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.