Section 37–5 The interference of electron waves

(Electron’s path / Complex numbers / Electron waves)

 

In this section, Feynman discusses the determination of electron’s path, use of complex numbers, and the interference of electron waves in the double slit experiment.

 

1. Electron’s path:

“Proposition A: Each electron either goes through hole 1 or it goes through hole 2. … … undoubtedly we should conclude that Proposition A is false. It is not true that the electrons go either through hole 1 or hole 2 (Feynman et al., 1963, p. 37–6).”

 

According to Feynman, the proposition "Each electron either goes through hole 1 or hole 2" is wrong when there is an interference pattern in the double-slit experiment. This proposition assumes that the electron follows a single, deterministic path, as if it were a classical particle. However, the presence of interference pattern implies that an electron’s path cannot be described by simply stating that it goes through one hole or the other. In quantum mechanics, the correct description involves a superposition of possible paths, where the electron’s wavefunction passes both holes simultaneously. In essence, the classical concept of an electron “either goes through one hole or the other” is inadequate, as it does not include the probabilistic nature of quantum mechanics.

 

In one of his Messenger lectures, Feynman says, “[i]f you have an apparatus which is capable of telling which hole the electron goes through (and you can have such an apparatus), then you can say that it either goes through one hole or the other. It does; it always is going through one hole or the other — when you look. But when you have no apparatus to determine through which hole the thing goes, then you cannot say that it either goes through one hole or the other. You can always say it - provided you stop thinking immediately and make no deductions from it. Physicists prefer not to say it, rather than stop thinking at the moment (Feynman, 1965, p. 144).” In short, physicists prefer not to say “the electron goes through both holes” when they lack apparatus that can directly make the observation. However, some may describe the electron behaves as if it passes through both holes simultaneously, which results in the interference pattern.

 

Some might conceptualize that an electron passes through both holes simultaneously from the perspective of the electron as a quantum field. However, the idea of “passing through both holes” is applicable to the electron’s wavefunction, but not to a classical particle. Theoretically, the wavefunction of the electron passes through both holes simultaneously, but the interference pattern observed is the result of the superposition of the “probability waves” emerging from both holes. In the next section, Feynman argues that the proposition "each electron either goes through hole 1 or hole 2” becomes correct if someone is watching the electron. The correctness of this proposition depends on whether the electron’s “path” is determined or remains undisturbed.

 

2. Complex numbers:

“Incidentally, when we were dealing with classical waves we defined the intensity as the mean over time of the square of the wave amplitude, and we used complex numbers as a mathematical trick to simplify the analysis. But in quantum mechanics it turns out that the amplitudes must be represented by complex numbers (Feynman et al., 1963, p. 37–6).”

 

Some might hope for Feynman to provide a geometric interpretation of complex numbers to justify their use in quantum mechanics. Bohm (1951), for instance, highlights the imaginary unit i in the Schrödinger equation, suggesting it as essential to the theory. Similarly, Sakurai (1967) emphasizes the usefulness of complex numbers in describing spin vectors, and linking them to the broader framework of Clifford algebra. Others may argue that complex numbers are well-suited to represent plane trigonometry and rotations in two dimensions, noting the significance of i in these contexts. Intuitively, a complex number acts like a rotating clock hand, where multiplying complex numbers follows a simple rule: “add angles and multiply lengths (distances from the origin)." Notably, a probability wave is a complex function, and thus it cannot itself be a probability (Penrose, 2004).

 

In a Berkeley Symposium, Feynman (1951) clarifies: “[a] more accurate equation valid for electrons of velocity arbitrarily close to the velocity of light is the Dirac Equation. In this case the probability amplitude is a kind of hypercomplex number (p. 539).” A hypercomplex number refers to an extension of complex numbers, where numbers have more than two components. Among the most common types of hypercomplex numbers are quaternions, which extend complex numbers to four dimensions, represented as a + bi + cj + dk, where i, j, k are imaginary units (Ö-1) with specific rules. Some may prefer Clifford or geometric algebra—a broader framework encompassing numbers with finite dimensional components—which is also used in neural networks and theoretical physics.

 

The question “Why is quantum physics based on complex numbers?” could be misleading. To begin with, the term complex number could be a misnomer; some might prefer “orthogonal number” to emphasize its component perpendicular to the real axis on the complex plane. Second, p-adic numbers—an extension of the rational numbers with a fractal-like structure—are arguably even more complex than complex numbers. Applications of p-adic numbers include not only quantum mechanics and quantum chaos but also extend to complex systems like spin glasses (Vladimirov et al., 1994). This suggests that complex numbers may not be strictly necessary for quantum mechanics, allowing for alternative mathematical frameworks.

3. Electron waves:

“The electrons arrive in lumps, like particles, and the probability of arrival of these lumps is distributed like the distribution of intensity of a wave. It is in this sense that an electron behaves ‘sometimes like a particle and sometimes like a wave’ (Feynman et al., 1963, p. 37–6).”

 

In his Messenger lecture, Feynman (1965) included a summary for the three different double slit experiments (as shown below). Importantly, electrons exhibit both particle and wave properties within the interference pattern. The wave behavior is evident in the interference pattern, which shows regions of constructive and destructive interference. The particle behavior, on the other hand, is observed as localized impacts or “lumps” where individual electrons hit the screen. This dual behavior distinguishes the electron double-slit experiment from the double-slit experiment with water waves, which shows purely wave properties, and from the double-slit experiment with bullets, which shows only particle properties. In this context, it could be misleading to say that “an electron behaves sometimes like a particle and sometimes like a wave.” Instead, one may describe the electron as guided by probability waves and interacting with the screen as a particle, reflecting the fundamental quantum nature that unifies both aspects within the same experiment.

Source: (Feynman, 1965, p. 139)

Some physicists might question the title of this section, “The interference of electron waves,” by citing Dirac’s (1958) dictum on interference: “Each photon interferes only with itself. Interference between different photons never occurs.” However, Glauber (1995) explains, “[t]he things that interfere in quantum mechanics are not particles. They are probability amplitudes for certain events. It is the fact that probability amplitudes add up like complex numbers that is responsible for all quantum mechanical interferences.” In this view, interference does not occur between particles themselves but between the probability amplitudes for the possible paths. Interestingly, photons of different colors (different energies) emitted one at a time would have different wavelengths, resulting in different interference patterns. Similarly, the interference pattern for single-electrons can vary depending on their energy (or wavelength).

 

Review Questions:

1. How would you explain the proposition “Each electron either goes through hole 1 or hole 2” in the context of double slit experiment?

2. Why is quantum mechanics based on complex numbers?

3. How would you explain the interference pattern of double slit experiment involving electrons (due to the interference of electron waves?)?

 

The moral of the lesson: the probability amplitude (or probability wave) of each photon interferes with itself based on its possible paths, and the interference pattern shows both particle-like and wave-like properties at the same time.

 

References:

1. Bohm, D. (1951). Quantum Theory. New York: Prentice-Hall.

2. Dirac, P. (1958). Quantum mechanics. 4th edition. Oxford: Oxford University Press.

3. Feynman, R. P. (1951). The concept of probability in quantum mechanics. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (Vol. 2, pp. 533-542). California: University of California Press.

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

5. 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.

6. Glauber, R. J. (1995). Dirac’s famous dictum on interference: one photon or two?. American Journal of Physics, 63(1), 12-12.

7. Penrose, R. (2006). The road to reality. London: Random house.

8. Sakurai, J. J. (1967). Advanced quantum mechanics. Boston: Addison-Wesley.

9. Vladimirov, V. S., Volovich, I. V., & Zelenov, E. I. (1994). p-adic Analysis and Mathematical Physics (Vol. 1). Singapore: World Scientific.

Section 37–4 An experiment with electrons

(Idealized conditions / Observing electrons arrival / Interference pattern)

 

In this section, Feynman discusses the idealized experimental conditions, observations of electrons arrival, and interference pattern of double slit experiment involving electrons (or single-electrons).

 

1. Idealized conditions:

“We make an electron gun which consists of a tungsten wire heated by an electric current and surrounded by a metal box with a hole in it. If the wire is at a negative voltage with respect to the box, electrons emitted by the wire will be accelerated toward the walls and some will pass through the hole. All the electrons which come out of the gun will have (nearly) the same energy. In front of the gun is again a wall (just a thin metal plate) with two holes in it. Beyond the wall is another plate which will serve as a “backstop.” In front of the backstop we place a movable detector. The detector might be a geiger counter or, perhaps better, an electron multiplier, which is connected to a loudspeaker (Feynman et al., 1963, p. 37–4).”

 

Feynman’s description of the double slit experiment involving electrons could be summarized as three idealized conditions: (1) Ideal electron gun: Electrons emitted from a heated tungsten wire are accelerated through a hole, ensuring they have nearly the same energy (or wavelength). (2) Ideal double slit: A thin metal plate with two holes is placed in front of the electron gun that allows an electron (or electrons) to have two alternative paths. (3) Ideal detector: The detector, such as a Geiger counter, registers the arrival of electrons as audible clicks. It seems that Feynman considered the three idealized conditions were difficult to be achieved with the technology available in his time. Remarkably, the double slit experiment involving electrons was performed in 1961 by Claus Jönsson, but he was not able to reduce the electron firing rate to one at a time.  

 

“We should say right away that you should not try to set up this experiment (as you could have done with the two we have already described). This experiment has never been done in just this way. The trouble is that the apparatus would have to be made on an impossibly small scale to show the effects we are interested in (Feynman et al., 1963, p. 37–5).”

 

Although Feynman suggested the concept of nanotechnology in 1959, he advised not to set up the experiment possibly because he was unaware of the latest technologies and Jönsson’s (1961) paper was published in German. However, to observe electrons’ interference, the slit width should be comparable to the de Broglie wavelength of the electrons, which is on the scale of nanometers to micrometers, depending on their energy. Jönsson created slits about 10 micrometers wide, small enough to cause interference when electrons passed through. In addition, electrons are sensitive to irregularities on the surface of the substrate or the edges of the slits. If the slits had any roughness or contamination, electrons could scatter unpredictably, blurring the interference pattern. Jönsson’s experience in clean materials ensured that the substrate and slits were free from microscopic debris or irregularities that could “distort” the path of the electrons.

Photograph of diffraction pattern of a single slit and interference pattern of two slits (Jönsson, 1974)

 

2. Observing electron arrival:

“As we move the detector around, the rate at which the clicks appear is faster or slower, but the size (loudness) of each click is always the same. If we lower the temperature of the wire in the gun the rate of clicking slows down, but still each click sounds the same. We would notice also that if we put two separate detectors at the backstop, one or the other would click, but never both at once (Feynman et al., 1963, p. 37–5).”

 

While hearing clicks (as suggested by Feynman) tells us that electrons are being detected, seeing the interference pattern provides spatial information that shows the probabilistic nature of quantum mechanics. Historically, Merli and his collaborators’ (1974) experiment used an electron biprism and TV image intensifiers to directly observe single-electron interference patterns. Firstly, by passing single-electrons through the electron biprism, which is essentially a very thin wire placed in the path of the electron beam. Applying a voltage to the biprism, it behaves like two virtual slits (instead of physical slits). Secondly, after passing through the biprism, the electrons were detected using image intensifiers that are highly sensitive and capable of detecting the impact of single-electron. These devices amplify the weak signal generated by each electron, which help to observe the gradual build-up of the interference patterns.

We would also notice that the “clicks” come very erratically. Something like: click ….. click-click … click …….. click …. click-click …… click …, etc., just as you have, no doubt, heard a geiger counter operating. If we count the clicks which arrive in a sufficiently long time—say for many minutes—and then count again for another equal period, we find that the two numbers are very nearly the same (Feynman et al., 1963, p. 37–5).”

 

Perhaps Feynman could have included Taylor’s (1909) double-slit experiment in his lecture because it serves as a historically significant and pedagogically valuable demonstration of “single photon interference.” (Taylor had to run the exposure for 3 months, i.e., counting the “clicks” for many minutes as suggested by Feynman may not seem to be a good time-scale.) Essentially, smoked glass screens were used as a filter to reduce the intensity of light to the point where it was dim enough that only one photon was emitted at a time. Although it significantly decreased the number of photons reaching the slits, the cumulative result of many photon impacts led to the formation of an interference pattern, similar to what would happen with continuous light. The feeblest light source could form the interference pattern led to Dirac’s dictum that “Each photon interferes only with itself. Interference between different photons never occurs.”

 

3. Interference pattern:

“Just as for our experiment with bullets, we can now proceed to find experimentally the answer to the question: “What is the relative probability that an electron ‘lump’ will arrive at the backstop at various distances x from the center?” As before, we obtain the relative probability by observing the rate of clicks, holding the operation of the gun constant. The probability that lumps will arrive at a particular x is proportional to the average rate of clicks at that x. The result of our experiment is the interesting curve marked P₁₂ in part (c) of Fig. 37–3 (Feynman et al., 1963, p. 37–5).”

Just like the interference pattern for water waves, 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 37-3 above). 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 30 above). In addition, Feynman’s (1965) diagram suggests that the diffraction pattern for a slit due to electron waves has no secondary maxima. It implies that the intensity distribution of the diffraction pattern would be different for electron waves and light waves. However, the diffraction pattern of electron waves for single slit has secondary maximas on either side of the central maximum. The intensity of the first maxima is not high as shown in the diagram below. 

(Young & Freedman, 1996, p. 1274)

In a Berkeley Symposium, Feynman (1951) writes: “The probability distribution must now be (d) instead of the interference pattern (a)… The uncertainty principle assures us that the vertical uncertainty Dq in the screen position must exceed h/Dp and hence exceed h/2dp so that the maxima and minima of the diffraction pattern (a) are completely smeared out and the resulting distribution is that of (d) (p. 538).” This shows that Feynman was inconsistent in describing the same probability distribution using the terms diffraction pattern and interference pattern. It is unsurprising because he explains that no one has ever been able to define the difference between interference and diffraction satisfactorily in chapter 30. Currently, physicists prefer the term interference pattern for the phenomenon involving two sources, but if there is a large number of sources (or only one slit), the term diffraction pattern is used instead. However, the probability distribution of arrival of electrons as drawn in the Berkeley Symposium (Fig 2 below) is slightly better than Fig 37-3 and Fig 30.

 

Review Questions:

1. Why did Feynman advise against setting up the double slit experiment involving electrons?

2. How would you observe the arrival of electrons?

3. How would you explain the interference pattern of the double slit experiment involving electrons?

 

The moral of the lesson: We can set up the double slit experiment involving single-electrons using two real slits or virtual slits (electron biprism or ultrathin wire) and TV image intensifiers.

Youtube Video: "Electron Interference" (english version) - YouTube

 

References:

1. Feynman, R. P. (1951). The concept of probability in quantum mechanics. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (Vol. 2, pp. 533-542). University of California Press.

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

3. 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.

4. Jönsson, C. (1961). Elektroneninterferenzen an mehreren künstlich hergestellten Feinspalten. Zeitschrift für Physik, 161(4), 454-474.

5. Jönsson, C. (1974). Electron diffraction at multiple slits. American Journal of Physics, 42(1), 4-11.

6. Merli, P. G., Missiroli, G. F., & Pozzi, G. (1974). Electron interferometry with the Elmiskop 101 electron microscope. Journal of Physics E: Scientific Instruments, 7(9), 729–732.

7. Taylor, G. I. (1909). Interference fringes with feeble light. Proceedings of the Cambridge Philosophical Society, 15(1), 114-115.

8. Young, H. D., & Freedman, R. A. (1996). University physics. Reading, MA: Addison-Wesley.

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.