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.