michelson, used an interferometer to attempt to determine the nature of the medium in which light waves propagated. Water waves propagate in water. Sound waves propagate in air. The Michelson-Morley experiment showed that light waves do not have an underlying medium, which had been called the aether. Light can propagate in a vacuum. There is no aether that pervades space. Light waves traveling to us from the stars are not traveling in a medium the way ocean waves and sound waves travel in water and air, respectively. This was an important step in recognizing that light waves are not waves in the same sense as sound waves. Here we only want to understand the classical description of what is observed with an interferometer.

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FIGURE 3.4. The incoming light wave hits a 50% reflecting mirror. Half of the light goes through the mirror and half reflects from it. The light in each leg of the interferometer reflects from the end mirrors. Part of each beam crosses in the overlap region at a small angle. To the right of the circled overlap region is a blowup of what is seen along the x direction when two beams cross. An interference pattern is formed in which the intensity varies along x from a maximum value to zero periodically.

In Figure 3.4, a beam of light, taken to be a light wave, enters the apparatus from the left. The light hits a partially reflecting “beam-splitting” mirror that reflects 50% of the light intensity and transmits 50% of the light intensity. In the wave description of light, there is no problem having part of the wave go one way and part the other way. The reflected light goes vertically up the page, reflects from end mirror 1, which is at a small angle so the reflected beam does not quite go right back along the same path. The reflected beam goes down the page and part of it goes right through the beam-splitting mirror. (Part of this beam reflects from the beam splitter, but we are not concerned with this portion.) This path is leg 1 of the interferometer. The 50% of the original beam that goes through the beam splitter hits end mirror 2, which is also at a small angle. This reflected beam travels back to the left, almost retracing its original path. It reflects from the beam splitter. (The portion that goes through the beam splitter is unimportant for our considerations.) The reflected portion heads down the page. This path is leg 2 of the interferometer. The result is that the two beams, one that traveled leg 1 and one that traveled leg 2, come together after traveling the same distance and cross at a small angle in the “overlap region” shown by the circle in Figure 3.4. This crossing of the light waves is like the crossing of the sound waves in Davies Symphony Hall that caused the interference problems.

In Figure 3.4 the light beams are drawn as lines, but in any real experiment the beams have a width. The x direction shown in the figure is perpendicular to the bisector of the angle (the line that splits the angle) made by the crossing beams. Since the angle is small, the x direction is basically perpendicular to the propagation direction of the beams, and in the figure it is the horizontal direction. A blowup of what is seen along the x direction in the overlap region is shown in the lower right portion of the figure. In the graph the vertical axis is the intensity of the light, I, and the horizontal axis is the position along x. Because the beams cross at a small angle, the phase relationship between them varies along the x direction, and there are alternating regions of constructive and destructive interference. The intensity of the light varies from a maximum value to zero back to the maximum, and again to zero, and so on. The crossed light waves form regions of constructive and destructive interference. At the intensity maxima, the light waves are in phase (0°—see Figure 3.2), and they add constructively to give increased amplitude. At the zeros of intensity, the light waves are 180° out of phase (see Figure 3.3), and they add destructively, to exactly cancel. This pattern can be observed by placing a piece of photographic film or a digital camera in the overlap region to measure the intensity at the different points along the x direction.

For a small angle, the fringe spacing, that is, the spacing, d, between a pair of intensity peaks or nulls is given by d = λ/θ, where λ is the wavelength of light, and θ is the angle between the beams in radians (1 radian = 57.3 degrees). If 700 nm red light is used, and the angle between the beams is 1°, the fringe spacing is 40 μm or 1.6 thousandths of an inch. These fringes can be seen with film or a good digital camera. If the angle is 0.1°, the fringe spacing is 0.4 mm, which you can see by eye. If the angle is 0.01° (an exceedingly small angle), the fringe spacing is 4 mm (about a sixth of an inch), which you can easily see by eye. To have 4 mm fringes, the beams that cross must be much larger in diameter than 4 mm.

As discussed, in the classical description, light is an electromagnetic wave, and the intensity is proportional to the square of the electric field amplitude (size of the wave in Figure 3.1). In the following, we are not going to worry about units. By including a lot of constants, the units in the following all work out, but they are unimportant for our purposes here. Take the electric field in one of the beams in one leg of the interferometer to have an amplitude of 10. Then the intensity is 100 (102 = 100 = 10×10). The other beam also has I = 100. These are the intensities when we are not observing in the beam overlap region. When the beams are separated, the sum of their intensities is 200. What happens in the overlap region? Waves constructively interfere in some places and destructively interfere in others (see Figure 3.4, lower right). Therefore, to determine the intensities in the overlap region, it is necessary to add the electric field amplitudes and then square the result to find the intensities. At an intensity maximum in the overlap region, the waves are perfectly in phase and add constructively. The electric field from beam 1 adds to the electric field from beam 2, that is, E = 10 + 10 = 20. Then the intensity in a peak in the interference pattern is I = E2 = 202 = 400. The intensity is 400, twice as great as the intensity of just the sum of the intensities of the two beams by themselves when they are not constructively interfering. In a null of the interference pattern, the waves destructively interfere perfectly. An electric field of +10 adds to an electric field of - 10, to give zero. The electric field equals zero, and I = 0. Therefore, the interference pattern is caused by alternating regions of constructive and destructive interference of electromagnetic waves. In some places the waves add, and we see a peak. In some places the waves subtract to give zero. Interference is a well-known property of waves, and the interference pattern produced by the interferometer seemed to be a perfect example of a wave phenomenon.

The interferometer and the interference pattern shown in Figure 3.4 can be described in complete detail using classical electromagnetic theory. The details of the interference pattern can be calculated with Maxwell’s equations. This and many other experiments, including the transmission of radio waves, can be described with classical theory. Therefore, classical theory, which treats light as a wave, appeared to be correct up to the beginning of the twentieth century. However, Chapter 4 shows how Einstein’s explanation of one phenomenon, the photoelectric effect, caused the beautiful and seemingly infallible edifice of classical electromagnetic theory to require fundamental rethinking.