Heisenberg’s uncertainty principle (definitions of uncertainty and error)

Question: When an electron passes through a slit, the uncertainty of its position Δx and the uncertainty of its momentum Δp are related by the expression, ΔxΔp ≈ h/2π. Which diagram below shows the correct position where Δx and Δp are defined? (A) (B)

Interestingly, the question asks for the location where Δx and Δp are defined in the diagram. Physics teachers may simply explain that the correct answer is (A) because Δx and Δp are in the same direction as well as the uncertainties are dependent on the experimental setup. However, the Heisenberg’s position-momentum uncertainty (indeterminacy) principle can be more precisely stated as ΔyΔp_y ≥ h/4π in the question. Firstly, the uncertainty in a particle’s momentum (Δp_y) should be in the same direction as the uncertainty in its position (Δy). Secondly, in Kennard’s formulation of the uncertainty principle, it is possible to have a lower theoretical limit h/4π instead of h/2π. Thirdly, physicists prefer the symbol ≥ instead of ≈ because of possible experimental errors. More importantly, we should be aware of different ways of defining Δx and Δp.

The definitions of Δx and Δp are still being revised. In his seminal paper titled, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Heisenberg (1927) derived the uncertainty relation q₁p₁ ∼ h, where q₁ is the “mean error” of the position measurement and p₁ is the “discontinuous change” of the momentum. However, his definitions of Δx and Δp are not sufficiently precise. Within the same year, Kennard (1927) derived the uncertainty inequality: σ_xσ_p ≥ h/4π, where σ_x and σ_p are the standard deviations of position and momentum. Alternatively, according to Ozawa (2003), Heisenberg’s uncertainty principle can also be expressed as ε(q)η(p) ≥ h/4π. It means that when you measure the position of an object with an error ε(q), you also alter the momentum of the same object by an amount of η(p) simultaneously. 

How would Feynman answer?

Firstly, you should be cognizant of three versions of Heisenberg's uncertainty principle in The Feynman Lectures on Physics.

1. ΔxΔp_x ≈ h: By using a photon of wavelength λ to locate a very small object, the position of this object can be measured with an uncertainty Δx ≈ λ. If the photon transfers its momentum to the same object, it may introduce an uncertainty in momentum (Δp_x) by approximately h/λ. Thus, ΔxΔp_x ≈ (λ)(h/λ) = h. (See Feynman et al., 1963, section 38–2 Measurement of position and momentum.)

2. ΔxΔp_x  ≥ h/2π: If a Gaussian wave packet has a spread in position Δx (or confined to a region Δx), the values of the wave numbers may have a spread of Δk. By using Fourier transform, we may deduce the inequality ΔxΔk ≥ 1. Since k = 2πp_x/h, we have the inequality ΔxΔp_x ≥ h/2π. In Volume III of The Feynman Lectures on Physics, Feynman explains that “[w]e have usually made the approximate statement that the minimum value of the product ΔpΔx is of the same order as h/2π (Feynman et al., 1966, section 16-3 States of definite momentum).”

3. ΔxΔp_x ≥ h/4π: We can define uncertainties as standard deviations of conjugate variables. By using Cauchy-Schwarz inequality, it is possible to deduce the inequality ΔxΔp_x ≥ h/4π. In Volume III of The Feynman Lectures on Physics, Feynman states that “Gaussian distribution gives the smallest possible value for the product of the root-mean-square widths (Feynman et al., 1966, section 16-3 States of definite momentum).” 

       There are at least two possibilities how Feynman would answer the question:

1. If Feynman was having a good mood, he would agree with the option (A) and explain that Δx and Δp cannot be exactly defined at a point in the diagram. Furthermore, he might elaborate that “[s]uppose we have a single slit, and particles are coming from very far away with a certain energy - so that they are all coming essentially horizontally (Fig. 38–2). We are going to concentrate on the vertical components of momentum. All of these particles have a certain horizontal momentum p₀, say, in a classical sense. So, in the classical sense, the vertical momentum p_y, before the particle goes through the hole, is definitely known. The particle is moving neither up nor down, because it came from a source that is far away - and so the vertical momentum is of course zero. But now let us suppose that it goes through a hole whose width is B. Then after it has come out through the hole, we know the position vertically - the y position - with considerable accuracy - namely ±B. That is, the uncertainty in position, Δy, is of order B... (Feynman et al., 1963, section 38–2 Measurement of position and momentum).”

It is worth mentioning that in the New Millennium Edition of The Feynman Lectures on Physics, the following footnote is included: “More precisely, the error in our knowledge of y is ±B/2. But we are now only interested in the general idea, so we won’t worry about factors of 2 (Feynman et al., 2011).” Thus, Feynman might admit that he goofed in drawing the uncertainty Δy in Fig. 38–2 just like the option (A). That is, there could be “two Δy” similar to the option (B). However, Feynman could suggest that the electron can penetrate the slit vertically. He might even add that if you insist upon a precise definition of uncertainty, you will never get it!

2. If Feynman was having a bad mood, he might refuse to choose the option (A) from the question. His explanation could be there is no need for an uncertainty principle. In QED: The Strange Theory of Light and Matter, Feynman (1990) states that “[i]f you get rid of all the old-fashioned ideas and instead use the ideas that I’m explaining in these lectures – adding arrows for all the ways an event can happen – there is no need for an uncertainty principle (p. 56)!” Feynman might also elaborate using his version of quantum electrodynamics that an electron “is able to get through the narrow slit goes to Q almost as much as to P, because there are not enough arrows representing the paths to Q to cancel each other out (p. 56).” (P and Q are two points on a screen at different y-positions; please read QED for more details.) 

In a sense, Feynman’s formulation of quantum electrodynamics supersedes the uncertainty principle. However, it was difficult for Bohr and many others to understand Feynman’s path integral approach during the Pocono Conference 1948. In the words of Feynman, “I said that in quantum mechanics one could describe the amplitude of each particle in such and such a way. Bohr got up and said: ‘Already in 1925, 1926, we knew that the classical idea of a trajectory or a path is not legitimate in quantum mechanics; one could not talk about the trajectory of an electron in the atom, because it was something not observable.’ In other words, he was telling me about the uncertainty principle. It became clear to me that there was no communication between what I was trying to say and what they were thinking. Bohr thought that I didn’t know the uncertainty principle … Bohr was concerned about the uncertainty principle and the proper use of quantum mechanics. To tell a guy that he doesn’t know quantum mechanics, — well, it didn’t make me angry, it just made me realize that he [Bohr] didn’t know what I was talking about, and it was hopeless to try to explain it further. I gave up, I simply gave up…  (Mehra, 1994, p. 248).”

Note: 

The validity of Heisenberg’s uncertainty principle has been questioned. For example, Ozawa (2003) formulates the error–disturbance uncertainty: ε(q)η(p) + σ(q)η(p) + σ(p)ε(q) ≥ h/4π, where ε(q) represents the measurement error of an observable q, η(p) represents the disturbance on an observable p due to the former measurement of q, and σ represents the standard deviation of the two observables (or conjugate variables, p and q). However, the definitions of “uncertainty,” “error,” and “disturbance” are still being debated. 

References:

1. Feynman, R. P. (1990). QED: The Strange Theory of Light and Matter. London: Penguin.

2. Feynman, R. P., Leighton, R. B., & Sands, M. L. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Feynman, R. P., Leighton, R. B., & Sands, M. L. (1966). The Feynman Lectures on Physics, Vol III: Quantum Mechanics. Reading, MA: Addison-Wesley.

4. Feynman, R. P., Leighton, R. B., & Sands, M. L. (2011). The Feynman Lectures on Physics, boxed set: The New Millennium Edition. New York: Basic Books.

5. Heisenberg, W. (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik, 43(3), 172-198.

6. Kennard, E. H. (1927). Zur Quantenmechanik einfacher Bewegungstypen. Zeitschrift für Physik, 44(4), 326-352.

7. Mehra, J. (1994). The Beat of a Different Drum: The life and science of Richard Feynman. Oxford: Oxford University Press. 

8. Ozawa, M. (2003). Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement. Physical Review A, 67(4), 042105.