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.