Section 38–2 Measurement of position and momentum

(Single slit / Diffraction grating / Uncertainty relation)

 

In this section, Feynman discusses the uncertainty of complementary variables involving a slit and diffraction grating, as well as the uncertainty relation in wave theory. A more fitting title for this section might be “Inherent Uncertainty of Position and Momentum.” It emphasizes that while measurement can increase uncertainty, the uncertainty itself is intrinsic to the system rather than solely a result of measurement (as implied by the title, “Measurement of Position and Momentum”).

 

1. Single slit:

“How does the pattern become spread? To say it is spread means that there is some chance for the particle to be moving up or down, that is, to have a component of momentum up or down. We say chance and particle because we can detect this diffraction pattern with a particle counter, and when the counter receives the particle, say at C in Fig. 38–2, it receives the entire particle, so that, in a classical sense, the particle has a vertical momentum, in order to get from the slit up to C (Feynman et al., 1963, p. 38–2).”

 

Feynman explains diffraction through a single slit using a particle-based perspective. While this explanation aligns with quantum mechanics’ probabilistic nature, it reflects Feynman’s preference for a particle-centric view. In wave theory, light can be represented as wave packets, which are superpositions of waves with slightly different wavelengths. The wave packets may provide a better understanding of diffraction, where the single slit acts as a device that spreads the wave’s wavelength components across a range of angles. The relationship between the slit width and the diffraction pattern can be understood via the uncertainty principle: narrowing the slit increases uncertainty in the momentum of wave packets, resulting in a broader angular spread, and vice versa.

 

The spreading of wave packets by the slit can be interpreted as an environmental interaction, but it is equivalent to a measurement. As light passes through a slit, it becomes localized, effectively corresponding to a position measurement. However, this localization in position introduces an uncertainty in the vertical wave number k_y​, which is directly related to the vertical momentum p_y​ (= ℏk_y​). Based on the wave theory, when the wave packet is highly localized, the momentum uncertainty (Δp_y) is large, causing greater spreading. In this framework, the spreading of the wave packet is not a result of the particle "moving up or down" in the classical sense, but rather an inherent property of wave-like behavior arising from the superposition of wave components.

 

“Sometimes people say quantum mechanics is all wrong. When the particle arrived from the left, its vertical momentum was zero. And now that it has gone through the slit, its position is known. Both position and momentum seem to be known with arbitrary accuracy (Feynman et al., 1963, p. 38–3).”

 

It is worth mentioning that Einstein did not say quantum mechanics was all wrong, but critiqued its completeness. His concern was whether quantum mechanics provides a complete description of reality or merely reflects statistical ignorance of some deeper, hidden reality. Instead of assuming a perfect correlation between a wave packet’s position and momentum, it is crucial to recognize the predictive limitations of quantum mechanics. Once a particle or wave packet interacts with the slit, its momentum is irreversibly disturbed, i.e., the act of measurement (position localization) fundamentally alters the system, increasing momentum uncertainty. However, questions about the nature of quantum correlations and their possible connections to deeper underlying mechanisms continue to drive theoretical and experimental investigations.

 

2. Diffraction grating:

“Suppose we have a grating with a large number of lines (Fig. 38–3), and send a beam of particles at the grating…… That is, the waves which form the diffraction pattern are waves which come from different parts of the grating. The first ones that arrive come from the bottom end of the grating, from the beginning of the wave train, and the rest of them come from later parts of the wave train, coming from different parts of the grating, until the last one finally arrives, and that involves a point in the wave train a distance L behind the first point (Feynman et al., 1963, p. 38–3).”

 

It is remarkable that Feynman used Rayleigh’s criterion of resolution to derive an uncertainty relation, but this criterion is not based on a fundamental physical principle—it is a guideline or convention for resolving two overlapping waves. The key factor in determining the diffraction pattern produced by a grating is the spread of wavelengths within the wave trains. The length of a wave train is crucial because it is directly related to its wavelength spread (Δλ). A longer wave train has a narrower spread of wavelengths, leading to sharper and distinct diffraction peaks, and vice versa. The term wave train is appropriate here because it conveys the idea of a longer wave, as compared to the shorter localized wave packet, which is more apt for modeling particles passing through a single slit.

 

The sharpness of diffraction peaks is not fundamentally limited by the grating itself but rather by the spectral composition (distribution of wavelengths and relative strengths) of the wave train.  If the wavelength spread (Δλ) of the wave train exceeds the grating’s resolving capability, the diffraction pattern will remain blurred, regardless of the number of lines on the grating or width of the slits. Interestingly, the distance L, corresponding to the wave train’s coherence length, can also be interpreted as the minimum length required for using the entire grating effectively. More important, the grating functions like a Fourier transform, decomposing the wave train into its constituent wavelengths. This is analogous to how the human ear—specifically, the cochlea, a fluid-filled spiral structure—distinguishes sounds of different frequencies (See below).

(Source: Parker, 2018)

 

3. Uncertainty relation:

“Now this property of waves, that the length of the wave train times the uncertainty of the wave number associated with it is at least 2π, is a property that is known to everyone who studies them. It has nothing to do with quantum mechanics. It is simply that if we have a finite train, we cannot count the waves in it very precisely (Feynman et al., 1963, p. 38–4).”

 

Feynman’s remark underscores a fundamental fact: a finite wave train does not have a precisely defined wavelength (or wave number). More generally, the product of uncertainties in wave number (Δk) and position (Δx, the length of a wave packet or wave train) can be normalized to a constant such as 2π or 1 for simplicity (see below), depending on the chosen units. This reflects the nature of waves rather than being tied to any specific measurement process. The spectral composition of a wave train depends on its length: a long wave train consists of a narrow spread of wavelengths, akin to playing a single note on a flute for an extended time—an almost pure tone. Conversely, a short wave train has a broad range of frequencies, similar to pressing all the keys on a piano simultaneously, thereby produces a short burst of sound. In short, the length of the wave train tells us about how “pure” or “mixed” the wave is (in terms of its wavelength).

Source: Wave Equation, Wave Packet Solution

The uncertainty relation in wave theory is closely related to Fourier series, where any wave can be represented as a sum of sinusoidal components. Though Feynman does not explicitly specify Fourier transforms, his discussion aligns with Fourier’s principles. Essentially, a shorter wave train needs more wave components of differing wavelengths to form its sharp edges and transient nature, whereas a longer wave train is dominated by a narrower set of frequencies, resulting in a smoother, more uniform waveform. This relation underscores the fundamental trade-off between localization in space and precision in wavelength, mirroring the uncertainty principle in quantum mechanics. This insight, deeply rooted in Fourier analysis, provides a mathematical foundation for understanding the intrinsic limits of measurement in both classical wave theory and quantum mechanics.

 

Note: The section title “Measurement of position and momentum” could be misleading, as the uncertainty relation applies universally to wave phenomena, not just quantum measurement. In a paper on the quantum postulate, Bohr (1928) expressed this clearly: “[r]igorously speaking, a limited wave-field can only be obtained by the superposition of a manifold of elementary waves corresponding to all values of ν and σ_x, σ_y, σ_z. But the order of magnitude of the mean difference between these values for two elementary waves in the group is given in the most favourable case by the condition Δt Δν = Δx Δσ_x = Δy Δσ_y = Δz Δσ_z = 1, where Δt, Δx, Δy, Δz denote the extension of the wave-field in time and in the directions of space corresponding to the co-ordinate axes. These relations — well known from the theory of optical instruments, especially from Rayleigh's investigation of the resolving power of spectral apparatus — express the condition that the wave-trains extinguish each other by interference at the space-time boundary of the wave-field.”

 

Review Questions:

1. How would you explain the spreading of a particle (or wave packet) through a slit?

2. Is it legitimate to derive an uncertainty relation using the Rayleigh’s criterion of resolution?

3. How would you explain inherent uncertainties using the Fourier Transform?

 

The moral of the lesson: The uncertainty of complementary variables in quantum mechanics—whether in a slit or diffraction grating—can be changed by environmental conditions or measurement, however, the uncertainty relation is already known as an inherent property of waves before the quantum revolution.

 

References:

1. Bohr, N. (1928). The Quantum Postulate and the Recent Development of Atomic Theory. Nature, 121, 580-590.

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. Parker, J. A. (2018). Image reconstruction in radiology. Boca Raton, FL: CRC press.

Section 38–1 Probability wave amplitudes

(Probability amplitude / Free particle / Wave packet)

 

In this section, Feynman discusses the concepts of probability amplitude, free particle, and wave packet (or wave train) within the framework of quantum mechanics. While Feynman’s use of the term “probability wave amplitude” might have been intended to make quantum mechanics more accessible, it diverges from standard quantum mechanical terminology. A more precise title, such as “Probability Amplitude of a Free Particle,” could better reflect the content.

 

1. Probability amplitude:

“The probability of finding the particle is then proportional to the absolute square of the amplitude. In general, the amplitude to find a particle in different places at different times varies with position and time (Feynman et al., 1963, p. 38–1).”

 

While the term "absolute square" is common in physics, some may argue it is imprecise. It could suggest multiplying a wavefunction by itself, rather than involving its complex conjugate. In quantum mechanics, the probability density is expressed as |ψ(x,t)|² = ψ*(x,t)ψ(x,t), where |ψ(x,t)| is the modulus of the wavefunction ψ(x,t) and ψ*(x,t) is its complex conjugate. This two-step operation involves both complex conjugation and multiplication. A more precise phrase, as used by Dirac (1981), is “the square of the modulus of the wavefunction.” Notably, Feynman has described the probability of an event as “the square of the absolute value of a complex number” in his previous lecture.

 

In his Alix G. Mautner Memorial Lectures on Quantum Electrodynamics, Feynman (1985) clarifies: “The probability of an event is equal to the square of the length of an arrow called the “probability amplitude (p. 37).” Essentially, he likens the probability amplitude to a clock hand (or complex vector) with magnitude and phase. The probability amplitude is a broader term applicable to both discrete and continuous quantum systems. While a wavefunction is a specific type of probability amplitude (a complex-valued function describing a quantum state in systems with continuous variables), but not all probability amplitudes form wavefunctions. Importantly, the probability amplitude is a mathematical construct (the square root of probability) and cannot be directly measured.

 

Historically, Max Born introduced the probabilistic interpretation of the wavefunction in 1926 through a footnote: “A more precise consideration shows that the probability is proportional to the square of the wave function.” Although square of the wave function lacks precision, Born’s insight laid the foundation for quantum mechanics. More importantly, Born credited Einstein for inspiring his probabilistic interpretation. In his Nobel lecture, Born (1955) recognizes Einstein’s idea of wave amplitude (squared) as the probability of detecting a photon: “… an idea of Einstein’s gave me the lead. He had tried to make the duality of particles - light quanta or photons - and waves comprehensible by interpreting the square of the optical wave amplitudes as probability density for the occurrence of photons.”

 

2. Free particle:

“In a special case the amplitude varies sinusoidally in space and time like e^i(ωt−kr) ... ... Then it turns out that this corresponds to a classical limiting situation where we would have believed that we have a particle whose energy E was known and is related to the frequency by E=ℏω… (Feynman et al., 1963, p. 38–1).”

 

Feynman’s description of a free particle’s amplitude prompts the question: “Why does a photon’s energy depend on frequency, while in classical wave theory, energy depends on wave amplitude?” The distinction lies in the models: in quantum mechanics, the probability amplitude corresponds to the likelihood of photon detection, whereas in classical wave theory, the electric field amplitude determines the wave’s energy. Interference patterns can be explained using electric field amplitudes (classical theory) or probability amplitudes (quantum mechanics), depending on whether the source is a laser or an electron gun. Thus, it is essential to distinguish between electric field amplitude and probability amplitude when describing a system.

 

Perhaps Feynman could have used the term free particle because its energy and momentum are constant, which means there is no external force acting on the particle. A free particle is often idealized as a wave packet, representing a localized particle with finite spatial extent and inherent momentum uncertainty. If the particle’s momentum is defined as p = ℏk, its position becomes completely uncertain, as represented by a plane wave*. Similarly, if the particle’s energy is E = ℏω, the particle’s transit time becomes completely uncertain. In other words, this is an idealization whereby there is a complete uncertainty of its location in space-time.

 

*In the Audio Recordings [4 min: 20 sec and 4 min: 30 sec] of this lecture (first try), Feynman mentions: “plane wave,” however, this term is absent from the corresponding text in his book. It is also worth noting that Feynman delivered this lecture twice, with slight variations in content between the two presentations.

 

“For instance, if an amplitude to find a particle at different places is given by e^i(ωt−kr), whose absolute square is a constant, that would mean that the probability of finding a particle is the same at all points. That means we do not know where it is—it can be anywhere—there is a great uncertainty in its location. …... Outside this region, the probability is zero (Feynman et al., 1963, p. 38–1).”

 

Feynman’s description of probability amplitude of a free particle reveals the limitations of the probability associated with the particle. For a truly free particle in infinite space, interpreting the constant probability (or plane wave) can be problematic, as it implies infinite extent and lacks a realistic connection to observable outcomes. To address this, physicists often introduce boundary conditions or approximate the particle with a localized wave packet. Furthermore, when Feynman says that the probability of finding a particle outside a certain region is "zero," this is a practical simplification. More accurately, the probability in these regions is better described as infinitesimally small, approaching but not exactly zero. This reflects the behavior of the particle** (or wave packet), which may extend to infinity but decay rapidly such that its probability becomes negligible.

 

**In the Audio Recordings [5 min: 20 sec and 6 min: 25 sec] of this lecture (first try), Feynman mentions: “photon,” however, this term is absent from the corresponding text in his book.

 

Top of Form3. Wave packet:

“It is something that anybody who works with waves, even if he knows no quantum mechanics, knows: namely, we cannot define a unique wavelength for a short wave train. Such a wave train does not have a definite wavelength; there is an indefiniteness in the wave number that is related to the finite length of the train, and thus there is an indefiniteness in the momentum (Feynman et al., 1963, p. 38–2).”

 

A wave packet is a localized disturbance formed by the superposition of harmonic waves with slightly differing wavelengths. It has three key features:

1.      Principle of Superposition: It is formed due to the superposition of waves, which differ from each other by infinitesimal increments of wavelengths.

2.      Bandwidth: It has a continuous range of wavelengths (Δλ) centered on a dominant wavelength (λ₀).

3.      Fourier Integral: It is represented by a Fourier integral, producing a modulated wave with finite spatial extent and central frequency.

In classical wave theory, wave trains propagate without changing shape in ideal, non-dispersive media. In quantum mechanics, wave packets spread over time due to their superposition of momentum states, but environmental interactions and experimental measurements can alter this process.

 

Historically, Schrödinger (1926) used terms like "wave parcel" and "parcel of waves" in his original works. The term “wave packet” became formalized later, notably by Dirac, who applied it in quantum mechanics. In Principles of Quantum Mechanics, Dirac (1981) writes: "It is interesting to apply (55) to a ket whose Schrödinger representative consists of what is called a wave packet," which suggests the use of wave packet in quantum mechanics. On the other hand, the term "wave-train" is sometimes attributed to E.L. Nichols and W.S. Franklin who first used in 1897 in a text titled The elements of physics. However, in Theory of Sound, Rayleigh (1877) showed that a “train of waves” would travel in a dispersive medium at a speed, u, different from that of the individual wave crests. 

 

Physicists commonly use the term wave packet to describe photons or electrons in quantum mechanics, as it captures their localized and probabilistic nature. In contrast, the probability amplitude of a free particle is represented by a plane wave, more accurately described as an “infinitely long wave train” of single frequency rather than a finite wave train. However, such an infinite wave train is not “square integrable” and cannot be normalized like a localized wave packet. Interestingly, Feynman’s use of terminology appears inconsistent. In this lecture, he mentions “a wave train whose length is Δx,” but the accompanying figure is labeled “A wave packet of length Δx.” Later, in Volume III, Feynman consistently uses the term wave packet, e.g., he explains: “we can make up a ‘wave packet’ with a predominant wave number k₀​, but with various other wave numbers near k₀​ (p. 13–6).” While both terms have their limitations, wave packet more accurately conveys the quantum mechanical reality of particles as localized entities.

 

Review Questions:

1. Would you explain the probability of finding a particle is proportional to the absolute square of the amplitude?

2. What are the limitations of describing a free particle in quantum mechanics?

3. How would you define wave packet in the context of quantum mechanics?

 

The moral of the lesson: The probability amplitude of a free particle can be modeled using a wave packet, which has both particle-like and wave-like properties. However, we must remain cognizant of the practical limitations of plane wave representations, particularly their lack of localization.

 

References:

1. Born, M. (1955). Statistical interpretation of quantum mechanics. Science, 122(3172), 675-679.

2. Dirac, P. A. M. (1981). The principles of quantum mechanics. Oxford university press.

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., Leighton, R. B., & Sands, M. (1966). The Feynman lectures on physics Vol III: Quantum Mechanics. Reading, MA: Addison-Wesley.

6. Nichols, E. L., & Franklin, W. S. (1898). The Elements of Physics: Electricity and magnetism (Vol. 2). Macmillan.

7. Rayleigh, J. W. S. (1877). The theory of sound (Vol. 1). Macmillan.

8. Schrödinger, E. (1926). An undulatory theory of the mechanics of atoms and molecules. Physical review, 28(6), 1049-1070.