Talk:Correspondence principle/Archive 1

Llamas
"macroscopic systems (springs, capacitors, llamas, and so forth)"

Surely llamas is a jarring and confusing example to use in this instance, if appropriate at all?

My first reaction was to believe it vandalism but it's been in since revision 1. --Air 13:02, 2 Nov 2004 (UTC)

Llamas are macroscopic systems, no? -- CYD


 * first, I appreciate the Monty Python element of the example. But the progression mechanical component, electrical component, wool-bearing quadraped makes it unclear what's being illustrated : )
 * is it better to instead say (springs, capacitors, trees, clouds, llamas, and so forth) to indicate we mean 'just about everything above the atomic level'? --Air 11:38, 3 Nov 2004 (UTC)

Rbj, I appreciate your taking the time to add the SR example, but it's unfortunately irrelevant. AFAIK, the correspondence principle is always used to refer to quantum mechanics reducing to classical (non-quantum) mechanics. The reason we care about this is that there's nothing in the postulates of quantum theory that says it should reduce to classical mechanics; you have to toss in the correspondence principle if you want to construct a physically correct quantum theory. While it is true that relativistic mechanics reduces to classical mechanics at low velocities, this happens automatically; you don't have to do any fiddling as you do with quantum theories. And it's certainly not referred to as the "correspondence principle". -- CYD


 * I believe that you are mistaken when you state " ... the correspondence principle is always used to refer to quantum mechanics  ". Despite its origin from Bohr, the definition of the correspondence principle has broader context.  This definition easily found when Googling pretty much agrees with my textbooks:


 * Correspondence Principle: any new theory, whatever its character--or details--should reduce to the well-established theory to which it corresponds when the new theory is applied to the circumstances for which the less general theory is known to hold. This principle was first applied to the theory of atomic structure by Niels Bohr in l923. (Weidner and Sells, l960, p. 29) The principle can be applied to great advantage in relativity theory and in quantum mechanics.


 * here's another from Encyclopedia Britannica:


 * Correspondence Principle: philosophical guideline for the selection of new theories in physical science, requiring that they explain all the phenomena for which a preceding theory was valid.


 * SR is a more modern theory than Newtonian Mechanics (and is GR) that has more accuracy than Newtonian mechanics as speeds get very large. When applying relativisitc theory to situations where speeds are lower (or space-time is flatter) should and does break down to the familiar Newtonian mechanics or gravitational model. That is the application of the correspondence principle and someday it will used to apply string theory (or something else as new) to default to traditional quantum mechanics or that new theory will fall by the wayside.  If what you say is true, why have that quote from Einstein at the bottom of the article?  Einstein didn't do quantum mechanics, why would his quote be relevant unless it applied more broadly than quantum mechanics? r b-j 19:22, 26 Nov 2004 (UTC)


 * Actually, for whatever it's worth, the majority of Google results points to the quantum mechanical "version" of the correspondence principle. If the Encyclopaedia Britannica uses the "broader context" definition, then it is wrong. It's mind-bogglingly obvious that newer theories of physics must reduce to older theories in the domains where the older theories agree well with experiment -- this is just saying that theories must agree with experiment! Thus, Newtonian dynamics reduces to statics in the limit of zero speeds, electromagnetism reduces to electricity and magnetism at low frequencies, statistical mechanics reduces to thermodynamics at large particle numbers, SR reduces to classical mechanics at low velocities, GR reduces to SR and Newtonian gravity for small spacetime curvatures, etc.; all this was known long before Bohr enunciated the correspondence principle in 1923.


 * The true significance of Bohr's remark lies in its application to quantum mechanics. In the early days of quantum mechanics, it seemed entirely possible that quantum theory applies only to the microscopic domain, i.e. individual atoms and molecules. This was especially the case before the invention of modern quantum mechanics by Heisenbery and Schrodinger, when quantum theories were constructed on a phenomenological basis (the most famous example being Bohr's atom). Bohr's real and decidedly non-obvious contribution was to state that these microscopic theories must match classical physics if, somehow, one can reach the limit of large quantum numbers -- e.g. if one can make an atom the size of a room! It is interesting to note that even today, no one has been able to satisfactorily show how quantum mechanics reduces to the classical world in a general context. By contrast, the way SR reduces to classical mechanics is obvious and completely "built-in" to the theory. -- CYD


 * It may be "mind-bogglingly obvious that newer theories of physics must reduce to older theories in the domains where the older theories agree well with experiment", but there have been new theories that fell by the wayside because they didn't pass that test. Before quantum mechanics could be completely accepted, there had to be some experiments (or thought experiments) that, after computing the expectation values, shown argeement, at least in perception, between the old and the new. That philosphical test should have a name and it is silly that some would weaken the language by giving different names to the same principle depending on where that principle is applied.


 * "it seemed entirely possible that quantum theory applies only to the microscopic domain", you could make the same claim regarding either general relativity or special relativity (that it only applies to high speeds or extremely strong gravitational fields). until someone (maybe the theory's developer) checks to see that the new agrees with the old where the old was known to be valid, then the theory is suspect.  For some reason, i suspect that there was a (natural) change in usage and sematic regarding this, extending the concept to all new theories and only some of the physics community have found out about it.  You still haven't answered my question regarding Einstein's quote.  What was Einstein talking about, when he said it?  Quantum mechanics?  Relativity?   I am still investigating this. r b-j 22:57, 29 Nov 2004 (UTC)


 * I have no idea where the Einstein quote came from, but notice that he does not use the term "correspondence principle" in it. Feel free to remove the quote from the article if you think that it is irrelevant.


 * I don't think it's irrelevant but it supports the notion that the correspondence principle (or something with a different name that I am still asking what that name would be) is a broader notion than just one that shows that quantum mechanics is not inconsistent with Newtonian mechanics (for large masses).


 * The point that you are steadfastly ignoring is that the way quantum mechanics reduces to classical physics is fundamentally different from the way that, e.g., SR reduces. The "philosphical test" that you describe is a trivial matter for SR and other theories, but a complicated (and still unsolved) matter for quantum mechanics! The point is that quantum theory is not so much a theory as a class of theories, and it is entirely possible to construct quantum mechanical models that have no classical limit. The simplest example of a useful quantum theory that does not possess a classical limit -- and to which the correspondence principle does not apply -- is spin. -- CYD


 * Nearly all of this is true (however there are models in SR and certainly GR that have no counterpart in Newtonian mechanics) but does not change the point. All modern theories, if they are to be accepted as "true" in some scientific sense, have to be able to agree with the older accepted theories in the domain where those theories were shown to be valid.  If that is not the case a hypothisized theory will eventually fail to gain acceptance.  Frankly, I have always been taught, that at least in physics, this was called the "correspondence principle".  To limit that semantically to only one area of science (or physics) is etimologically (sp??) not economical.  Applying the term, with the same meaning of the term, to other areas, does not change its meaning for the original area.  However restricting it artificially (or unnecessarily), does change its meaning. r b-j 17:47, 30 Nov 2004 (UTC)


 * It is even more economical to adopt the principle that theories have to agree with experiment. If old theories agree with experiment in certain domains, this would imply that new theories have to agree with the old theories in those domains, obviating the need for a "correspondence principle" in the sense you describe. Frankly, I've never come across the term except in the quantum mechanical context. String theorists, for example, never refer to the limit in which string theory is supposed to reduce to Standard Model physics as the "correspondence limit". The term classical limit is often used to describe the regime in which both quantum mechanics and relativity reduce to classical physics, and the term thermodynamic limit is used to describe the regime of large particle numbers in statistical physics, but that's about it. -- CYD

I've read all the above comments - quite revealing. The fact is that the term 'correspondence principle' means (at least) 2 things: firstly, the exclusively quantum mechanical one that CYD is referring to and the more general one referring to the reduction of scientific theories to less accurate ones (albeit possibly precursors to the more accurate one). One cannot argue that just because the term arose from quantum mechanics, then it must only be used in that context. The point is that this is an encyclopedia and both meanings must be accommodated. If that means creating another article (and a disambiguation page) then so be it, but for the time being I've added a new section: 'Other uses of the term'.

Ehrenfest's theorem
Shouldn't Ehrenfest's theorem be mentioned somewhere? My understanding is that the correspondence principle relies heavily on Ehrenfest's theorem.--Michael C Price 11:53, 13 June 2006 (UTC)

Einstein quote misleading
I'm not sure what the purpose of the Einstein quote at the bottom is. The importance of the correspondence principle in respects to Einstein is not that he accepted that correspondence between QM and classical values was necessary, but because he did not think it was sufficient. Einstein thought that it was obvious that any QM result had to accord with experiment at a bare minimum, but unlike Bohr he did not think that should be the only limiting factor on a new theory — he believed that it should correspond with certain axiomatic and generalized principles, i.e. that it should make physical sense (Bohr considered this to be too limiting a constraint).

I think the invocation of Einstein here is a bit misleading both about Einstein's own opinions, to say the least, but also because nowhere in this article is it reflected why the correspondence principle was considered a radical methodological approach in its time. It is not because of what the principle says, but what it does not say. --Fastfission 00:17, 21 September 2006 (UTC)


 * I agree, I don't see the relevance of the quote. --Michael C. Price talk 06:30, 21 September 2006 (UTC)

Formulas (Bohr atom)
I think it would be more helpful to substitute all those "\propto" with some real formula. What I mean is that someone could read this article to understand where does the "L=\hbar n" formula came out from, and could be a little bothered reading "If you keep track of the constants, the spacing is \hbar". Of course the page would look a little bit more complecated. In alternative, all this stuff could be put in the "Bohr Model" page.

Here I write the "missing" steps (hoping to have "filled in the gaps" correctly...):

Being $$\scriptstyle{m{4\pi^2\over T^2}r={ke^2 \over r^2}}$$ and $$\scriptstyle|E|={ke^2\over 2r}$$ for large values of $$n$$ we have

\Delta E_n= { h\over T(E_n) } = {\hbar {\sqrt{ke^2\over m}} r^{-3/2}_n } = {{\hbar \over {ke^2}} \sqrt{8\over m} E_n^{3/2}} $$

Being also $$\scriptstyle{mv^2 \over r}={ke^2\over r^2} $$

L^2=(mvr)^2=mke^2r={m(ke^2)^2\over 2E} $$

And supposing $$L_{n+1}-L_n=\Delta L$$, for large values of $$n$$, we have

\Delta E_n = {m(ke^2)^2\over 2}\left( { {1\over L^2_n}-{1\over L^2_{n+1} } }\right)\approx {m(ke^2)^2{\Delta L \over L^3}}={{\Delta L \over {ke^2}} \sqrt{8\over m} E_n^{3/2}} $$

So, putting everything together again, we find:

\Delta E_n={{\hbar \over {ke^2}} \sqrt{8\over m} E_n^{3/2}}={{\Delta L \over {ke^2}} \sqrt{8\over m} E_n^{3/2}}\Rightarrow \Delta L=\hbar\Rightarrow L_n=n\hbar $$ Secondopremio (talk) 11:57, 1 February 2008 (UTC)


 * I didn't do this explicitly for two reasons: 1. it's really good for the reader to check for themselves, as you did above, and this introduces a lot of extraneous symbology which is hard for anyone except the checker to keep straight. 2. because the discussion goes on to introduce action-angle variables, which solve the correspondence condition for all motions. With action-angle, you can see that 2\pi L=nh is the right condition without any lengthy calculation. This gives a conceptual explaination for why all the constants just "fell out" like that leaving a simple result in terms of the angular momentum.Likebox (talk) 14:00, 2 February 2008 (UTC)

Ptolmey's Theory
I used this example as a contrast to Aristotle's physics. Aristotle's physics was a bunch of philosophical wankery, so it could be totally ignored. But Ptolmey actually had a theory--- he introduced the equant, and this was very important--- it meant that the planetary motion is nonuniform, and that the orbits are not centered on the sun, but on a mathematical point displaced from the sun. In heliocentric language, reproducing Ptolmey meant that the sun needed to be off center. So Copernicus and Kepler had to reproduce Ptolmey, but Newton did not have to reproduce Aristotle. This is a big difference, and I thought it would be nice to highlight.Likebox (talk) 20:10, 4 February 2008 (UTC)

Relativity
In the first section above (strangely entitled Llamas), there's a discussion on whether or not the correspondence principle has a more general secondary meaning (in addition to the specific quantum theory sense) in which it refers to the requirement that ANY new theory should reproduce the results of the old theory in the realm in which the old theory is known to be successful.

Currently, we have a self-contradictory page. The intro here refers exclusively to quantum theory; but later we have an example from special relativity - an example of this general sense. The disambiguation page indicates that both senses are valid.

The discussion above doesn't seem to come to a conclusion. And it seems to date from 2004. I don't want to jump straight in and alter things without discussion, as I've not been involved with this page, so perhaps someone could clear this up - with citations - if there are clear views on the matter? If not, I'll just copy the text from the disambig page into the introduction, as - whether correctly or not - both senses are certainly in use to some extent.

(The quantum sense is clearly the primary and most important use of the term, and should be preserved as such.)

Comments welcome. Bobathon (talk) 11:43, 28 May 2009 (UTC)


 * This is an interesting question. The notion of correspondence has different meanings in different disciplines. Let us first restrict ourselves to physics, although even there the notion does not have an unambiguous meaning. Within physics the correspondence principle has its origin in quantum physics before the advent of quantum mechanics. In the Old quantum theory it served as a kind of reassurance that the quantum rules of that theory (added to classical mechanics) did not have any influence on its predictions within the domain of application of classical mechanics (i.e. reproduced the results of the latter theory if applied to macroscopic objects). During the period 1900-1925 the correspondence principle can be seen either as a relation between theories (viz. the Old quantum theory and classical mechanics), or as a relation between a theory (viz. the Old quantum theory) and physics (viz. the physics described by conventional classical mechanics). With the advent in 1925 of quantum mechanics the correspondence principle obtained similar meanings with respect to the relation between quantum mechanics and classical mechanics c.q. the physics described by classical mechanics. In this sense the principle has been applied also to relativity theory.


 * In the philosophical literature, especially in work by empiricist philosophers, correspondence has a rather different meaning, viz. a relation between terms of a theory and physical entities the terms refer to. Contrary to the above-mentioned application of correspondence it is not a relation between theories. Correspondence is meant to give a physical meaning to a theoretical term of the theory. It should be valid on the whole domain of application of the theory, not just on a part of it (on which a previous theory might be applicable).


 * Within Bohr's views the physical and philosophical meanings of correspondence have been amalgamated. Bohr maintained the relation with classical mechanics by requiring that measurement arrangements and results of measurement should be described classically. On the other hand, he adopted also the philosophical notion of correspondence by requiring it to define what is a quantum mechanical observable. However, reference to classical mechanics is virtually absent in Heisenberg's approach and that of most other physicists, and has died out completely after the formalism of quantum mechanics had been firmly established. Whether Bohr's idea of correspondence with a measurement arrangement as definition of a quantum mechanical observable should be maintained without relying on classical mechanics, is a matter of controversy (e.g. http://www.phys.tue.nl/ktn/Wim/muynck.htm).WMdeMuynck (talk) 13:28, 22 August 2009 (UTC)


 * Thanks - interesting! Bobathon (talk) 11:59, 24 August 2009 (UTC)

I have never seen a section on correspondence principle in a relativity text book. I believe all uses of the phrase "correspondence principle" has been only for relating quantum mechanics to classical mechanics. The same reason I believe, is why people group classical mechanics and relativity together, separate from quantum mechanics. The relationship between relativity and classical mechanics is much clearer than between QM and classical mechanics - in the former a Taylor expansion is all that is required to show the two are related, whereas in the latter usually different scenarios have to be posed, e.g. WKB in slow varying potential, high n = classical energies, etc, that are much less general. I think the relativity example is confusing and should be removed in this article. —Preceding unsigned comment added by 134.174.140.104 (talk) 21:57, 16 November 2010 (UTC)

Aristotle's mechanics, although academically viable for 18 centuries, do not have any domain of validity
Is this true? Take for example the idea that things don't move unless you push them. To some order, a nice rough block of stone sitting on a rough surface fits William M. Connolley (talk) 09:31, 13 October 2012 (UTC)

Reduction
Reduction is asymmetric: 'x reduces to y' does not mean the same as 'y reduces to x'. More than once in the article and in Talk, it seems to me, writers have reversed what they intend. 'Quantum mechanics reduces to classical mechanics': No, the special and logically derived case reduces to the general and fundamental case. Aubrey Bardo (talk) 12:32, 25 May 2022 (UTC)
 * You are using "reduce" in a recondite and technical term barely understood by non-logicians. Throughout in science, and in this article, "reduces" means "collapses by losing information and properties"; this is the standard, and intended, usage. So a quantum theory, like other, different theories, reduces to a common classical limit upon loss of its quantum information, like sea-water reduces to salt upon boiling.Cuzkatzimhut (talk) 18:29, 25 May 2022 (UTC)

Criticism to the correspondence principle
In this book we are told how Einstein, Heisenberg and Pauli dismissed the correspondence principle, specially  after the creation of matrix mechanics. This article sees the principle in a positive way, what's up with that? ReyHahn (talk) 00:06, 7 March 2024 (UTC)


 * I think we should be a bit cautious with
 * Beller, M. (1999). Quantum dialogue: The making of a revolution. University of Chicago Press.
 * From the reviews on
 * '...an excellent, though controversial, book."
 * "Of course, one may not agree with all the conclusions. "
 * And one in Science by Daniel Greenberger
 * " I think Beller is guilty of trying to force the worst possible conclusions out of very slight and ambiguous evidence."
 * Johnjbarton (talk) 03:17, 12 March 2024 (UTC)

Oh I see, there are three different correspondence principles: Bohr's, the naive one, and the generalized principle used here. This article should be reworked to include that. See .--ReyHahn (talk) 00:15, 7 March 2024 (UTC)


 * I think you are on the right track. As I understand it, the principle worked ok in the Old quantum era, but fell out when two level quantum spin projection showed no continuum limit.
 * I think there is an important issue: the Bohr era intended the principle to be a form of boundary condition to fix values in theories. As I understand it, all the later uses were more like interpretation analysis, trying to show that certain quantum parameters relate to certain classical parameters when comparing extreme values. The "principle" part only really applies to the former.  On this count most of the article is off base, esp. the generalized bit. Johnjbarton (talk) 16:56, 7 March 2024 (UTC)


 * So far I think the article should start stating the generalized principle which is the important one and what most people care for. Then the modern quantum mechanics principle with all its caveats. And finally Bohr's old quantum theory one as an undercover history section. Nobody should go through Bohr first to get the overall idea. What do you think of the examples?--ReyHahn (talk) 21:38, 7 March 2024 (UTC)
 * Sorry, but I disagree. The section here on generalized principle is just an unreferenced rehash of classical limit. It is not related to the correspondence principle.
 * If we look at Pais Inward Bound pg 247 he describes the Balmer formula with n>>1 as used by Bohr to determine the Rydberg constant. Pais: "This correspondence has predictive power; it determines R.". On 498 he discusses Heisenberg's work on QFT for S matrix theory and describes the principle as "what long distance aspects of the theory would survive a change of it short distance properties".
 * I know that Messiah v1 has a chapter on Correspondence principle, but I don't have with me just now. I can find a ref on how spin messed with the Correspondence principle.
 * It looks like the current article follows https://plato.stanford.edu/entries/bohr-correspondence/ which has lot of references. My synopsis would be: despite some early success, Bohr's correspondence principle did not pan out. A lot of historians have tried to make it into something, but the physics does not back it up. I suppose we could report on specific references that mutate the concept into something about the classical limit. Johnjbarton (talk) 22:35, 7 March 2024 (UTC)
 * Part of the current article was just me trying to make it look more like the Standford entry. If I am reading you right do you prefer to start the article with Bohr's account? I think Bohr's version is very old quantum like and no longer useful. A modern version was reworked after new quantum theory but it is far from being perfect.--ReyHahn (talk) 02:32, 8 March 2024 (UTC)
 * Bohr invented and famously used the Correspondence principle; he literally wrote the book on it. Later other folks came along, could not figure out Bohr so they took up a similar idea and called it "Correspondence principle". Bohr's work should be judged by what it said, not by mis-alignment with an idea of the same name that came along later.
 * My reading of the "Generalized Correspondence Principle" section of the Stanford work is that there are two different things: Bohr's Correspondence Principle and the classical limit correspondence. I think the article could reflect this rather than pick one. (But if one must be picked it has to Bohr's). Failures of each are instructive FWIW.
 * The Stanford article actually has Bohr's reaction to the generalize form:
 * When Rosenfeld off-handedly suggested to Bohr that the correspondence principle was about the asymptotic agreement of quantum and classical predictions, Bohr emphatically protested and replied, “It is not the correspondence argument. The requirement that the quantum theory should go over to the classical description for low modes of frequency, is not at all a principle. It is an obvious requirement for the theory” (Rosenfeld 1973, p. 690).
 * Johnjbarton (talk) 03:08, 8 March 2024 (UTC)
 * Aside from Bohr's principle and generalized principle, I was under the impression that there is an additional modern one. In the Standford source it is under "modern literature". The correspondence principle should have died together with old quantum theory. What we have now are the semiclassical limit where we can recover some of the results of old quantum theory and the result of classical physics. I will try to read on it over the weekend. Cheers.--ReyHahn (talk) 08:45, 8 March 2024 (UTC)

"while referring back to this link"
The current article says: The sentence has two references, a Physics Today article by Liboff which I don't have and Bohr's lecture which is available: The way I read Bohr's lecture he does not refer back to a link. Rather he says: Bohr does not view correspondence as an asymptotic limit to classical theory. Johnjbarton (talk) 23:15, 13 March 2024 (UTC)
 * Bohr introduced the term "correspondence principle" during a lecture in 1920, while referring back to this link between classical and quantum mechanics.
 * https://www.gutenberg.org/files/47464/47464-pdf.pdf
 * "This peculiar relation suggests a general law for the occurrence of transitions between stationary states. Thus we shall assume that even when the quantum numbers are small the possibility of transition between two stationary states is connected with the presence of a certain harmonic component in the motion of the system."
 * I agree. I removed "this link". --ReyHahn (talk) 09:26, 14 March 2024 (UTC)

Classical limit
The article Classical limit is essentially a better version of this without all of Borh's problematic take. I wonder if it would be just better to merge correspondence principle there. If not what would be the best way to distinguish both?--ReyHahn (talk) 10:09, 13 March 2024 (UTC)


 * I would strong oppose merger. Bohr did not win a Nobel prize for a classical limit. In my opinion the correspondence principle can't possibly be considered exclusively a classical limit of quantum theory: during the height of its application there was no quantum theory. There was only the Bohr-Sommerfeld atom and a hypothesis of quantum radiation transitions. The correspondence principle was Bohr's attempt to provide a radiation theory. With the emergence of quantum mechanics and QED, the correspondence principle had no value.
 * If not what would be the best way to distinguish both?
 * I would have a section in each article explaining that application of the correspondence principle sometimes results in relationships that were interpreted as asymptotic relationships between quantum and classical. Post 1926 this was the only use physics had for the correspondence principle so that use got more attention the remaining 100 years. Johnjbarton (talk) 15:55, 13 March 2024 (UTC)
 * I think it's a good idea to have an article called correspondence principle that is mostly about the history (Bohr and immediate reactions). The classical limit article should cover how full-fledged quantum theory is used today. XOR&#39;easter (talk) 16:18, 13 March 2024 (UTC)
 * I think we should use Rosenfeld's report that Bohr was adamant about the principle not being a limit as a transition to a section on how many sources (most modern QM books) use the name for a $$\hbar \rightarrow 0$$ limit. Johnjbarton (talk) 22:56, 13 March 2024 (UTC)
 * Thanks for the input. I am still lost here, (1) do we need the correspondence principle in some form or another for modern quantum mechanics? or can this article be just a history article for Bohr's take? (2) aside from Bohr's account is there any other principle? Should we just throw out the limit description (quatum->classical) from this article? If the answer is no to the latter, then how we define it? --ReyHahn (talk) 11:04, 14 March 2024 (UTC)
 * There are, I believe, three topics that fit into an article called "correspondence principle". First, there is Bohr's view (and the attempts by historians to figure out exactly what his view was). Second, there is the different but well-established practice of using "correspondence principle" to mean an $$\hbar \to 0$$ kind of limit. Third, there is the "generalized" meaning, where people talk about one theory reducing to another. I would organize this article accordingly: one section, probably divided into subsections, about readings of Bohr; another, probably shorter, about asymptotic limits and with a main tag pointing to classical limit; and a third about the "generalized correspondence" notion, which would resemble the section we have now but condensed to be less textbook-ish. XOR&#39;easter (talk) 14:38, 14 March 2024 (UTC)
 * Yes, that is what I was thinking. The generalized section needs much better references. I've never heard of it, but then again I thought it was "common sense" so someone must have written about it. Johnjbarton (talk) 16:09, 14 March 2024 (UTC)
 * I added a small section on classical limit. Needs to be merged with the one following it. My main goals were to connect to Ehrenfest and WKB. Johnjbarton (talk) 22:27, 14 March 2024 (UTC)