Section 37–6 Watching the electrons

 (Observer effect / Complementarity principle / Complementary descriptions)

 

In this section, Feynman discusses Proposition A, uncertainty principle, and logical tightropes in communicating quantum mechanics. However, the three main points of the section could be observer effect, complementarity principle, and complementary descriptions in quantum mechanics.

 

1. Observer effect:

“Here is what we see: every time that we hear a “click” from our electron detector (at the backstop), we also see a flash of light either near hole 1 or near hole 2, but never both at once! And we observe the same result no matter where we put the detector. From this observation we conclude that when we look at the electrons we find that the electrons go either through one hole or the other. Experimentally, Proposition A is necessarily true (Feynman et al., 1963, p. 37–7).”

 

According to Feynman, Proposition A—"each electron either goes through hole 1 or hole 2"—becomes true only under experimental conditions that forces the electron to choose a definite path. This reveals a deeper truth: the validity of Proposition A is dependent on the act of observation. In the absence of observation, the electron is described by a wavefunction representing a superposition of paths, leading to interference. When an observation is made, the act of measurement restricts the wavefunction to a new quantum state (a different function), allowing the electron to exhibit particle-like behavior and follow a specific path—either through slit 1 or slit 2. This transition from wave-like to particle-like behavior upon observation is often attributed to the observer effect, a concept sometimes linked to Heisenberg’s microscope. It challenges classical notions of objective reality, illustrating that the properties of quantum systems are not pre-determined but emerge through interaction with the apparatus.

 

“You remember that when we discussed the microscope we pointed out that, due to the wave nature of the light, there is a limitation on how close two spots can be and still be seen as two separate spots. This distance is of the order of the wavelength of light. So now, when we make the wavelength longer than the distance between our holes, we see a big fuzzy flash when the light is scattered by the electrons (Feynman et al., 1963, p. 37–9).”

 

Traditionally, physicists would use Heisenberg’s microscope to explain the uncertainty principle involved in the experiment. However, Feynman’s mention of microscope is related to Rayleigh criterion, which states the minimum angular resolution of an image-forming system. Notably, the experiment could be analyzed using the concepts of visibility and distinguishability: 1. Visibility refers to the clarity of the interference pattern, with high visibility indicating sharp fringes and low visibility indicating blurred or no patterns. 2. Distinguishability refers to the ability to determine which hole a particle passed through. There is a quantitative relation, sometimes known as distinguishability-visibility relation, which expresses a trade-off between the visibility of interference pattern and the distinguishability of the particle’s path.

 

2. Complementarity principle:

“He proposed, as a general principle, his uncertainty principle, which we can state in terms of our experiment as follows: ‘It is impossible to design an apparatus to determine which hole the electron passes through, that will not at the same time disturb the electrons enough to destroy the interference pattern’ (Feynman et al., 1963, p. 37–9).”

 

Feynman’s statement of the uncertainty principle suggests that the principle arises from the disturbance caused by the act of measurement. Furthermore, it implies that the main problem is one of apparatus design, but one may consider the possibility of determining the electron’s path using an observation system (instead of physical apparatus). More important, the uncertainty principle is now understood as intrinsic uncertainties of quantum systems, that are independent of any disturbance. Bohr’s complementarity principle offers an alternative explanation: we cannot observe both the wave and particle properties simultaneously. When we observe the electron’s path (particle-like behavior), the interference pattern (wave-like behavior) disappears, and vice versa.

 

Feynman’s perspective is related to Bohr-Einstein debates on the epistemological problems of quantum mechanics. In Bohr’s (1949) words, “[I]t is only the circumstance that we are presented with a choice of either tracing the path of a particle or observing interference effects, which allows us to escape from the paradoxical necessity of concluding that the behavior of an electron or a photon should depend on the presence of a slit in the diaphragm through which it could be proved not to pass. We have here to do with a typical example of how the complementary phenomena appear under mutually exclusive experimental arrangements and are just faced with the impossibility, in the analysis of quantum effects, of drawing any sharp separation between an independent behavior of atomic objects and their interaction with the measuring instruments which serve to define the conditions under which the phenomena occur (p. 46).” Remarkably, Bohr explains the experiment using the complementarity principle. Complementarity is the concept that a single entity can exhibit different (or seemingly contradictory) properties depending on the perspectives or context in which it is observed.

 

Strictly speaking, Feynman’s statement is not a general formulation of the uncertainty principle but rather a specific application to the double-slit experiment. An alternative phrasing of the principle, avoiding terms like “apparatus” and "disturb," might be: It is impossible to determine which hole the electron passes through without affecting the visibility of the interference pattern. Specifically, the distinguishability-visibility relation in a modern double slit experiment may be expressed as: D² + V² ≤ 1 (Jaeger, 2007). This inequality means that as path distinguishability (D) increases, the visibility (V) of the interference pattern must decrease, and vice versa. In the double-slit experiment, when D = 1 (complete distinguishability or complete path knowledge), V = 0, meaning no interference pattern will appear. Conversely, when D = 0 (no path knowledge), V = 1, meaning a full interference pattern.

 

3. Complementary descriptions:

“If one looks at the holes or, more accurately, if one has a piece of apparatus which is capable of determining whether the electrons go through hole 1 or hole 2, then one can say that it goes either through hole 1 or hole 2. But, when one does not try to tell which way the electron goes, when there is nothing in the experiment to disturb the electrons, then one may not say that an electron goes either through hole 1 or hole 2. If one does say that, and starts to make any deductions from the statement, he will make errors in the analysis. This is the logical tightrope on which we must walk if we wish to describe nature successfully (Feynman et al., 1963, p. 37–9).”

 

Feynman’s “logical tightrope” could be related to complementary descriptions, which include classical and quantum descriptions. Proposition A—"each electron either goes through hole 1 or hole 2" is a classical description, which can be determined by an experiment. Perhaps Feynman could have included Proposition B: “each electron exists as a superposition of different states (or possible paths),” which is a quantum description. This describes the electron’s quantum state, where its path remains undefined until measured. In essence, the classical description captures the electron’s particle-like behavior, emphasizing a definite path when observed, while the quantum description highlights its wave-like nature, characterized by an indefinite path when unobserved. This duality illustrates the principle of complementarity, where the wave and particle descriptions are not contradictory but dependent on the experimental setup.

 

In the Audio Recordings* [39 min: 30 sec] of this lecture, Feynman says something like: “The world must be entirely quantum mechanical. It cannot be half classical and half quantum mechanical. If, for example, shmootrinos, a new particle, would all go exactly like waves according to classical physics, we can make any intensity we want. Then, we can just use them to watch electrons, we can cut the intensity down because they don’t have the rules of quantum mechanics, then it would be a paradox...” Feynman also used the word shmootrinos in his lecture on gravitation delivered to advanced graduate students. In Feynman’s (1995) words, “These we might think of as photons or gravitons or neutrinos, or maybe some new particles, some shmootrinos which don't worry about baryon conservation. When they meet another shmootrino dropping in from the other side with opposite momentum, these can have enough energy to create a hydrogen atom.” This playful invention serves to emphasize the necessity of quantum mechanical rules, suggesting the inconsistency that would arise if a new (classical?) particle existed in a quantum world.

*The Feynman Lectures Audio Collection: https://www.feynmanlectures.caltech.edu/flptapes.html

 

“In Fig. 37–5 we have tried to indicate schematically what happens with large-scale objects. Part (a) of the figure shows the probability distribution one might predict for bullets, using quantum mechanics. The rapid wiggles are supposed to represent the interference pattern one gets for waves of very short wavelength. Any physical detector, however, straddles several wiggles of the probability curve, so that the measurements show the smooth curve drawn in part (b) of the figure (Feynman et al., 1963, p. 37–10).”

 

It might seem surprising that the double-slit experiment involving bullets could produce an interference pattern as shown in Fig. 37–5(a). This can be explained by the de Broglie wavelength of bullets, which is extremely small compared to objects like electrons. The rapid wiggles in the interference pattern predicted by quantum mechanics for bullets could be observed if a sufficiently large number of bullets is fired under highly controlled conditions. In practical experiments, however, real-world factors such as slight imperfections in the slit edges (e.g., sharp versus smooth edges) and the granularity of the bullet impacts would likely introduce significant noise, obscuring the interference pattern. If the separation between the two slits were increased, the probability distribution might display two distinct peaks instead of one. Feynman’s analysis also implicitly assumes that the bullets are highly correlated—that is, they possess nearly identical energies or de Broglie wavelengths. Without this assumption, the underlying quantum mechanical effects would be overwhelmed by classical randomness.

Review Questions:

1. Would you explain the uncertainty principle using the observer effect? (Do you agree with Feynman’s statement of uncertainty principle)?

2. Would you explain the double slit experiment using the uncertainty principle or complementarity principle?

3. How would you explain the logical tightrope on which we must walk if we wish to describe nature successfully?

 

The moral of the lesson: The test of a first-rate intelligence is the ability to hold two opposed ideas in the mind at the same time, and still retain the ability to function (Fitzgerald, 2009).

 

References:

1. Bohr, N. (1949). Discussion with Einstein on Epistemological Problems in Atomic Physics, In N. Bohr, Philosophical Writings of Niels Bohr, 3 vols. Woodbridge: Ox Bow Press.

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.

3. Feynman, R. P., Morinigo, F. B., & Wagner, W. G. (1995). Feynman Lectures on gravitation (B. Hatfield, ed.). Reading, MA: Addison-Wesley.

4. Jaeger, G. (2007). Quantum information. New York: Springer.

5. Fitzgerald, F. S. (2009). The crack-up. New Directions Publishing.

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.