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.