Absolutely Small: How Quantum Theory Explains Our Everyday World - Michael D. Fayer (2010)
Chapter 5. Light: Waves or Particles?
THE EXPLANATION OF THE photoelectric effect discussed in Chapter 4 required a new theoretical description of the interferometer experiment discussed in connection with Figure 3.4. Understanding the interferometer experiment in a manner that does not contradict the description of the photoelectric effect requires the big leap into thinking quantum mechanically rather than thinking classically. In discussing absolute size in Chapter 2, the idea was introduced that for a system that is small in an absolute sense, a measurement will make an unavoidable nonnegligible disturbance. However, we did not discuss the nature or consequences of such a disturbance. Now, we need to come to grips with the true character of matter and what happens when we make measurements.
The problem we have is that light waves were used to explain the interference phenomenon in Figure 3.4, but “particles of light,” quanta called photons, were used to explain the photoelectric effect in connection with Figures 4.3 and 4.4. The classical mathematical description of light waves employed Maxwell’s equations to quantitatively describe interference. The mathematical entity that represented a light wave in the theory is called a wavefunction. A function gives a mathematical description of something, in this case a light wave. It describes the amplitude, frequency, and the spatial location of a light wave. The incoming light wave is described by a single wavefunction. In the classical description, after the light wave hits the 50% beam splitter, half of the wave goes into each leg of the interferometer (see Figure 3.4). There are now two waves and two wavefunctions, one for each wave. These wavefunctions describe waves that are each half of the intensity of the original incoming wave and are in different locations, the two legs of the interferometer. When these two wavefunctions are combined mathematically to describe the nature of the overlap region inside the circle in Figure 3.4, the interference pattern can be calculated. All of this worked so well that it was thought that the same math must apply to photons.
CLASSICAL DESCRIPTION OF INTERFERENCE DOESN’T WORK FOR PHOTONS
Figure 5.1 shows the interferometer again with everything exactly the same as in Figure 3.4 except that the incoming light beam is now composed of photons. Initially, it was thought that when the beam of photons hits the 50% beam splitter, half of the photons go into leg 1 of the apparatus and head toward end mirror 1, while the other half of the photons go into leg 2 of the apparatus and head for end mirror 2. The photons would then reflect from the end mirrors and after again hitting the beam splitter, half of the photons from each leg would cross in the overlap region. The inference pattern was thought to develop when photons from one leg of the apparatus interfered with photons from the other leg of the apparatus. This thinking proved to be incorrect.
To describe the interference effect, the mathematical formulation in terms of Maxwell’s wavefunctions was not changed at all. However, the physical meaning of the wavefunction was redefined. Instead of the amplitude of an electromagnetic wave in a certain region of space, such as leg 1 or leg 2 of the interferometer, the wavefunction was redefined as describing the number of photons in a region of space. Previously, the wavefunction was taken to give the amplitude of the wave in a region of space, and this amplitude could be used to calculate the intensity. After the redefinition, the wavefunction was taken to tell how many photons were in a region of space, say leg 1 of the interferometer, and the intensity could still be calculated. This redefinition seemed perfectly reasonable, but it was wrong! The entire description of half of the photons going into each leg of the interferometer is fundamentally wrong. The correct description required the leap to thinking quantum mechanically.
FIGURE 5.1. The beam of light is composed of photons that hit a 50% reflecting mirror. In the initial incorrect description of the interference effect in terms of photons, it was thought that half of the photons go into each leg of the interferometer. The photons from each leg cross in the overlap region, and it was believed that the photons from one leg interfere with photons from the other leg to produce the interference pattern. The idea that photons from one leg interfere with photons from the other leg is not correct.
Many things are wrong with the picture that half of the photons go into each leg of the interferometer and then come together and interfere with each other. The simplest experiment that shows the problem with this description is the intensity dependence of the interference pattern (the lower right portion of Figure 5.1). The shape of the interference pattern observed in the overlap region of the interferometer is independent of the intensity used to create it. For a given method of detection, a piece of photographic film or a digital camera, note that if the intensity is turned up it will take less time to acquire a high-quality pattern, but the shape of the pattern is unchanged. That is, the separation and shapes of the peaks and nulls in the pattern are unchanged. As discussed in Chapter 3, the periodicity of the pattern depends on the crossing angle of the beams and the wavelength of light. It does not depend on the intensity. If the intensity is reduced, it will take more time to collect the pattern, but the pattern will not change shape. A standard red laser pointer puts out 1 mW (milliwatt), which is one thousandth of a watt, or 0.001 J/s (joules per second). The red color is approximately 650 nm, that is, the wavelength λ = 650 nm. Using λν = c and E = hν where h is Planck’s constant, ν is the frequency of the light, and c is the speed of light, one photon of 650 nm light is about 3 × 10-19 J. So the 1 mW laser pointer is emitting about 3 × 1015 photons per second, that is 3 thousand trillion photons per second. If this is the input beam for the interferometer, the interference pattern will be very easy to record and, in fact, if the fringe spacing is big enough (see Chapter 3, the discussion of fringe spacing following Figure 3.4), you will be able to see the interference pattern with your eyes.
Imagine turning the intensity down and down. Soon, you will not be able to see the interference pattern because your eyes are not very sensitive light detectors, but you can still record it with photographic film or a digital camera. Once recorded, the pattern is the same. Turn the intensity down a factor of 3000 to a trillion photons a second, and the pattern is unchanged. In the description in which half of the photons go into each leg of the apparatus, half a trillion photons per second go into each leg of the interferometer. Turn down the intensity to a billion photons per second, and the pattern is the same. Further reduce the intensity to a million photons per second, and there is still no change. Here is where the fallacy in the description becomes obvious. Turn down the intensity until there is only one photon per second entering the apparatus. Again, the pattern is unchanged. At one photon per second, it will take a long time to record enough signal to see the interference pattern, but if you wait long enough the pattern is the same.
When only one photon per second is entering the interferometer, there is only one photon at a time in the apparatus. A photon will take on the order of one hundred millionth of a second (10 -8 s) to traverse the interferometer. With one photon per second, there is virtually no chance that there is more than one photon at a time in the instrument, yet the interference pattern, once recorded, is the same. But the modified classical description of the interference effect in terms of photons said that half of the photons go into leg 1 and half of the photons go into leg 2. The photons in leg 1 interfere with the photons in leg 2 to produce the interference pattern. If there is only one photon in the apparatus at a time, there is no other photon for it to interfere with. The “half of the photons go into each leg of the apparatus model” predicts that the interference pattern should vanish at low enough intensity. The interference pattern does not disappear at low intensity. The model is wrong!
A NEW DESCRIPTION OF PHOTONS IN THE INTERFEROMETER
Here is where the complete change in thinking is required that will bring us back to Schrödinger’s Cats. How is it possible to have an interference pattern when only one photon enters the interferometer at a time? Our understanding of this problem and the nature of quantum mechanics in general is based on the conceptual interpretation of the mathematical formalism that is strongly associated with the work of Max Born (1882-1970). Born won the Nobel Prize in Physics in 1954 “for his fundamental research in quantum mechanics, especially for his statistical interpretation of the wavefunction.” This interpretation is frequently referred to as the Copenhagen interpretation.
The correct description of the interferometer experiment is that each photon goes into both legs of the interferometer. This is the big leap. A single photon encounters the 50% beam splitter. That means there is a 50% chance that the photon will be reflected and go into leg 1 of the interferometer (see Figure 5.1) and a 50% chance it will go into leg 2. Classically one would say the photon must go one way or the other, that is, it either goes into leg 1 or it goes into leg 2. This is not correct. When the photon encounters the beam splitter, its state is changed. If a photon is in fact moving in leg 1, call this state of motion “translation state 1,” abbreviated T1. If a photon is moving in leg 2, call this state of motion “translation state 2,” abbreviated T2. After a photon interacts with the beam splitter, it is not in T1 or T2. The state of the system after the beam splitter is referred to as a superposition state. It is an equal mixture of T1 and T2. In some sense, the photon is simultaneously in both T1 and T2. This sounds really strange. The single photon is in two regions of space simultaneously. It is in a superposition translation state T = T1 + T2. It is in a state that is an equal mixture of T1 and T2.
The photon is in the superposition translation state T = T1 + T2 because this is what is known about it. It has a 50% chance of being in leg 1 (T1) and a 50% chance of being in leg 2 (T2). The Born interpretation of the wavefunction states that the wave is not a real wave in the sense of the amplitude of an oscillating electromagnetic field. Rather, the wavefunction describes a “probability amplitude wave.” The incorrect interpretation of the wavefunction in terms of photons is that it tells how many photons are in each leg of the apparatus, that is, how many photons are in a region of space. The correct interpretation is that the wavefunction is related to the probability of finding a photon in a region of space. The difference between the incorrect and correct interpretation may not seem like a major difference, but, as shown in detail below, it is a fundamental change in our view of nature. In the classical description of light, the intensity is proportional to the absolute value squared of the amplitude of the electric field, which in turn was given by the amplitude of the wavefunction. In the Born interpretation, the absolute value squared of the wavefunction for a certain region of space gives the probability of finding a particle, in this case a photon, in that region of space.
A PHOTON INTERFERES WITH ITSELF
When the photon meets the beam splitter, two probability amplitude waves are created, one in leg 1 and one in leg 2. The total probability amplitude wave T is the superposition of the probability amplitude waves T1 and T2. After encountering the beam splitter, each and every photon is in a state T1 + T2. Because there are two probability amplitude waves after the beam splitter, they cross in the overlap region. A single photon inside the interferometer has two waves, T1 and T2, associated with it. The interference of these two waves determines the probability of finding a photon near the peak, which is high, and finding a photon near the null, which is low. A photon interferes with itself because it is composed of two waves in the interferometer, and two waves can interfere with each other. Because every single photon is placed in the T1 + T2 superposition state after meeting the beam splitter, there is no problem with turning down the intensity. A single photon entering the apparatus produces two waves, probability amplitude waves, in the interferometer. Therefore, there is always a pair of waves to produce an interference pattern.
A PHOTON CAN BE IN TWO PLACES AT ONCE
The first natural response of a classical thinker to the Born interpretation is “this is nuts.” Are we really to believe that a single photon can be in two places at once? After meeting the beam splitter, the state that is produced is T1 + T2. The state T1 + T2 means that in some sense the photon is simultaneously in both legs of the apparatus. If this is true, why don’t we just make some measurements to see where the photon is? It doesn’t do much good to make the measurement with trillions of photons going into the apparatus. If we put an instrument in leg 1 to see how much light there is, we will find half of the light. However, that doesn’t tell us what we want to know. Maybe half of the photons go in each leg and we see half, or maybe there is a 50% chance that each photon goes into each leg. We will still see half. The correct experiment is to use such low-intensity light so that only one photon is in the apparatus at a time.
Consider the experiment in which we shoot one photon at a time into the interferometer. We use a photodetector that is so sensitive that it can detect a single photon. This is readily doable with the scientific equivalent of a superdigital camera. We place the detector in leg 1 of the interferometer. A photon enters the apparatus, and we detect it. We see an entire photon. We don’t see half a photon. Another photon goes in the apparatus, and we don’t see a photon. Five more photons enter the apparatus. We detect two of them and do not detect the other three. After doing this for a long time, it is found that 50% of the photons are observed by the detector placed in leg 1 of the apparatus. We also find that no interference pattern was produced. In fact, what is observed is a single bright spot (no oscillatory pattern) in the region in which the interference pattern occurred previously.
Observation Causes a Nonnegligible Disturbance Causing a Change of State
What is going on? After the photon meets the beam splitter, it is in a superposition state, T1 + T2. However, photons are particles that are small in the absolute sense. The act of making an observation causes a nonnegligible disturbance. When we place the photodetector in leg 1 of the apparatus, we are making an observation of the location of the photon. The act of making the observation causes the system to jump from being in the superposition state T1 + T2 into either a pure state T1 or a pure state T2. The superposition wavefunction has been “collapsed” into one of the pure states that make up the superposition. If the system makes the jump into the state T1, the photon is detected. Of course, once it slams into the photodetector, it does not continue to propagate through the interferometer. If the photon jumps into the state T2, it is not detected with the photodetector that is located in leg 1, and it continues to propagate, eventually reaching the region in which the instrumentation is set up to observe the interference pattern. However, because the photon is in the pure state T2, there is only a single probability amplitude wave. When it reaches the “overlap” region (see the bottom of Figure 5.1), there is no other probability amplitude wave to interfere with. Therefore, no interference pattern is created. A single spot is formed as each photon that traverses the apparatus in the pure T2 state hits a single spot on the detector like a bullet aimed at that spot. The spot has the same size (diameter) as the initial light beam that entered the apparatus, but no spatially oscillatory interference pattern.
BACK TO SCHRÖDINGER’S CATS
Making the observation of the location of the photon with the photodetector in leg 1 of the interferometer causes the photon to jump from the mixed superposition state T1 + T2 into a pure state, either T1 or T2. However, on a single measurement it is not possible to know which state will be produced by the observation. The chance is 50/50 that it is in T1 and 50/50 that it is in T2. After many measurements, we know that the probability of making the jump to T1 is 50%, but it is impossible to say in advance what will happen for any single observation. This is a true physical manifestation of the issues raised with Schrödinger’s Cats in Chapter 1, where we had 1000 boxes with a cat in each one. Each cat was in a superposition state that was 50% alive and 50% dead. In this very nonphysical heuristic scenario, when a box is opened an observation is made as to the health of the cat. Sometimes the cat is perfectly healthy and sometimes the cat is dead. After opening all of the boxes, it was determined that the probability of finding a live cat was 50%, but there is no way of predicting ahead of time when opening a particular box, that is, on making a single observation, whether a live or dead cat will be found. Before opening the box, the cat is in a superposition state that is a 50/50 mixture of live and dead. The act of making the observation makes a nonnegligible disturbance and causes the superposition state to jump into either a pure live state or a pure dead state. As discussed in Chapter 1, a live/dead cat superposition state does not and cannot exist, but the interferometer is a real example of the ideas illustrated by Schrödinger’s Cats.
A photon can easily be placed into a superposition state consisting of a 50/50 mixture of two translations states using the 50% beam splitter. When it is in the superposition state, it is not possible to say whether the photon is in leg 1 or leg 2 of the apparatus. It is only possible to say that if we make a measurement to see where the photon is, the measurement will make a nonnegligible disturbance. The disturbance will cause the state of the system to change from having equal probability of being in the two legs of the interferometer to being in either one or the other. The interference pattern is produced when the photon’s probability amplitude waves interfere with each other. The two pieces of the superposition state, T1 and T2, which comprise the total probability amplitude wave for a photon in the interferometer, interfere with each other. If an observation is made to see where the photon is, it will be found to be either in leg 1 or leg 2 of the apparatus. However, the act of observation changes the system so that it is no longer in a superposition state. There are no longer two parts of the probability amplitude wave to interfere with each other, and the interference pattern vanishes. Thus, a photon in an interferometer is a real manifestation of the ideas relating to Schrödinger’s Cats.
BACK TO THE PHOTOELECTRIC EFFECT
In Chapter 4, the photoelectric effect was described in terms of photons, which are particles that behave in some sense like bullets of light. One photon strikes one electron and knocks it out of a piece of metal (see Figure 4.3). The description of the photoelectric effect showed that the classical description of light as an electromagnetic wave was incorrect. A new concept had to be introduced to explain both the photoelectric effect and the fact that photons could produce an interference pattern. The Born interpretation of the wavefunction as a probability amplitude wave gave the photon the necessary wavelike characteristics, so that photons could produce an interference pattern. However, in discussing the probability amplitude waves in connection with the interferometer, we only considered the location of the photon in terms of two rather large regions of space; a photon was in a superposition state, T1 + T2, with equal probabilities of being in leg 1 and leg 2 of the interferometer. The photoelectric effect implied that a photon is quite small. Chapter 6 will show how a superposition of probability amplitude waves can produce a photon that is very small in size. The ideas will lead to one of the central and most nonclassical aspects of quantum mechanics, the Heisenberg Uncertainty Principle.