User talk:Johnjbarton/sandbox/introduction to eigenstates

Comments

@Johnjbarton: Thanks for trying to fix that problematic article. I only have some short comments on the draft.

I think the introduction should be longer, much more concrete, and define with rough precision what an eigenstate is. I think this definition by Merriam-Webster has some merit:

Eigenstate: "a state of a quantized dynamic system (such as an atom, molecule, or crystal) in which one of the variables defining the state (such as energy or angular momentum) has a determinate fixed value".

Perhaps start with something similar (but hopefully with less parentheses)? I would also suggest combining the section with the title "Eigenstates cannot be seen" to the introduction, since that gives the connection to the measurements which is something that should be established already in the introduction.

Eigenvalues could, in my opinion, be introduced already in the section "Eigenstates from measurements" to connect them better with the results of the measurement.

Each section title should not repeat the word "eigenstate".

About the origin of the word Eigenstate. Do you have source for the claim that it originates from Schrödinger? The article Eigenvalues and eigenvectors#History attributes the Eigen-words to Hilbert instead.

Jähmefyysikko (talk) 11:14, 26 June 2023 (UTC)[reply]

You appear to have misquoted Penrose in you apparent eagerness to apply a "relatable" interpretation. I can understand Penrose; I can't figure out what this text means. I'd suggest avoiding the term "measurement". Penrose does a decent job of retaining accuracy. —Quondum 13:24, 26 June 2023 (UTC)[reply]

I'm a bit unsure how to apply your comment. I used the Penrose reference for 'Physicists also use the word "eigenstate" as a synonym for "eigenfunction"'. He says on page 506: "Eigenfunctions (or eigenstates, as they are frequently called in quantum mechanics),..." Can you help me understand how that is a misquote?

The complexity here is that "eigenstate" refers to a state of a physical system which can be either in the lab or in math. I don't recall ever hearing a physical system in lab called an "eigenfunction/eigenvector": those words are used exclusively for math. Regrettably there is no word used exclusively for quantum things in a lab: the word is "eigenstate". Unfortunately I have no reference for this common knowledge so I used Penrose as best I could.

Since you also suggest avoiding "measurement", I will read between the lines: you believe that eigenstate only means 'eigenvector/function', a mathematical construct relating to linear spaces? So eigenstates are just math. I do think that is a common (mis)understanding. However, eigenstates are the result of measurements and measurements give eigenstates. That relationship is true in the lab and in the math. An article avoiding the term "measurement" would just have to substitute another word. Can you help me understand why you think it should be avoided?

For example the Merriam-Webster quote in the comment of @Jähmefyysikko says "one of the variables...has a determinate fixed value". How does this value become fixed? By measuring it.

Thanks Johnjbarton (talk) 16:25, 26 June 2023 (UTC)[reply]

The placement of the Penrose reference suggests that it is being used for the whole sentence or the latter half of it. I would say it belongs at the comma.

I terms of avoiding the term "measurement", it is a mystical term bandied about in quantum mechanics, heavily loaded with conflicting interpretations (pun intended). There is nothing in this article that clarifies this. An eigenstate is not a measurement, but the relationship is not clear. Also, what is the eigenstate of? Of a particular operator: you cannot just say "The waveform is in an eigenstate". If I pass a photon through a polarizor, the photons that are transmitted could be in an eigenstate of the specific polarizor, I guess, and by your definition might be said to have been measured, but in fact we have not made any measurement in the classical sense: the photon is propagating or not, but we have no information about it until we measure this using a sensor. So in an introductory article without fleshing out what is meant by "measurement" in some detail, its use is likely just confusing to the intended reader. And on your interpretation of "eigenstate", I'm not convinced that it does not just have the one meaning; even in the lab, it seems to refer to quantum states that have a particular relationship in relationship to a linear operator. "Measurements" that produce, say, only vertically polarized photon have left the (transmitted) photons into an eigenstate (of the polarizer) in the mathematical sense, and I think that is exactly what is generally meant by the term in every case. —Quondum 16:56, 26 June 2023 (UTC)[reply]

I agree that "measurement" is often used in a broad and vague way. That is why I tried to include concrete examples in each of the sections here. I will work on some definition of measurement. Your polarizer clearly measures the polarization of the light. All quantum measurements are filtration so if I can get that in a definition/explanation maybe that will help. So yes, "measurement" alone is confusing because it conjures up a tape measure and a stopwatch rather than a sieve.

I don't know how much more clearly I can connect eigenstate and measurement than what I say: "A quantum measurement produces an eigenstate, but more correctly, particular measurements produce particular eigenstates." As far as I can tell that is the same thing you conclude at the end of your comment.

Let me work on measurement and see if that helps. Johnjbarton (talk) 18:43, 26 June 2023 (UTC)[reply]

Yes, it is not easy to put things to fit this context (introductory, quantum mechanics). You do actually give quite a bit of explanatory detail. It is also not important to get it "perfect", and mine is but one (possibly overhasty) opinion. —Quondum 18:54, 26 June 2023 (UTC)[reply]

Please take a fresh look. I added a section to outline quantum measurements. Thanks for you comments. Johnjbarton (talk) 01:31, 28 June 2023 (UTC)[reply]

Eigenvector is definitely used to refer to the quantum state of a physical system. See, for example, p. 221 of Cohen-Tannoudji, Diu and Laloë, volume 1. Eigenfunction can be used that way too [1]. I don't think physicists really draw a distinction between these terms, although (speculating wildly) there might be a slight bias in favor of -function for infinite-dimensional Hilbert spaces and -vector for finite-dimensional ones. Shankar seems to use eigenstate and eigenvector interchangeably; Sakurai appears to use mostly eigenstate but also occasionally eigenvector with no discernible difference in meaning. The Dictionary of Material Science and High Energy Physics just says that eigenvector is "equivalent to eigenstate" [2]. XOR'easter (talk) 22:29, 26 June 2023 (UTC)[reply]

Yes, I would agree with your assessment about -function and -vector. These endings are mathematical. Eigenstate stands a bit apart because "state" is more physical. In my experience theoreticians will call anything an eigenfunction/vector, but experimentalists would not refer to something they created in lab as an "eigenvector". Of course I have no reference for this claim. However, Messiah who has one of the best books discussing empirical evidence for quantum states uses eigenstate when discussing experimental results.

But I don't think these distinctions are something we can draw out, let alone in an intro article.

I'm almost tempted to not mention eigenvector/function; but I think with comments like the one I referenced in Penrose a reader may wonder and then be left with the very much more complex articles as resources. Johnjbarton (talk) 23:07, 26 June 2023 (UTC)[reply]

Ok I tried to give a definition up front. I'm a bit concerned that the result is a barrage of words possibly unknown to the reader. That is the reason for the parenthesis in Merriam-Webster. I tried defining the words after, inverted pyramid-ish. Let me know what you think.

I removed the repetition of "eigenstate" but I really want to avoid having a section on, say, "Uncertainty principle" that drifts off the relation to eigenstate. That was the problem with the previous article.

I'll work on eigenvalues. The correct relation to eigenstate is less clear than popular articles make out.

Thanks for the correction on "eigen". I'll clean up the paragraph when I work on eigenvalue. Johnjbarton (talk) 18:30, 26 June 2023 (UTC)[reply]

Its definitely better. What do you think, could the somewhat vague word 'property' be replaced by quantum number? This word usually comes up in elementary expotitions that use hydrogen atom as a model system. And as the prototypical atom model it is often discussed in high school or similar level. Jähmefyysikko (talk) 14:55, 27 June 2023 (UTC)[reply]

I made some changes around 'property', brought up eigenvalues in a new section on theory point of view, and I added a section on quantum number. Let me know what you think. Johnjbarton (talk) 01:33, 28 June 2023 (UTC)[reply]