Nature of plane mirror image

Question: What are the characteristics of an image formed by a plane mirror?

In general, the characteristics of a plane mirror image can be described as follows:

1. The image is left-right reversed or front-back reversed in relation to the object.

2. The image is virtual because it cannot be formed on a screen.

3. The image has the same size as the object.

4. The image is upright.

5. The image is located at the same distance behind the mirror as the object is in front of the mirror. (Image distance = object distance)

6. The image has the same color as the object.

It has been controversial whether the nature of image formed by a plane mirror should be described as “left-right reversed” or “front-back reversed.” This is related to an interesting question, why does a mirror reverse left and right but not up and down?Currently, the plane mirror image could be specified in textbooks as “lateral inverted” (Muncaster, 1993), “left-right reversed” (Cutnell & Johnson, 2004), “appears left-right reversed” (Giancoli, 2005), “front-back reversed” (Knight, 2004), or “depth inverted” (Tipler & Mosca, 2004). Interestingly, Tomonaga, who shared the 1965 Nobel (Physics) Prize with Feynman and Schwinger, discussed the mirror reflection problem with his colleagues and believed that the top–bottom and front–back axes had absoluteness in a “psychological space” (Tabata & Okuda, 2000; Tomonaga, 1965). However, there is no agreement on the description and explanation of the plane mirror image.

The concept of the plane mirror image is related to terms such as “parity,” “enantiomorph,” and “chirality.” For example, Lord Kelvin defines the concept of chirality in a footnote of a lecture, titled The molecular tactics of a crystal. This famous footnote reads:

“I call any geometrical figure, or group of points, chiral, and say that it has chirality, if its image in a plane mirror, ideally realized, cannot be brought to coincide with itself. Two equal and similar right hands are homochirally similar. Equal and similar right and left hands are heterochirally similar or ‘allochirally’ similar (but heterochirally is better). These are also called ‘enantiomorphs,’ after a usage introduced, I believe, by German writers. Any chiral object and its image in a plane mirror are heterochirally similar (Kelvin, 1894, p. 27).”

The term chirality is derived from the Greek word “hand.” Naturally, human hands are chiral objects because the left hand, for example, is a non-superimposable mirror image of the right hand. In other words, an object is chiral if it cannot be brought to coincide with itself by rotations and translations alone.

How would Feynman answer?

Feynman’s answer may include the concepts “front-back reversed” and “handedness of an object” pertaining to the nature of plane mirror image. However, it is more meaningful to understand his explanations on “front-back reversed,” “handedness of an object,” and “definitions of left and right.”

1. Front-back reversed: During a BBC Television interview, Feynman (1994) explains that “if you wave one hand, then the hand in the mirror that waves is opposite it – the hand on the ‘east’ is the hand on the ‘east,’ and the hand on the ‘west’ is the hand on the ‘west.’ The head that’s up is up, and the feet that are down are down. So everything’s really all right. But what’s wrong is that if this is ‘north,’ then your nose is to the back of your head, but in the image, the nose is to the ‘south’ of the back of your head. What happen is, the image has neither the right nor the left mixed up with the top and the bottom, but the front and the back have been reversed, you see (p. 37).” In a sense, there is a semantic problem in describing the nature of plane mirror image (Ansbacher, 1992). It is remarkable that Feynman is not constrained by the words, “left” and “right,” and he is able to replace them by “east” and “west.” Thus, the plane mirror image can be simply described as either “north-south reversed” or “front-back reversed.”

In general, the choice of phrase such as “front-back reversed” is imprecise. To be more precise, “[a] mirror image reproduces exactly all object points in two spatial directions parallel to the mirror surface, but reverses the sequential ordering of object points in the direction of the third spatial axis, perpendicular to the mirror plane (Galili & Goldberg, 1993, p. 463).” In other words, a plane mirror does not vary the coordinates such as y and z in the two-dimensional planes that are parallel to the mirror, but it reverses the “x” coordinates, for example, that are in the same direction as the axis of the mirror. However, physics teachers may find it cumbersome to describe the nature of plane mirror image in greater detail.

In short, the nature of plane mirror image may appear as “left-right reversed,” “top-bottom reversed,” and “front-back reversed” (See Fig 1a, Fig 1b, and Fig 1c). According to Feynman (1994), “we say left and right are interchanged, but really the symmetrical way is it’s along the axis of the mirror that things get interchanged (p. 38).” The descriptions of the plane mirror image are dependent on the “axis of the mirror.” In Fig 1a, the plane mirror image of a right-handed glove appears as “left-right reversed.” The axis of this mirror is in a horizontal direction and the mirror is placed beside the glove. In Fig 1b, the plane mirror image of the upright glove appears as “top-bottom reversed.” The axis of the mirror is in a vertical direction and the mirror is placed below the glove. In Fig 1c, the plane mirror image of the glove appears as “front-back reversed.” The axis of this mirror is horizontal and the mirror is placed in front of the glove. Essentially, the location of the mirror relative to the object affects descriptions of the plane mirror image.

Fig 1a. Mirror beside the object (“left-right reversed”)

Fig 1b. Mirror below the object        (“top-bottom reversed”)

Fig 1c. Mirror in front of the object          (“front-back reversed”)

2. The handedness of an object: Although Feynman explains that a characteristic of the plane mirror image is front-back reversed, he also describes the handedness of an object. Feynman provides the following example: “[t]he first molecule, the one that comes from the living thing, is called L-alanine. The other one, which is the same chemically, in that it has the same kinds of atoms and the same connections of the atoms, is a ‘right-hand’ molecule, compared with the ‘left-hand’ L-alanine, and it is called D-alanine… (Left-handed sugar tastes sweet, but not the same as right-handed sugar.) So it looks as though the phenomena of life permit a distinction between ‘right’ and ‘left,’ or chemistry permits a distinction because the two molecules are chemically different (Feynman et al., 1963, section 52–4 Mirror reflections).” Simply phrased, the chemical and physical properties of the object are dependent on its handedness.

The handedness of a particle is an important concept which helps to understand the principle of the conservation of parity (mirror symmetry). As an analogy, Swedish physicist Cecilia Jarlskog identified a similarity between left-handed neutrinos and “vampire”: they do not have a mirror image (t’Hooft, 1997). In other words, the plane mirror changes the handedness of a particle, and this “mirror particle” may or may not be observed in nature. In essence, nature has a preference on the handedness of the particle, and it does not conform to the conservation of parity principle. Importantly, T. D. Lee and C. N. Yang (1956) predicted three experiments that illustrate the non-conservation of parity in weak interactions. It resolves the famous “tau-theta puzzle” pertaining to the decay of kaons, which supposedly have the same mass but they can decay into products of opposite parity.

The non-conservation of parity should not be simply illustrated by the handedness of a particle. It also involves physical conditions such as very low temperature and strong magnetic field. To quote Feynman, “When we put cobalt atoms in an extremely strong magnetic field, more disintegration electrons go down than up. Therefore, if we were to put it in a corresponding experiment in a “mirror,” in which the cobalt atoms would be lined up in the opposite direction, they would spit their electrons up, not down; the action is unsymmetrical (Feynman et al., 1963, section 52-7 Parity is not conserved!).” In this experiment, the observation of a preferred direction of decays helps to establish the violation of parity. It is pertinent to understand how the “mirror condition” such as the magnetic field is related to the handedness of the object (e.g. cobalt atoms).

3. Definitions of “left” and “right”: Feynman would discuss the problems of defining “left” and “right” and explain that “the world does not have to be symmetrical. For example, using what we may call ‘geography,’ surely ‘right’ can be defined. For instance, we stand in New Orleans and look at Chicago, and Florida is to our right… (Feynman et al., 1963, section 52–4 Mirror reflections).” That is, it is possible to define “right” and “left” by using geography because there is no symmetry between two locations such as Chicago and Florida. However, the directions for up-down, left-right and front-back are arbitrarily defined depending on one’s orientation and perspective. Generally speaking, definitions of left and right are ambiguous due to possible rotations of an observer about a vertical axis. Thus, one should initially define the directions of “up,” “down,” “front,” and “back.”

Interestingly, Feynman would explore how to tell a Martian the definitions of “left” and “right.” During a lecture at Cornell University, he mentions the following procedure: “take a radioactive stuff, a neutron, and look at the electron which comes from such a beta-decay. If the electron is going up as it comes out, the direction of its spin is into the body from the back on the left side. That defines left. That is where the heart goes (Feynman, 1965, p. 103).” Alternatively, in The Feynman Lectures on Physics, he describes the Wu et al. (1957) experiment: “build yourself a magnet, and put the coils in, and put the current on, and then take some cobalt and lower the temperature. Arrange the experiment so the electrons go from the foot to the head, then the direction in which the current goes through the coils is the direction that goes in on what we call the right and comes out on the left (Feynman et al., 1963, section 52-7 Parity is not conserved!).” Nevertheless, Feynman also defines the direction of “top” and “bottom” in this experiment. 

Ideally, the definitions of “right” and “left” should not be dependent on history and convention (Feynman et al., 1963, section 52-4 Mirror reflections). As an example, most screws have right-handed threads which are arbitrarily determined. In fact, it is possible to have left-handed screws which are traditionally used for coffins (McManus, 2002). On the other hand, the right-handed rule for magnetic fields and definition of neutrinos as left-handed are merely conventions. Physicists could also define electric field as a pseudo-vector and magnetic field to be a vector (Griffiths, 2004). Similarly, physicists could have renamed neutrinos as anti-neutrinos and vice versa, thus changing their handedness. However, an interesting question now is whether right-handed neutrinos can be detected, and thus, they exist not only in the mirror world but also the real world.

       In summary, we may describe the nature of a plane mirror image as “front-back reversed,” “left-right reversed,” and “top-bottom reversed.” The descriptions of the plane mirror image are dependent on the “handedness of an object,” and the “axis of the mirror.” However, we should understand Feynman’s reasonings pertaining to the concept of “front-back reversed,” “handedness of an object,” and “definitions of left and right.”

Note: 
In an article titled Theory of the Fermi interaction, Feynman and Gell-Mann (1958) state that “only neutrinos with left-hand spin can exist (p. 195).”

References:

1. Ansbacher, T. H. (1992). Left-Right Semantics. The Physics Teacher, 30(2), 70.

2. Cutnell J. D., & Johnson, K. W. (2004). Physics (6th ed.). New Jersey: John Wiley & Sons. 
3. Feynman, R. P. (1965). The character of physical law. Cambridge: MIT Press.

4. Feynman, R. P. (1994). No Ordinary Genius - The Illustrated Richard Feynman. New York: W. W. Norton and Company.

5. Feynman, R. P., & Gell-Mann, M. (1958). Theory of the Fermi interaction. Physical Review, 109(1), 193-198.
6. 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.

7. Galili I., & Goldberg F. (1993). Left-right Conversions in a Plane Mirror. The Physics Teacher, 31(8), 463–466.

8. Giancoli, D. C. (2005). Physics: Principles with Applications (6th ed.). Upper Saddle River, N. J.: Prentice Hall.

9. Griffiths, D. (2004). Introduction to Elementary Particles. Weinheim: Wiley-VCH.

10. Kelvin, W. T. (1894). The molecular tactics of a crystal. Oxford: Clarendon Press.

11. Knight, R. D. (2004). Physics for Scientists and Engineers with Modern Physics: A Strategic Approach. Boston: Addison Wesley.

12. Lee, T. D. & Yang, C. N. (1956). Question of parity conservation in weak interactions. Physical review, 104(1), 254–258.

13. McManus, C. (2002). Right Hand, Left Hand: The Origins of Asymmetry in Brains, Bodies, Atoms and Cultures. Cambridge: Harvard University Press.

14. Muncaster, R. (1993). A Level Physics (4th ed). Cheltenham: Nelson Thornes.

15. Tabata, T., & Okuda, S. (2000). Mirror reversal simply explained without recourse to psychological processes. Psychonomic Bulletin & Review, 7(1), 170–173.

16. t’Hooft, G. (1997). In Search of the Ultimate Building Blocks. Cambridge: Cambridge University Press.

17. Tipler, P. A., & Mosca, G. P. (2004). Physics for Scientists and Engineers (5th ed.). New York: W. H. Freeman.

18. Tomonaga, S. (1965). Kagaminonaka no sekai [The world in the mirror]. Tokyo: Misuzu-Shobo.

19. Wu, C. S., Ambler, E., Hayward, R. W., Hoppes, D. D., & Hudson, R. P. (1957). Experimental test of parity conservation in beta decay. Physical review, 105(4), 1413-1415.

Newton’s First Law of Motion

Question: A physics teacher provides the following statement of Newton’s first law of motion: Every body persists in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed on it. Explain two problems of this statement and rewrite it with the appropriate amendments.

This “examination question” is crafted based on an article by Williams (1999), titled Semantics in teaching introductory physics. He proposes two improvements in the above statement of Newton’s first law of motion: (1) the term “motion” should be replaced by “velocity” because motion does not have a quantitative definition; (2) the term “a net force” is more precise than “forces impressed” because a vector sum of forces acting on an object does not necessarily result in acceleration. However, there are still three problematic terms “isolated object,” “straight line,” and “inertial frame” which are commonly found in the modern formulation of Newton’s first law.

1. Isolated object: In Relativity: The Special and The General Theory, Einstein (1961/1916) states the law of inertia as “[a] body removed sufficiently far from other bodies continues in a state of rest or of uniform motion in a straight line (p. 13).” That is, the law of inertia is applicable to an isolated object or free particle that is very far from other bodies. In other words, the term isolated object replaces the phrase “net force impressed on an object.” However, it is difficult to ensure an object to be completely free of external interactions such as gravitational forces and electromagnetic forces in an experiment. An isolated object is an idealized concept and it does not exist in the real physical world.

2. Straight line: In Motte’s translation of Newton's first law, it was stated that “every body perseveres in its state of rest, or of uniform motion in a right line (Newton, 1687/1995).” Currently, the term “straight line” is commonly used rather than “right line.” Importantly, a straight line may be defined as a path of a light ray (Poincaré, 1952). Thus, a straight line can be observed as a curve dependent on an observer’s frame of reference. Historically speaking, Galilei seems to be open to the idea of circular inertia. Furthermore, Newton (1687/1995) explains his first law by stating that “[t]he greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time (p. 19).”

3. Inertial frame: In The Science of Mechanics, Mach (1888/1989) writes that “I have remained to the present day the only one who insists upon referring the law of inertia to the earth and, in the case of motions of great spatial and temporal extent, to the fixed stars (p. 336-337).” On the other hand, Einstein (1961/1916) states that the laws of the mechanics of Galilei-Newton are valid only for a Galileian system of co-ordinates. However, current physics textbooks may specify this condition of validity as an inertial frame of reference. Thus, there is a problem of circularity if the inertial frame of reference is defined as the reference frame in which Newton’s first law holds. To resolve this problem, the inertial frame of reference can be defined as the reference frame in which space is homogeneous and isotropic, and time is homogeneous (Landau & Lifshitz, 1976).

How would Feynman answer?

Feynman’s father had an influence on him in understanding the concept of inertia. In the words of Feynman, “[m]y father taught me to notice things. One day, I was playing with an ‘express wagon,’ a little wagon with a railing around it. It had a ball in it, and when I pulled the wagon, I noticed something about the way the ball moved. I went to my father and said, ‘Say, Pop, I noticed something. When I pull the wagon, the ball rolls to the back of the wagon. And when I’m pulling it along and I suddenly stop, the ball rolls to the front of the wagon. Why is that?’ ‘That, nobody knows,’ he said. ‘The general principle is that things which are moving tend to keep on moving, and things which are standing still tend to stand still, unless you push them hard. This tendency is called ‘inertia,’ but nobody knows why it’s true.’ Now, that’s a deep understanding. He didn’t just give me the name (Feynman, 1988, p. 16).”

In short, Feynman prefers this general principle to be known as the “principle of inertia” rather than “Newton’s first law of motion” or “Galilei-Newton’s law of inertia.” It is possible that Feynman would explain the three problematic concepts: isolated object, straight line, and inertial frame as follows:

1. Isolated object: The principle of inertia can be stated as follows: “if an object is left alone and is not disturbed, it continues to move with a constant velocity in a straight line if it was originally moving, or it continues to stand still if it was just standing still.” That is, this law refers to an isolated object (or a free particle) that is completely undisturbed and has no potential energy at all. However, it is difficult to define a free particle or isolated object because there are gravitational fields everywhere. Importantly, by using the principle of least action, a free particle will move from one point to another at a constant speed such that the kinetic energy integral is least. If the particle were to go any other way, the velocities would be sometimes higher and sometimes lower than the average. The average velocity of the particle is the same for every case because it has to get from ‘here’ to ‘there’ in a given amount of time.

2. Straight line: One important aspect of the principle of inertia is the “physical” straight line of a moving particle. However, the concept of straight line can be ambiguous. It is possible to have different definitions of straight line on a plane, a sphere, or a hot plane. In general, the straight line can be defined as the shortest line between two points. Interestingly, the curve of shortest distance in space corresponds in space-time not to the path of shortest time, but to the one of longest time, because of the funny things that happen to signs of the t-terms in relativity. Therefore, we should be aware of possible problems of defining a straight line. In essence, the concept of the straight line is dependent on an observer’s frame of reference.

3. Inertial frame: The idea that inertia represents the effects of interactions with faraway matter was first developed by Ernst Mach in the nineteenth century. Mach felt that the concept of an absolute acceleration relative to “space” was not meaningful; the usual absolute accelerations of classical physics should be rephrased as accelerations relative to the distant nebulae. In addition, an inertial frame of reference can also be automatically determined from the nebulae. However, a motion relative to the nebulae is a mysterious question that can be answered only by experiment. Thus, there are also difficulties in defining an inertial frame or reference frame which is moving with a constant velocity in a straight line.

In summary, Feynman would discuss the problems of defining isolated object, straight line, and inertial frame. However, he states that “if an object is left alone, is not disturbed, it continues to move with a constant velocity in a straight line if it was originally moving, or it continues to stand still if it was just standing still (Feynman et al., 1963, section 9–1 Momentum and force).” Furthermore, Feynman (1995) explains that “[t]he principle of inertia is a statement that the time scale is independent of coordinates X; the classical trajectories are interpreted to follow the normal lines of constant phase (p. 72).” Essentially, Feynman has a deep understanding of the principle of inertia even though he says that he does not know the reason behind it (Feynman et al., 1963, section 7–3 Development of dynamics).

“Feynman’s answers” pertaining to isolated object, straight line, and inertial frame are based on the following compilation of his statements: 1. Isolated object: That is the principle of inertia—if something is moving, with nothing touching it and completely undisturbed, it will go on forever, coasting at a uniform speed in a straight line. (Why does it keep on coasting? We do not know, but that is the way it is.) (Feynman et al., 1963, section 7–3 Development of dynamics). Galileo made a great advance in the understanding of motion when he discovered the principle of inertia: if an object is left alone, is not disturbed, it continues to move with a constant velocity in a straight line if it was originally moving, or it continues to stand still if it was just standing still (Feynman et al., 1963, section 9–1 Momentum and force). First, suppose we take the case of a free particle for which there is no potential energy at all. Then the rule says that in going from one point to another in a given amount of time, the kinetic energy integral is least, so it must go at a uniform speed. (We know that’s the right answer—to go at a uniform speed.) Why is that? Because if the particle were to go any other way, the velocities would be sometimes higher and sometimes lower than the average. The average velocity is the same for every case because it has to get from ‘here’ to ‘there’ in a given amount of time (Feynman et al., 1964, chapter 19 The Principle of Least Action).  But, as soon as we allow the presence of gravitating masses anywhere in the universe, concept of such truly unaccelerated motion becomes impossible, because there will be gravitational fields everywhere (Feynman, 1995, p. 93). 2. Straight line: Finally, our third bug — the one in Fig. 42-3 — will also draw “straight lines” that look like curves to us. For instance, the shortest distance between A and B in Fig. 42-6 would be on a curve like the one shown. Why? Because when his line curves out toward the warmer parts of his hot plate, the rulers get longer (from our omniscient point of view) and it takes fewer “yardsticks” laid end-to-end to get from A to B. So for him the line is straight — he has no way of knowing that there could be someone out in a strange three-dimensional world who would call a different line “straight” (Feynman et al., 1964, section 42–1 Curved spaces with two dimensions). The point of all this is that we can use the idea to define “a straight line” in space-time. The analog of a straight line in space is for space-time a motion at uniform velocity in a constant direction. The curve of shortest distance in space corresponds in space-time not to the path of shortest time, but to the one of longest time, because of the funny things that happen to signs of the t-terms in relativity. “Straight-line” motion—the analog of “uniform velocity along a straight line”—is then that motion which takes a watch from one place at one time to another place at another time in the way that gives the longest time reading for the watch. This will be our definition for the analog of a straight line in space-time (Feynman et al., 1964, section 42-4 Geometry in space-time). Clearly, the vector (dxμ/ds) along the geodesic represents a tangential velocity, Δtμ, along the geodesic, which is the “physical” straight line (Feynman, 1995, p. 130). 3. Inertial frame: The idea that inertia represents the effects of interactions with faraway matter was first developed by Ernst Mach in the nineteenth century, and it was one of the powerful ideas that Einstein had in mind as he constructed his theory of gravitation. Mach felt that the concept of an absolute acceleration relative to “space” was not meaningful; that instead, the usual absolute accelerations of classical physics should be rephrased as accelerations relative to the distant nebulae (Feynman, 1995, p. 70).  We shall now show that the inertial frame is now also automatically determined from the nebulae, and the phenomena of inertia for accelerations relative to the nebulae can be understood if the “length determining principle” is accepted (Feynman, 1995, p. 72). In special relativity, extensive use is made of reference frames which are moving with a uniform velocity in a straight line (Feynman, 1995, p. 92).  Then we say to him, “Now, my friend, is it or is it not obvious that uniform velocity in a straight line, relative to the nebulae should produce no effects inside a car?” Now that the motion is no longer absolute, but is a motion relative to the nebulae, it becomes a mysterious question, and a question that can be answered only by experiment (Feynman et al., 1963, section 16–1 Relativity and the philosophers).

Note: 

1. An example of examination question pertaining to Newton’s first law of motion is shown below: A student makes the following statements of Newton’s laws of motion: “First Law: Every body continues in its state of motion unless it is acted upon by a resultant external force.”… The statement of the First Law is incomplete in two aspects. Identify the two aspects in which it is incomplete and hence rewrite it with the appropriate amendments (Council for the Curriculum, Examinations & Assessment, 1998). 

2. You may want to take a look at this website: 

http://physicsassessment.blogspot.sg/2016/05/ib-physics-2015-higher-level-paper-2_25.html

References:

1. Einstein, A. (1961/1916). Relativity: The Special and The General Theory. New York: Random House.
2. Feynman, R. P. (1988). What Do You Care What Other People Think? New York: W W Norton.
3. Feynman, R. P. (1995). Lectures on gravitation (B. Hatfield, Ed.). Reading, MA: Addison-Wesley.
4. 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.
5. Feynman, R. P., Leighton, R. B., & Sands, M. L. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.
6. Landau, L. D., & Lifshitz, E. M. (1976). Mechanics (3rd ed.). Oxford: Pergamon Press.
7. Mach, E. (1888/1989). The Science of Mechanics - A Critical and Historical Account of its Development. La Salle: Open Court.
8. Newton, I. (1687/1995). The Principia. Translated by Andrew Motte. New York: Prometheus.
9. Poincaré, H. (1952). Science and hypothesis. Mineola, NY: Dover.
10. Williams, H. T. (1999). Semantics in teaching introductory physics. American Journal of Physics, 67(8), 670–680. 

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.