Section 37–8 The uncertainty principle

 Indeterminacy principle / Thought experiment / Protect quantum mechanics

 

In this section, Feynman discusses the uncertainty (or indeterminacy) principle, using thought experiments to illustrate its role in quantum mechanics. He presents a modified version of Bohr’s thought experiment to demonstrate how the uncertainty principle protects quantum mechanics from Einstein’s critiques. The section could aptly be titled The Indeterminacy Principle, aligning with Heisenberg’s original terminology.

 

1. Indeterminacy principle:

“This is the way Heisenberg stated the uncertainty principle originally: If you make the measurement on any object, and you can determine the x-component of its momentum with an uncertainty Δp, you cannot, at the same time, know its x-position more accurately than Δx≥ℏ/2Δp (Feynman et al., 1963, p. 37–11).”

 

In his paper titled The actual content of quantum theoretical kinematics and mechanics, Heisenberg (1927) originally states the principle as “According to the basic equations of the Compton effect, the relation between p₁ and q₁ is then p₁q₁ ~ h.” This relation, using Planck’s constant (h), was an approximation and lacked the precise inequality ΔxΔp≥ℏ/2. The formal mathematical expression emerged later through Kennard (1927) and Weyl (1928), who defined Δx and Δp as the standard deviations of position and momentum, providing precision to the concept. Heisenberg originally termed this principle as the indeterminacy principle (German: Unbestimmtheit), reflects its meaning morning accurately than “uncertainty,” which is often associated with measurement inaccuracies. The principle is not merely a consequence of measurement limitations but it is intrinsic to the very nature of quantum systems.

Note: In Fundamentals: Ten Keys to Reality, Wilczek (2021) writes: “[t]he two principles we mentioned above - that observation is an active process and that observation is invasive - were bedrock foundation of Heisenberg analysis (p. 211).”

 

“This is a special case of the uncertainty principle that was stated above more generally. The more general statement was that one cannot design equipment in any way to determine which of two alternatives is taken, without, at the same time, destroying the pattern of interference (Feynman et al., 1963, p. 37–11).”

Feynman’s phrasing—“one cannot design equipment in any way”— may give the impression that the principle arises from practical limitations like equipment design or observer interference. This interpretation aligns with Heisenberg’s initial framing of the principle, which attributed it to the physical disturbance caused by measurement. In Heisenberg’s gamma-ray microscope thought experiment, he proposed that attempting to measure an electron’s position would inevitably disturb its momentum due to the recoil imparted by the interacting photon. In section 38-2 of The Feynman Lectures on Physics, Feynman revisits the uncertainty principle (or bandwidth theorem) from the perspective of wave mechanics. He explains that the uncertainties associated with a wave packet, such as the trade-off between position and momentum precision, are intrinsic to the wave theory, but have nothing to do with quantum mechanics.

 

Historical Context: Pauli’s correspondence with Heisenberg inspired much of the latter’s work. In a letter dated October 19, 1926, Pauli writes: “one can look at the world with the p-eye and one can look at it with the q-eye. But if one wants to open both eyes at the same time, one goes crazy (Hermann et al., 1979).” In other words, he metaphorically described the uncertainty principle as akin to observing the world with the “p-eye” (momentum) and “q-eye” (position). It also succinctly captures the essence of quantum indeterminacy.

 

2. Thought experiment:

“We imagine a modification of the experiment of Fig. 37–3, in which the wall with the holes consists of a plate mounted on rollers so that it can move freely up and down (in the x-direction), as shown in Fig. 37–6. By watching the motion of the plate carefully we can try to tell which hole an electron goes through (Feynman et al., 1963, p. 37–11).”

 

Feynman’s thought experiment builds upon Bohr’s defense of quantum mechanics at the 1927 Solvay Conference. It modifies Bohr’s thought experiment, in which the wall (or screen) with two slits as a plate mounted on rollers. While this thought experiment is a conceptual exercise designed to explore quantum principles, it is practically impossible to perform. The recoil momentum of the plate would be extremely small due to the electron’s negligible mass and measuring such changes in momentum with the necessary precision is beyond current experimental capabilities. This illustrates how any attempt to determine “which-path” information inevitably disrupts interference patterns, affirming the principle of quantum indeterminacy.

 

Bohr’s thought experiment involved a screen suspended by springs, where reading the scale on the screen required illumination (see below). The scattering of photons during illumination led to an uncontrollable transfer of momentum, introducing uncertainty in the position of the slit in the first screen. This destroys the coherence of the particle’s associated wave, effectively erasing the interference pattern. In Principles of the Quantum Theory, Bohr writes, “It would seem that any theory capable of an explanation of the photoelectric effect as well as the interference phenomena must involve a departure from the ordinary theorem of conservation of energy as regards the interaction between radiation and matter (Kragh, 2012).” Bohr’s words suggest that, in quantum mechanics, the conservation of energy might be momentarily violated.

 

(Source: Bohr, 1996)

Einstein’s thought experiment involves a particle passing through a slit in the first screen, and the screen’s movement is observed to determine the particle’s deflection direction. This observation helps infer which slit in the second screen the particle passed through. By observing the particle’s position on a third screen, its path through the entire setup could be reconstructed. According to Einstein, this setup would reveal particle-like properties (a definite trajectory) and wave-like properties (interference patterns), challenging the validity of the uncertainty principle. Einstein’s argument was based on both the conservation of energy and his philosophical position on realism, the view that physical properties exist independently of measurement.

 

3. Protect quantum mechanics:

“The uncertainty principle “protects” quantum mechanics. Heisenberg recognized that if it were possible to measure the momentum and the position simultaneously with a greater accuracy, the quantum mechanics would collapse. So he proposed that it must be impossible. Then people sat down and tried to figure out ways of doing it, and nobody could figure out a way to measure the position and the momentum of anything—a screen, an electron, a billiard ball, anything—with any greater accuracy (Feynman et al., 1963, p. 37–12).”

 

Feynman’s assertion that the uncertainty principle “protects” quantum mechanics highlights its role in preserving the theory’s consistency. However, in a footnote of his book titled QED, Feynman writes: “I would like to put the uncertainty principle in its historical place: When the revolutionary ideas of quantum physics were first coming out, people still tried to understand them in terms of old-fashioned ideas…… If you get rid of all the old-fashioned ideas and instead use the ideas that I’m expounding in these lectures—adding arrows for all the ways an event can happen—there is no need for an uncertainty principle!’’ (Feynman 1985, pp. 55, n. 3). This reflects Feynman’s belief that the uncertainty principle is not fundamental in his path integral formulation of quantum mechanics. In this framework, the probability amplitude for an event is calculated as the sum over all possible paths, each weighted by a phase factor. However, it contrasts with Bohr’s insistence on the need of uncertainty principle, which questioned the legitimacy of a "path" in quantum mechanics.

 

In Lectures on Gravitation delivered to graduate students, Feynman proposed a double-slit thought experiment involving a gravity detector to determine which slit an electron passes through. In Feynman’s words, “I would like to suggest that it is possible that quantum mechanics fails at large distances and for large objects…… We must therefore not neglect to consider that it is possible for quantum mechanics to be wrong on a large scale, to fail for objects of ordinary size” (Feynman et al, 1995, pp. 12–13). Feynman’s caution shows an important point: while quantum mechanics has been successful within its domain of applicability, its prediction for a large scale could be unreliable. This open-mindedness is significant, particularly regarding the unresolved challenges of reconciling quantum mechanics with general relativity.

 

Not all physicists share the view that the uncertainty principle "protects" quantum mechanics. Einstein’s critiques of quantum mechanics, for example, spurred the development of the second quantum revolution. His famous Einstein-Podolsky-Rosen (EPR) paradox questioned whether quantum mechanics could provide a complete description of reality. In response to the EPR paradox, Bell (1964) showed that if quantum mechanics is correct, nature must allow for non-local correlations. It has driven advances in quantum technologies, including quantum computing, cryptography, and teleportation—developments far beyond what Bohr could have envisioned.

 

Review Questions:

1. How would you state the uncertainty (or indeterminacy) principle)?

2. How does Feynman’s thought experiment reinforce the principle?

3. Why might the uncertainty principle be considered essential for protecting quantum mechanics?

 

The moral of the lesson: Heisenberg’s uncertainty principle remains a cornerstone of quantum mechanics, but Feynman’s path integral formulation offers an alternative perspective, challenging its necessity.

(The evolution of quantum theory—from foundational debates to breakthroughs in quantum entanglement—demonstrates that quantum mechanics advances not by “protecting” it, but by probing its boundaries and exploring new horizons.)

 

References:

1. Bell, J. S. (1964). On the einstein podolsky rosen paradox. Physics Physique Fizika, 1(3), 195-200.

2. Bohr, N. (1996). Discussion with Einstein on epistemological problems in atomic physics. In Niels Bohr Collected Works (Vol. 7, pp. 339-381). Amsterdam: Elsevier.

3. Feynman, R. P. (1985). QED: The strange theory of light and matter. Princeton: Princeton University Press.

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

6. Heisenberg, W. (1983). The actual content of quantum theoretical kinematics and mechanics (No. NAS 1.15: 77379).

7. Hermann, A., v. Meyenn, K., & Weisskopf, V. F. (Eds.). (1979). Wolfgang Pauli: Wissenschaftlicher Briefwechsel mit Bohr, Einstein, Heisenberg ua Band I: 1919–1929. Berlin, Heidelberg: Springer Berlin Heidelberg.

8. Kennard, E. H. (1927). Zur quantenmechanik einfacher bewegungstypen. Zeitschrift für Physik, 44(4), 326-352.

9. Kragh, H. (2012). Niels Bohr and the quantum atom: The Bohr model of atomic structure 1913-1925. Oxford: OUP.

10. Wilczek, F. (2022). Fundamentals: Ten keys to reality. New York: Penguin.

Section 37–7 First principles of quantum mechanics

 (Ideal experiment / Probabilistic predictions / Hidden variables)

 

In this section, Feynman discusses an ideal experiment, probabilistic predictions, and hidden variables in quantum mechanics. Perhaps a more fitting title would be “Probabilistic Determinism in Quantum Mechanics” or “Probabilistic Predictions and Hidden Variables.” The section addresses the conceptual shift from classical determinism to probabilistic determinism in quantum mechanics.

 

1. Ideal experiment:

“We can write our summary more simply if we first define an “ideal experiment” as one in which there are no uncertain external influences, i.e., no jiggling or other things going on that we cannot take into account. We would be quite precise if we said: ‘An ideal experiment is one in which all of the initial and final conditions of the experiment are completely specified’ (Feynman et al., 1963, p. 37–10).”

 

Feynman’s definition of an ideal experiment—where all initial and final conditions are completely specified and free of uncertain influences—has notable issues in quantum mechanics: 1. The act of measurement itself alters the system, making it impossible to exactly specify final conditions without disturbing the initial state. 2. The uncertainty principle limits the simultaneous specification of complementary variables, such as position and momentum. 3. The external influences (such as thermal fluctuations) cannot be entirely eliminated or accounted for in practice. While Feynman’s definition is an idealization, the concept remains a valuable theoretical construct for analyzing probabilistic relationships in quantum systems.

 

In quantum mechanics, an ideal experiment would need to account for probabilistic nature of quantum systems. It can be redefined to include the following:

• Experimental Design: Control conditions (e.g., laser voltage), manipulated variables (e.g., slit separation), and environmental factors (e.g., a dark room) are specified.
• Minimization of External Influences: The experiment must minimize or account for unintended environmental interactions that disturb the quantum state.
• Probabilistic Outcomes: The experiment tests quantum probabilities rather than deterministic causality.

Importantly, the experiment must be highly accurate and repeatable, i.e., the probability of an event remains constant across repeated measurements.

 

Note: In one of his Messenger Lectures, Feynman (1965) added: “A philosopher once said 'It is necessary for the very existence of science that the same conditions always produce the same results'. Well, they do not. You set up the circumstances, with the same conditions every time, and you cannot predict behind which hole you will see the electron…...” In short, Feynman emphasizes that in quantum mechanics, the deterministic predictability of outcomes gives way to probabilistic predictions. Furthermore, he clarifies that we do not know how to predict what would happen in a given circumstance and the only thing that can be predicted is the probability of different events. This shift from classical determinism to probabilistic determinism underscores the need for redefining the ideal experiment in quantum mechanics, acknowledging its fundamentally probabilistic nature.

 

2. Probabilistic predictions:

“Yes! Physics has given up. We do not know how to predict what would happen in a given circumstance, and we believe now that it is impossible, that the only thing that can be predicted is the probability of different events. It must be recognized that this is a retrenchment in our earlier ideal of understanding nature. It may be a backward step, but no one has seen a way to avoid it (Feynman et al., 1963, p. 37–10).”

 

Feynman’s statement that “physics has given up” may convey an unintended sense of pessimism, which is potentially misleading. In reality, quantum mechanics represents a profound advancement in our understanding of nature. Unlike classical physics, quantum mechanics does not predict specific outcomes of individual events but instead provides precise probabilistic predictions.

For instance, in the double-slit experiment:

• We cannot predict the exact position where a single photon or electron will land.
• However, we can accurately predict the probability distribution of many such particles, which manifests as an interference pattern.

Thus, quantum mechanics can be described as probabilistically deterministic: while individual outcomes appear random, the probabilistic behavior of a large number of events is determined by the wavefunction.

 

Feynman’s explanations should not be viewed as pessimistic; quantum mechanics underpins integrated circuits (ICs) technologies, particularly at the nanoscale. For example:

• In ICs, the probabilistic behavior of electrons determines quantum tunneling through barriers in transistors as devices shrink toward quantum limits.
• Accurate quantum models help optimize transistor performance and improve IC yield rates—the proportion of functional chips produced. Even minor yield improvements can translate into substantial profits, especially in high-demand chips used for consumer electronics, data centers, and AI systems.

Thus, far from “giving up,” quantum mechanics has enabled us to harness the probabilistic nature in ways that profoundly affect both science and industry.

 

3. Hidden variables:

“We make now a few remarks on a suggestion that has sometimes been made to try to avoid the description we have given: “Perhaps the electron has some kind of internal works—some inner variables—that we do not yet know about. Perhaps that is why we cannot predict what will happen (Feynman et al., 1963, p. 37–11).”

 

The term “inner variables” is not a standard term in quantum mechanics. The correct term in this context is “hidden variables,” which refers to theoretical variables not accounted for in the conventional quantum mechanical framework. In his Messenger lectures, Feynman (1965) explains: “One theory is that the reason you cannot tell through which hole you are going to see the electron is that it is determined by some very complicated things back at the source: it has internal wheels, internal gears, and so forth, to determine which hole it goes through ... … That is called the hidden variable theory.” Furthermore, in the Audio Recordings* [48 min: 35 sec] of this lecture, Feynman says: “internal conditions— hidden variables—…” instead of inner variables. Thus, this seems to be an editorial problem.

 

Feynman (1965) describes hidden variables as “internal wheels, internal gears, and so forth” that determine which path an electron takes, without references to properties like spin or angular momentum. Formally, hidden variables (local or non-local) are hypothetical, unobservable parameters introduced to explain the outcomes of quantum mechanics. Experiments testing Bell’s inequalities have conclusively ruled out the local hidden variables, which assume no faster-than-light communication between entangled particles. However, the term “hidden” itself is arguably misleading or even a misnomer. In the derivation of Bell’s inequality, there is no hidden pre-established agreement among particles (Scarani et al., 2010). Interestingly, some interpret the term “hidden variables” as variables that are hidden from the eyes of quantum pioneers (Belinfante, 2014).

 

According to Belinfante (2014), hidden-variable theories can be categorized:

1. First Kind: Deterministic theories (e.g., Bohmian mechanics) have the same probability predictions as a conventional quantum theory.

2.  Second Kind: They aim for theories that look like causal theories when applied to spatially separated systems that interacted in the past. Each theory (e.g., “local” theory) has a deterministic mechanism underlying quantum phenomena.

3. Zeroth Kind: They include non-conventional approaches that attempt to explain quantum phenomena. For example, von Neumann defines an “impossible” hidden variables theory.

 

Review Questions:

1. How would you redefine an ideal experiment in the context of quantum mechanics?

2. How would you explain that quantum mechanics predicts probabilities rather than specific outcomes? Do you agree that physics has given up?

3. What are hidden variables and how do they relate to quantum theory?

 

The moral of the lesson: In quantum mechanics, we cannot predict the outcome of an individual event. The only thing that can be predicted—reliably and precisely—is the probability of different events. This shift from classical determinism to probabilistic determinism marks a fundamental change in how we view the natural world.

 

References:

1. Belinfante, F. J. (2014). A Survey of Hidden-Variables Theories: International Series of Monographs in Natural Philosophy (Vol. 55). Philadelphia: Elsevier.

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

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. Scarani, V., Lynn, C., & Liu, S. (2010). Six quantum pieces: A first course in quantum physics. Singapore: World Scientific.