Section 37–3 An experiment with waves

 (Idealized conditions / Measure wave intensity / Interference pattern)

 

In this section, Feynman discusses the idealized experimental conditions, measurement of wave intensity, and interference pattern of double slit experiment involving water waves.

 

1. Idealized conditions:

“A small object labeled the ‘wave source’ is jiggled up and down by a motor and makes circular waves. To the right of the source we have again a wall with two holes, and beyond that is a second wall, which, to keep things simple, is an ‘absorber,’ so that there is no reflection of the waves that arrive there. This can be done by building a gradual sand ‘beach’ (Feynman et al., 1963, p. 37–3).”

 

The double slit experiment involving water waves can be described in terms of idealized experimental conditions. Based on Feynman’s description of the experiment, there are three possible idealizations: 1. Ideal Water Wave Generator: The wave source produces circular water waves with a constant wavelength. 2. Ideal wall: The first wall (with two holes) allows coherent water waves with the same amplitude to pass through, fulfilling the conditions for stable interference pattern. 3. Perfect absorber (second wall): The wall absorbs the water waves completely without any reflection; the interference pattern is formed due to the primary wave interactions without reflected waves. In summary, it is important to have coherent waves (or constant phase difference) such that the interference pattern is stable and visible.

 

It is possible that Feynman realized some shortcomings of his Caltech lecture and included two improvements in his Cornell lecture. Firstly, he introduces a single hole before the double-slit setup: the single hole acts as a source of coherent waves, ensuring the waves coming through the first barrier are of equal amplitude. The use of a single hole also ensures the waves through the double-slit are coherent (or same frequency), allowing the formation of clear interference pattern. Secondly, he excludes the absorber (gradual sand beach) possibly because it is not an ideal place for making measurements. The interference pattern near the absorber is less visible due to the lower wave intensity. However, Feynman’s placing of a cork in water only gives a basic visualization of the wave motion, but it is not an accurate method to measure wave intensity.

In his Cornell lecture, Feynman (1965) explains that, “[t]he source is now a big mass of stuff which is being shaken up and down in the water. The armour plate becomes a long line of barges or jetties with a gap in the water in between. Perhaps it would be better to do it with ripples than with big ocean waves; it sounds more sensible. I wiggle my finger up and down to make waves, and I have a little piece of wood as a barrier with a hole for the ripples to come through. Then I have a second barrier with two holes, and finally a detector. What do I do with the detector? What the detector detects is how much the water is jiggling. For instance, I put a cork in the water and measure how it moves up and down, and what I am going to measure in fact is the energy of the agitation of the cork, which is exactly proportional to the energy carried by the waves. One other thing: the jiggling is made very regular and perfect so that the waves are all the same space from one another (pp. 133-134).”

 

2. Measure wave intensity:

“In front of the beach we place a detector which can be moved back and forth in the x-direction, as before. The detector is now a device which measures the “intensity” of the wave motion... If we cover one hole at a time and measure the intensity distribution at the absorber we find the rather simple intensity curves shown in part (b) of the figure (Feynman et al., 1963, p. 37–3).”

 

It does not seem appropriate to measure the intensity of water waves at the absorber in the double slit experiment. By the time the waves reach the absorber (sand beach), much of their energy is absorbed, and therefore, the intensity measured at the absorber would be significantly lower than the initial wave intensity at earlier points along its path. Surely, Feynman was joking? The absorber is supposed to dissipate the energy of the incoming waves and prevent their reflection. Thus, the ideal place to measure the wave intensity should be farther away from the absorber, where the waves have a higher intensity. However, the placement of a detector in the water can create reflected waves, which could interfere with the incoming waves and distort the measurement results.

 

Water waves are an example of waves that involve a combination of both longitudinal and transverse motions.  Generally speaking, the height (vertical displacement) of water waves can be represented mathematically using complex functions or simply sine functions. In other words, the amplitude of the wave is a real number if it is expressed using a sine function, but the amplitude can be a complex number when it is expressed as a complex function. Importantly, the amplitude of water waves does not remain constant when they are moving in the direction of the gradual sand “beach.” It is better to use high-speed cameras combined with image processing software to accurately capture and analyze the wave amplitudes and intensities in real time.

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3. Interference pattern:

“The intensity I₁₂ observed when both holes are open is certainly not the sum of I₁ and I₂. We say that there is ‘interference’ of the two waves (Feynman et al., 1963, p. 37–4).”

  

Perhaps Feynman could have explained how the separation between two slits may affect the interference pattern. When the two slits are closer to each other, the overlapping water waves form a higher central peak as shown in the middle of the graph (See Fig 37-2 below). In his Cornell lecture, the two slits are drawn further apart, thus there is no central peak due to the reduced overlapping in the middle, but there are two separate peaks corresponding to each of the two slits (See Fig 29 below). Interestingly, another difference between the two interference patterns is that one decreases to almost zero as the distance x increases, but the one drawn in Cornell remains at almost half of the maximum intensity. Perhaps the two wave intensities were drawn differently because Feynman considered the detector was at the absorber in Fig 37-2, but the detector was far from the absorber in Fig 29.

 

For the purpose of visualizing interference patterns, Feynman has chosen water waves instead of light waves. In Feynman’s words, “the next waves of interest, that are easily seen by everyone and which are usually used as an example of waves in elementary courses, are water waves. As we shall soon see, they are the worst possible example, because they are in no respects like sound and light; they have all the complications that waves can have (Feynman et al., 1963, p. 51–7).” Thus, we should not expect the diffraction pattern of water wave from a single slit to be accurately represented by Fig 37-2 or Fig 29 as shown above. To be precise, the diffraction pattern of light waves through a single slit may be represented by the diagram (see below), but some may assume the water waves can have a similar diffraction pattern as light waves.

 

Source: Single Slit Diffraction: Definition, Formula, Types and Examples (geeksforgeeks.org)

The intensity I₁₂ observed when both holes are open is certainly not the sum of I₁ and I₂. We say that there is “interference” of the two waves. At some places (where the curve I₁₂ has its maxima) the waves are “in phase” and the wave peaks add together to give a large amplitude and, therefore, a large intensity. We say that the two waves are “interfering constructively” at such places. There will be such constructive interference wherever the distance from the detector to one hole is a whole number of wavelengths larger (or shorter) than the distance from the detector to the other hole (Feynman et al., 1963, p. 37–4).”

 

It should be worth mentioning that constructive interference is more than just an addition of two peaks; it is also about the combining of whole waveform (including peaks and troughs). Furthermore, it affects the rate of flow of energy, i.e., it is not only an increase in amplitude. We may define constructive interference as a phenomenon involving maximum fluctuation in intensity depending on the energy from both waves adding up, which leads to a greater energy flow in the regions where the waves reinforce each other. On the contrary, destructive interference refers to minimal fluctuation in intensity due to the energy contributions from two waves cancelling out. In essence, the wave energy is continuously being redistributed by interfering waves, but the intensity at regions of constructive interference could momentarily reach zero due to zero displacement for both waves, which depends on their phase relationship.

 

Review Questions:

1. How would you explain the idealized conditions of the double-slit experiment involving water waves?

2. How would you measure the wave intensity (at the absorber) in the context of double-slit experiment involving water waves?

3. How would you explain the interference pattern formed by the water waves?

 

The moral of the lesson: The interference pattern formed by water waves can be explained by the superposition of two waves, where the wave intensities combine non-linearly, creating regions of constructive interference (maximum fluctuation in intensity) and destructive interference (minimal fluctuation in intensity).

 

References:

1. Feynman, R. P. (1965). The character of physical law. Cambridge: MIT Press Feynman,

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

Section 37–2 An experiment with bullets

(Idealized conditions / Measure probability / Probability curve)

 

In this section, Feynman discusses the idealization conditions, measurement of particle’s arrival probability, and probability distribution curve (or interference pattern) of double slit experiment involving indestructible bullets.

 

1. Idealized conditions:

“… imagine a somewhat idealized experiment in which the bullets are not real bullets, but are indestructible bullets—they cannot break in half... If the rate at which the machine gun fires is made very low, we find that at any given moment either nothing arrives, or one and only one—exactly one—bullet arrives at the backstop. Also, the size of the lump certainly does not depend on the rate of firing of the gun. We shall say: ‘Bullets always arrive in identical lumps’ (Feynman et al., 1963, p. 37–2).”

 

Perhaps Feynman could have clarified that there are three idealized conditions in the double-slit experiment using bullets. Feynman’s descriptions of the experiment can be interpreted in terms of three idealizations: 1. Indestructible Bullets: It mirrors the quantum entities like electrons that are indivisible. 2. Low Firing Rate: It simulates the scenario where one particle is emitted one at a time, without any interference with another particle. 3. Identical lumps: This is analogous to the concept of identical particles, where each particle (such as a photon*) is indistinguishable from another in terms of its intrinsic properties. It allows consistent and repeatable measurements, similar to how identical photons in experiments lead to predictable outcomes. Together, these three idealizations enable the experiment to have comparison with other forms of double-slit experiment.

 

* “The size of the lump certainly does not depend on the rate of firing of the gun” may correspond to the experiment whereby the energy of each photon emitted does not depend on the rate of light emission.

 

It is possible that Feynman realized a few shortcomings of his lecture and made some improvements in his Cornell lecture as follows: “First I would like to make a few modifications from real bullets, in three idealizations. The first is that the machine gun is very shaky and wobbly and the bullets go in various directions, not just exactly straight on; they can ricochet off the edges of the holes in the armor plate. Secondly, we should say, although this is not very important, that the bullets have all the same speed or energy. Thirdly, the most important idealization in which this situation differs from real bullets is that I want these bullets to be absolutely indestructible, so that what we find in the box is not pieces of lead, of some bullet that broke in half, but we get the whole bullet (Feynman, 1965, p. 131).” Alternatively, Feynman could have included two more idealizations, e.g., low firing rate and identical lumps, but the lecture would be more complicated for the non-science students.

 

The three idealizations mentioned in his Cornell lecture help to provide a simple model that allows for a comparison with three aspects of quantum behavior: (1). Random nature: The shaky machine gun results in the random nature of particles’ paths, where their exact trajectory is uncertain and influenced by various factors. (2). Same wavelength: “Same energy bullets” ensure uniformity in the experiment, analogous to quantum particles having the same wavelength, which allows for an analysis of the interference patterns. (3). Quantization: The indestructible bullet is analogous to the indivisibility (or countable quanta, such as photons or electrons) of quantum particles when observed. In essence, these idealizations help us understand how quantum behavior, especially in experiments where the paths are uncertain, is influenced by external factors, including slow emissions of photons or electrons.

 

2. Measure probability:

“We shall say: “Bullets always arrive in identical lumps.” What we measure with our detector is the probability of arrival of a lump. And we measure the probability as a function of x (Feynman et al., 1963, p. 37–2).”

 

Feynman (1965) improves his explanation of probability in his Cornell lecture as follows: “we can call that the probability of arrival … the number that I have plotted does not come in lumps. It can have any size it wants. It can be two and a half bullets in an hour, in spite of the fact that bullets come in lumps. All I mean by two and a half bullets per hour is that if you run for ten hours you will get twenty-five bullets, so on the average it is two and a half bullets. I am sure you are all familiar with the joke about the average family in the United States seeming to have two and a half children. It does not mean that there is a half child in any family - children come in lumps (p. 132).” However, one may use the term empirical probability to describe the concept that is related to the measurement of bullet’s arrival. It distinguishes this observed probability from theoretical probability, which is calculated by known models without requiring actual experiment. The empirical probability is not a constant, but it depends on the experimental conditions.

 

In a Berkeley Symposium, Feynman (1951) says: “[w]e shall see that the quantum mechanical laws of the physical world approach very closely the laws of Laplace as the size of the objects involved in the experiments increases. Therefore, the laws of probabilities which are conventionally applied are quite satisfactory in analyzing the behavior of the roulette wheel but not the behavior of a single electron or a photon of light.” Laplace’s theory of probability is based on the principle of indifference: “if there is no known reason for predicating of our subject one rather than another of several alternatives, then relatively to such knowledge the assertions of each of these alternatives have an equal probability (Keynes, 1921, pp. 52–53).” One limitation of the classical definition of probability is that it is primarily applicable to situations (e.g., flipping a coin) where there is a finite number of equally likely outcomes. This theory does not account for updating probabilities based on experimental results, i.e., once the setup is completed, the probabilities are fixed.

 

In chapter 6, Feynman** explains subjective probability as follows: “[p]robabilities need not, however, be “absolute” numbers. Since they depend on our ignorance, they may become different if our knowledge changes (Feynman et al., 1963, p. 6–7).” This concept of probability is dependent on our knowledge, but they may vary in accordance with the experimental setup. If there is uncertainty or variability in the bullet paths (e.g., a wobbly gun or changing environmental conditions), some may adopt Bayesian probability, which allows for updating the likelihood of bullets passing through each slit based on observed outcomes. Specifically, two experimental physicists may set up the apparatus, e.g., adjusting slit separations whereby the probability curves (or interference patterns) would be varied. Interestingly, Feynman drew two different probability curves for the same double slit experiment in his lecture delivered in Caltech and Cornell University.

 

**The lecture was delivered by Matthew Sands.

 

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3. Probability curve:

“We call the probability P₁₂ because the bullets may have come either through hole 1 or through hole 2. You will not be surprised that P₁₂ is large near the middle of the graph but gets small if x is very large. You may wonder, however, why P₁₂ has its maximum value at x = 0 (Feynman et al., 1963, p. 37–2).”

  

Perhaps Feynman could have explained how the separation between two slits may affect the probability curve​ on the detection screen. When the two slits are closer to each other, the overlapping probability distributions cause a higher central peak in the middle of the graph (See Fig 37-1 below). In his Cornell lecture, the two slits are drawn further apart, thus there is no central peak due to the reduced overlapping in the middle, but there are two separate peaks corresponding to the each of the two slits (See Fig 28 below). The two different probability curves are not really empirical probability obtained from the experiments, but deduced by Feynman. To obtain a smooth probability curve, the experimenter would need to fire almost an infinity number of bullets. In a sense, the two resultant curves show the subjective probability that is dependent on the slit separation conceived by Feynman at different times.

 

“The probabilities just add together. The effect with both holes open is the sum of the effects with each hole open alone. We shall call this result an observation of “no interference,” for a reason that you will see later. So much for bullets. They come in lumps, and their probability of arrival shows no interference (Feynman et al., 1963, p. 37–3).”

 

There are three simplifications that result in the smooth probability distribution curve. Firstly, there could be rapid wiggles in the interference pattern due to the de Broglie wavelength of bullets (see Fig. 37-5 below), but Feynman explains it in section 37-6. Secondly, there could be diffraction of the bullets for single slit due to the de Broglie wavelength (see Fig. 38-2 below), but this is shown in the next chapter. Thirdly, the probability curve for each slit would be asymmetrical because the machine gun is off-center with respect to each of the two slits. Some bullets may pass through the slit without deflection, resulting in a skewed distribution on one side of the screen, but bullets traveling at certain oblique angles may hit parts of the slit edge unevenly, further contributing to the asymmetry. The probability curve is likely more messy in the real experiment, however, Feynman preferred to reveal the complications due to interference later (possibly pedagogical reasons, but it may still be confusing…).

 

“By ‘probability’ we mean the chance that the bullet will arrive at the detector, which we can measure by counting the number which arrive at the detector in a certain time and then taking the ratio of this number to the total number that hit the backstop during that time (Feynman et al., 1963, p. 37–2).”

 

There are at least four views of probability: classical, frequency, subjective, and propensity (de Elía & Laprise, 2005). Feynman’s explanations suggest three views: 1. Classical: In the Berkeley Symposium, Feynman (1951) mentions the Laplace’s classical theory of probability that is based on the principle of indifference, which is applicable to classical physics. 2. Frequency: In this lecture, Feynman measures probability by taking the ratio of the number of bullets that hit a detector at a location to the total number during that time is related to “frequentist probability” (based on the frequency of occurrence). 3. Subjective: In chapter 6, Feynman emphasizes subjective probability: “[i]t is probably better to realize that the probability concept is in a sense subjective, that it is always based on uncertain knowledge… (Feynman et al., 1963, p. 6–7).” Perhaps some mathematicians do not agree with Feynman’s explanations of probability. However, the probability of particle’s arrival in the experiment corresponds to the propensity theory of probability because it reflects a physical propensity for certain outcomes inherent in the experimental setup.

Review Questions:

1. How would you explain the idealized conditions of the double-slit experiment involving bullets?

2. How would you explain the concept of probability in the context of double-slit experiment involving bullets?

3. How would you explain the simplifications needed to achieve the smooth probability distribution curve (or interference patterns)?

 

The moral of the lesson: There is no interference pattern for bullets because they are classical particles that travel along well-defined trajectories in the double slit experiment; the probability distribution curve due to the two slits can be deduced by simply adding the probability curve for each slit (slit 1 and slit 2) linearly.

 

References:

1. de Elía, R., & Laprise, R. (2005). Diversity in interpretations of probability: Implications for weather forecasting. Monthly Weather Review, 133(5), 1129-1143.

2. Feynman, R. P. (1951). The concept of probability in quantum mechanics. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (Vol. 533).

3. Feynman, R. P. (1965). The character of physical law. Cambridge: MIT Press Feynman,

4. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

5. Keynes, J. M., 1921, A Treatise on Probability, London: Macmillan and Co.