# The electron always waves at you

This post has reference to Aatish Bhatia’s post on the subject matter bearing the title: “Hey There Little Electron, Why Won’t You Tell Me Where You Came From?” dated 27th September 2014 and published at the “Empirical Zeal” blog [^].

The post being referred to may immediately be perused.

“… Here’s the setup. On the table in front of me there’s a box with two thin slit-like openings at one end. We’re shooting particles into this box through these slits. I did the experiment with photons, i.e. chunks of light, but others have done it with electrons and […] For convenience, I’m going to call the objects in this experiment electrons but think of that word as a stand-in for any kind of stuff that comes in chunks, really. [bold emphases added] …”

Concerning the differences between electrons and photons, it would appear that it is not essential to look into all details of the proceedings that occurred when the present author defended his PhD thesis in mechanical engineering; however inasmuch as an inclusion of a reference increases the total number of references being cited for this post, the same [^] may perhaps be found in order.

“… And indeed, if you do this experiment with only one slit open, they behave just like baseballs, hitting the wall in a single band behind the open slit. …”

Further research is needed to develop a deeper understanding of the term: “single band.”

“… You can watch the electrons coming in one at a time in this video produced by scientists at Hitachi in 1989. …”

It is with great pleasure that the present author wishes to recall the inclusion of this video clip at the time of his conference presentation; the simulation he presented however was for photons.

“…Did the electron go through the left slit?

No! Because when you cover up the right slit, the stripey pattern disappears and you get a boring single band instead. …”

As has been mentioned above, more research is needed to develop a deeper understanding of the term: “single band.”

“…Did the electron go through the right slit?

No! For the same reason as above. When you cover up the left slit, instead of the stripey pattern you get a single band. …”

However, in the light of the further and deeper study of the reference post, it would appear that when the term “single band” is being used, the meaning being indicated is that of only one band as would be produced by a classical i.e. non-quantum mechanical particle.

“…As MIT professor Allan Adams puts it, that pretty much exhausts all the logical possibilities!…”

The quantitative logical closure contained in the salient reference which has been alluded to in the main reference post would be of great theoretical interest in general.

Therefore, it is one of the intermediate and urgent proposals of the present post to peruse this reference in an expeditious manner.

As Heisenberg and others taught us, although language fails us, it’s possible to come up with rules that correctly predict how tiny things behave. Those rules are quantum mechanics. You can learn these rules for yourself by reading Richard Feynman’s classic book QED…

It is further proposed to also include this reference in the literature review on a priority basis.

Kindly refer to Appendix A for details of the funds and employment opportunities needed. Kindly refer to Appendix B for details of justifications thereof.

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OK. I have had enough of this“research” talk (i.e. write). So let me switch back to writing in my usual blogsome way.

I think that Aatish makes a conceptual mistake here.

The electron always, say, “waves at you.” It does so even when it goes through only a single slit.

What you observe with two (or more) slits is an interference pattern. There of course is no interference pattern when only one slit is kept open. But even in this case, what you observe is not a single classical band (as for a heap of grains), but a diffraction pattern containing many fringes.

Many people, esp. science popularizers, make this error. They present QM as if a single— and by implication, classical—band is observed when one the two slits is closed.

… What could be the source of this error? Where might have it begun?

It was Feynman who highlighted the conceptual importance of the double-slit interference arrangement. He also wrote his text book in a very informal and attractive style. He has enormously influenced some two generations by now, including the pop-sci book writers. May be there was something to the way he presented this material, which misled people?

Check out the informal diagrams Feynman includes in his Lectures, esp. fig. 1.3, part (b) [^].

He draws each of the two single-slit probability curves with only single humps. Look at the curve for P2 more carefully. It gradually decreases in magnitude and becomes zero as you go up (i.e., along the positive x-axis; here the x-axis is taken vertically). However, no hint is given that this P2 curve would then once again increase in magnitude (or go to the right in this diagram) as you continue going further up, and it will thus have another, relatively much smaller peak—and then, an entire infinite series of similar (and progressively more and more faint) peaks.

In diffraction, these other peaks often are almost undetectably faint. For instance, see the two diagrams that appear just above the “Problem” section, here [^]. In fact, in terms of brightness/faintness, the first diagram just above the “Problem” section is only schematic; it depicts the other fringes with much more brightness than actually is the case. The diagram above it (i.e. the graph) is a better representation of the relative magnitudes involved. (Another point: These outer fringes in the single-slit diffraction also happen to be relatively very faint when compared to the intensity of the interference bands which appear when both the slits are kept open.)

However, in the context of the single-slit, Feynman doesn’t explicitly say anything at all about the diffraction phenomenon—either in the text or in the diagram.

People then must have over-interpreted his diagram, and wrongly thought that (i) when an electron goes through a single slit, it behaves exactly similar to how grains falling through a chute behave, and (ii) when both slits are open, the electron somehow begins to behave something like a wave, something like a superposing quantum particle.

Thus, the idea being advanced is: single-slit means classical grain nature (Bhatia uses the example of baseballs); double-slit means in part a wave nature, as in QM. This characterization is wrong.

The faulty interpretation also makes QM sound more mysterious than it actually is.

Since Feynman’s double-slit diagrams anyway were only schematic (observe that the curves for P1 and P2 are manually drawn and therefore they are not precisely symmetrical), he could have shown a bit of the fringing effect in the single-slit too, with the usual note: diagram not drawn to scale. That single feature would have saved a significant error in so many popular expositions.

Feynman himself does explicitly note the fact of appearances of fringes even in the single-slit diffraction, but only in his later QED book—and only by way of a footnote. (Sorry, can’t locate it for you. I don’t have the book ready with me right now—it is still packed in a carton when I moved from Mumbai to Pune after my job-loss in January. But I do remember that it is in the early parts of the book, very probably in chapter i.e. lecture 1.)

Since I have exhausted my ‘net bandwidth for this month, I couldn’t go through the MIT professor Allan Adam’s video that Aatish Bhatia refers to. (It’s more than 1 hour long.) Instead, I checked out his PDF course notes, to see if he too makes this common mistake. … Well, that way, I didn’t actually expect Adam to repeat this mistake, but since Bhatia makes an enthusiastic reference to it, I wanted to check out. The relevant course notes are here, L2 [(.PDF) ^]. As expected, the notes don’t actually commit the mistake, but still, they repeat the same omission (of diffraction). Adam says in his notes (p. 7–8)

Hence, determining through which slit an electron passes does away with the interference pattern.

He could add an explicit mention of the diffraction pattern.

Though both electrons and photons would show a similar behaviour, it should be easiest to demonstrate the diffraction effect using light rather than electrons. My unpublished simulation of the PhD times showed a gradual “morphing” from a full double-slit interference pattern to a full single-slit diffraction pattern, as the detection efficiency of the photon detector placed near only one of the two slits was increased. Very natural. Check out Adam’s maths on p. 6:

$A(y) = A_0 \left( e^{i\theta_1(y)} + e^{i\theta_2(y)} \right)$

Split it up keeping two different terms $A_1$ and $A_2$, even if we assume $A_1 = A_2$:

$A(y) = A_1 e^{i\theta_1(y)} + A_2 e^{i\theta_2(y)}$

Gradually take, say $A_1$, to zero. You gradually transition from interference to diffraction.

Why might Bhatia have made the error? Here is a speculation.

He mentions the experiment with photons, which he did have an opportunity to perform, as an undergraduate student.

It’s possible that the geometrical scaling of the experimental arrangement was such that the outer diffraction fringes got placed outside the detection limits of the CCD camera. It’s also possible that they ran the experiment mainly to bring out the particle nature well, and so, they effectively “expanded” the time axis a great deal (by using a very low flux rate). Now, for the single-slit version, the outer diffraction fringes are very faint, as compared to the big fringe in the middle. Therefore, in the actual experimentation, enough particles might not have been registered at the locations of these outer diffraction fringes, at least over the relatively shorter duration of the experiment.

Eleven years later, as he began sharing the joy and excitement of seeing the quantum nature in action from memory, he might have focused a bit more on the dramatic features of the experimental measurements and forgotten the correct theory that helps explain it. Possible.

Or, it is also possible that this bit simply skipped his attention. Quantum theory actually is a very big theory—it has a very large scope. As a typical university student, your goal is to master its mathematical tools. It’s easily possible that you skip over some of the rudiments rather quickly. It happens to almost everyone. The initially unintuitive nature of QM isn’t the only thing that people get used to; they also keep adding many small details in their understanding, because no one book can possibly cover all such details, applications, etc.

Anyway, he would know best.

… I have already dropped a comment at his blog, mentioning the concepts-wise serious nature of this mistake. It sets people—the newcomers, the layman—on a wrong path. It did that to me, for more than 5 years (in fact almost a decade, perhaps more, if you count the time from the very first exposure to the wave-particle duality, which was sometime in XI or XII standard in my case).

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A Song I Like:

[A very careful study of this song is advisable. There will be examination, once the song is over.]

(Marathi) “hee gulaabee hawaa, veD laavi jeevaa…”
Singer: Vaishali Samant