Talk:Quantum state

"Quantum" but not "state".

The lede is pretty good here, but all of the rest of the article fall strong on "quantum". There is no beginner level discussion of "state" and no link to address this question either.

The disambiguation page has "State, a complete description of a system in classical mechanics," which as exactly zero content on "state". There is Phase space and it has pretty pictures, but again nothing at a beginner level for "state".

Since almost 100% of QM discussions are about "state", it's easy to overlook the fundamental strangeness of the concept for beginners. I wonder if anyone has any good resources to use as references to write more about "state"? Johnjbarton (talk) 16:28, 14 June 2023 (UTC)

I hijacked the lead to address this issue. Please review.

The hijacking did some damage to the pure/mixed content which was already a muddle. I'll take that up next. Johnjbarton (talk) 03:25, 16 June 2023 (UTC)

I ended the lead with a summary paragraph that serves as an outline and thus as a plan for revising the article. The concept that emerged was to use this article as an overview of concepts and as gateway, a summary for the many other pages discussing "quantum state". I will proceed along that plan until someone stops me ;-) Johnjbarton (talk) 14:49, 16 June 2023 (UTC)

I reorganized along the lines I outlined, with major sections "Wave function" and "Formalism" then pure and mixed underneath. Some content in the wave function part still needs to move to Formalism.

Broadly I hope the Wave function part will be more approachable matching how QM is mostly taught. I'm leaning pretty heavy on Messiah's classic which might not be everyone's favorite but reliable.

I feel like I have cleaned up the damage I did earlier; I believe the material that was in the article previous is in as good a shape as it was and now it sits in a better context. Suggestions and comments welcome. Johnjbarton (talk) 17:30, 16 June 2023 (UTC)

photon travel

The current revision has this (unreferenced) sentence in the section on superposition:

The photon state is a superposition of two different states, one corresponding to the photon travel through the left slit, and the other corresponding to travel through the right slit.

As is typical in many cases where "photon" appears, this description misleads readers to imagine tiny balls flying through slits. "travel" implies time and the interference effect in the double-slit experiment is not based on time. Or tiny balls.

I will correct this and try to re-point the content towards addressing "quantum state". Johnjbarton (talk) 23:55, 17 June 2023 (UTC)

Pure states

@Roffaduft I reverted a couple of edits related to pure states. You may have a valid point but I don't think changes help the article. If you have a reference maybe we can agree on a better solution. Johnjbarton (talk) 17:33, 8 June 2024 (UTC)

Regarding pure states, please read the first answer on this stackexchange page. It’s explained in full the point I was trying to get across Roffaduft (talk) 18:57, 8 June 2024 (UTC)

Much of the content on stackexchange is excellent, but it is often obscure and difficult to confirm with reliable sources. The page you point illustrates this point. The issue of pure states is a side comment to a discussion on another topic. I don't think it help us.

I think the "Formalism in quantum physics" section of the article is poorly presented. I almost deleted several sections with zero refs. The article is confused on "pure states", referring to them in different ways, mostly unsourced. I put the Messiah ref in to at least give one ref that discusses the topic. Johnjbarton (talk) 19:24, 8 June 2024 (UTC)

I do agree that the section “Formalism..” could do with a good cleanup. Mixed states are discussed in two different subsections throughout the article for example, which makes it a bit messy overall.

I personally prefer to have a clear “mathematical definition” and “quantum physical definition” when it comes to topics in quantum mechanics. That is because quite ofter there is a lot of ambiguous use of terminology (something we’ve discussed earlier on the topic of Wave function collapse).

Regardless, all I wanted to do with the clarification was to emphasize the mathematical nuance between pure states and bound states.

If you feel it is too technical, I’m ok with removing the subsection “pure states vs bound states” all together and just include the notion of pure states being bound states in the subsection “eigenstates” Roffaduft (talk) 19:46, 8 June 2024 (UTC)

@Tercer instead of making a rude remark, you could perhaps point out where on page 204 of Messiah it is stated that “Any state that is not pure is called a mixed state”

I couldn’t find it. The reason I subsequently removed the claim is because the terminology can become quite ambigous, i.e., use of “eigenstates”, “states”, “pure states” etc.

The phrasing would suggest that any eigenstate that is not a pure state is automatically a mixed state for example, which doesn’t make much sense Roffaduft (talk) 19:36, 8 June 2024 (UTC)

I'm not going to waste my time leafing through references to find an exact quote to support the blinding obvious. The quote is true by definition.

As for "eigenstate", the author must have meant any state (pure or mixed) belonging to the (possibly degenerate) eigenspace of the measured eigenvalue. So yeah, any eigenstate which is not a pure state is automatically a mixed state. Tercer (talk) 19:55, 8 June 2024 (UTC)

so the eigenstate of the position operator; the dirac delta function, is a mixed state then? As it is clearly not in $L^{2}$, not bounded and hence not a pure state. Roffaduft (talk) 20:04, 8 June 2024 (UTC)

It is my understanding that you cannot answer the question "is the eigenstate of the position operator a mixed state?" A pure state requires all compatible observables to be eigenvalues. Pure states are common in theory and rare in experiment. The section "Eigenstates and pure states" was attempting to get that across so if there is some way to clarify that would be great.

Many presentations of QM delay pure/mixed states discussions, I suppose for these reasons of confusion. They are something of a different axis than issues about eigenstates/values. For example you can discuss energy levels and transitions with just eigenstates. To do any comparison with experiment or discuss statistics you have to move on to pure/mixed states. Johnjbarton (talk) 21:20, 8 June 2024 (UTC)

Just to be clear, it's not that I don't know what a mixed state is. The only issue I had is with the phrasing of the sentence: "Any state that is not pure is called a mixed state" in addition to this claim not being mentionned in Messiah p.204. The phrasing is just a bit too strong and context dependent to my liking.

For example, I prefer the definition used in "Hall, Brian C. (2013). Quantum theory for mathematicians"

• A density matrix ρ ∈ B(H) is a pure state if there exists a unit vector ψ ∈ H such that ρ is equal to the orthogonal projection onto the span of ψ. The density matrix ρ is called a mixed state if no such unit vector ψ exists.

It just leaves a lot less room for ambiguity.

The reason I reverted your reversion was because you summarized it as: "it has a references and page number". Yeah, I know. I read the reference, which is what led me to making the edit in the first place.

The example of the position operator eigenstate was just in response to the rather rude and dismissive remarks by @Tercer Roffaduft (talk) 04:24, 9 June 2024 (UTC)

The eigenvectors of the position operator are not normalizable, and hence they are not states at all. Not eigenstates, not pure states, and not mixed states. Tercer (talk) 22:28, 8 June 2024 (UTC)

This is just nonsense. The position operator has an eigenfunction: the Dirac delta, which is also referred to as a (generalized) eigenstate.

It sounds like "leafing throough references" would not be such a waste of your time at all. Roffaduft (talk) 03:49, 9 June 2024 (UTC)

It's not my job to teach you quantum mechanics. If you want to wallow in your ignorance, be my guest. Don't try to put it in Wikipedia, though. Tercer (talk) 06:43, 9 June 2024 (UTC)

Thank god it's not your job indeed. Given that you've contributed absolutely nothing to this discussion other than being very rude and dismissive, I don't feel I am missing out on much.

As you have no intention to engage in polite conversation, I am not going to waste my time continuing this discussion with you. Turning a talk page into a toxic environment doesn't benefit anyone.

I wish you all the best. Roffaduft (talk) 07:33, 9 June 2024 (UTC)

I think the misunderstanding here is whether you regard operators and their eigenvalues and eigenfunctions (i.e. eigenstates) as purely mathematical constructs or subject them to the axioms of quantum mechanics first.

Roughly speaking, I consider QM as an application of more general mathematical theory, with QM borrowing a lot of terminology from mathematics. That is perfectly fine, but a consequence is that certain words have a much more restrictive meaning in QM than they do in mathematics.

A strong claim like: "Any state that is not pure is called a mixed state" might make complete sense from a QM point of view, but it may also lead to ambiguity about the definition of a "state" down the line.

I used the eigenfunction of the position operator as an example, where the physicist has to jump through hoops in order to explain why the dirac delta function is "somewhat like" a normalizable eigenvector in the "continuous sense" Roffaduft (talk) 06:48, 9 June 2024 (UTC)

Perhaps we need to change the section "Eigenstates and pure states" to "Eigenstates and mixed states". The section needs to help readers relate eigenstates, the result of a measurement, into the overall "quantum state" topic. Pure states are not directly needed for that goal. Johnjbarton (talk) 15:51, 9 June 2024 (UTC)

At this point I’m honestly fine with whatever. My initial edit was primarily based on the reference not backing the “strong claim” (mathematically speaking) and only secondly the on “strong claim” itself. An extra reference has been added which resolves most of the issue I had. That is, if one wants extra context, one can look up the reference. Roffaduft (talk) 16:48, 9 June 2024 (UTC)

Superposition section.

@PyetroPy edited the Superposition section, then I tried again. The questionable section was I think trying to do too much. My version uses more math. Alternatively we could just state that the overall phase is ambiguous and leave it at that. Johnjbarton (talk) 23:21, 4 September 2024 (UTC)

@Johnjbarton I like the final version. In my comment I was just trying to link the two previous concepts together (the multiplication with complex number and the fact that only the relative phases count) and to summarize, sorry if it sounded dismissive. PyetroPy (talk) 22:14, 7 September 2024 (UTC)