Talk:Introduction to quantum mechanics
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[edit] Use in Wikibooks?
Perhaps articles like this would have a better place as part of a Wikibook rather than an encyclopedia entry. Fyorl (talk) 09:19, 14 February 2008 (UTC)
[edit] Shimony and "passion at a distance"?
This is in response to a recent edit by Cmeyer1969. You say you "Removed confusing joke and added informative reference." I appreciate the reference to the Westmoreland and Schumacher paper; I will read it. Shimony's idea of "passion at a distance," however poorly it may be named, is not a joke. See Abner Shimony and the reference to Sandu Popescu's essay in this article. If you are under the impression that someone added the sentence you removed as a joke or as vandalism, you are mistaken. By the way, I have no personal stake and I did not write that sentence; someone called for a citation to back up the sentence and I found one. — SWWrightTalk 03:20, 8 March 2007 (UTC)
Hi, thanks for the citation. I did see the paper before I removed the sentences. The sentence before it "though it is possible to use it to increase the probability of success in a conflict situation where a number of allies must collaborate against a joint attack without information on what their common enemy is doing at each of their allies' separate locations (except their own)" is clearly garbage. However, I felt that the comment by Abner Shimony was confusing at best. I think it might be best to refer that entire section to the actual Quantum entanglement page and add the "passion at a distance" information there. What do you think? Feel free to edit the sentence back in if you disagree, with perhaps a bit more explanation as to what it is. Cmeyer1969
Cmeyer1969, thank you for responding. I may do so, but first I have to study his writings some more. While I was able to find a citation, I do not yet understand the idea. I do know that Shimony is a highly-regarded physicist; a friend remarked to me that Shimony is the 'S' in "CHSH." This same friend also told me that Shimony is one of the few real physicists who are willing to tackle the hard questions (such as whether quantum nonlocality is real or just a mathematical construct) -- though there are many pseudoscientists willing to muddy those waters. His idea of "passion at a distance" appears to be such an attempt, and the awkward name shows the difficulty of putting it into words at all. — SWWrightTalk 20:46, 8 March 2007 (UTC)
[edit] fact?
The text currently says: "In May 1926 Schrödinger published a proof that Heisenberg's matrix mechanics and his own wave mechanics gave equivalent results: mathematically they were the same theory. Both men claimed to have the superior theory." Somebody has put a "fact" tag on this claim. It is true that Schrödinger published such a proof. If one theory can be derived from another theory, i.e., if they are "mathematically the same theory," then one cannot be superior to the other. About the only thing one of them might claim would be that his way for formulating the mathicatical relationships is easier to compute under certain conditions. I would be in favor of deleting the claim. It does not help the reader to understand quantum physics. P0M 21:13, 30 April 2007 (UTC)
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- I put that tag there, not doubting that Schroedinger gave the proof (Eckart did too), but doubting that both men claimed their own theory to be superior. I have never before read that. As far as I am aware Heisenberg understood immediately the equivalence proved by Sch. Equivalent theories rank equally as you, H., and Sch. know, but not the author of the sentence. So, I am in favor of deleting the sentence + tag, too. --P.wormer 15:26, 16 May 2007 (UTC)
Upon reading it again I think that the original author intended to refer to the interpretation dispute. Schroedinger felt uneasy about the probabilistic character of QM, while Heisenberg had no problems with it. I adapted the text a little to clarify this.--P.wormer 11:12, 20 May 2007 (UTC)
[edit] spin
Spin was discovered by Uhlenbeck and Goudsmit in 1925 as stated in most textbooks of QM. Not by Kronig as myth has it. See http://www.lorentz.leidenuniv.nl/history/spin/goudsmit.html --P.wormer 15:10, 16 May 2007 (UTC)
[edit] contradictory?
"These ideas [waves and particles] seem mutually contradictory, because neither idea by itself can explain electromagnetic radiation".
This doesn't make sense. It's like saying wheels and engines are contradictory because neither alone can explain vehicle motion.
Should the two parts of this sentence instead be separated by but? --217.18.21.2 14:30, 29 May 2007 (UTC)
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- I agree, but couldn't find the sentence--P.wormer 14:41, 29 May 2007 (UTC)
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- Right next to the constructive and destructive interference image. Note that when quoting the sentence, I added the portion in square brackets to make it easier for people to follow the point I was making.--217.18.21.2 15:07, 29 May 2007 (UTC)
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- OK I found it. I would write something like: In classical physics these ideas are mutually contradictory. Ever since the early days of quantum mechanics we know that neither idea by itself can explain electromagnetic radiation. --P.wormer 15:26, 29 May 2007 (UTC)
[edit] h*hertz=joules
I see that this has already been reverted 3 times in few days - so, before it becomes an edit war, let's spell it out clearly: Planck constant is 6.6260693 × 10E-34 joule seconds; light's frequency is hertz = 1/seconds; the resulting unit for h * \nu is joule seconds / seconds = joules, so the result is in joules, a unit of energy like the electronvolt, ok ? :-) -- Sergio Ballestrero 17:55, 30 June 2007 (UTC)
Sergio, if the eV is an energy unit then Hz is it too (as is kelvin, cm-1, etc.). The only SI energy unit is joule. All the other "energy units" have a natural constant hidden in it. For the eV it is the elementary charge, for Hz it is Planck's constant, for kelvin it is Boltzmannn's constant, etc. It very much depends on the subfield of physics what "energy unit" is preferred. --P.wormer 07:18, 1 July 2007 (UTC)
- Paul, you're right that the only SI energy unit is the Joule, and that you can use other units; but unlike the others you mentioned, eV is a proper energy unit, with no "hidden constant", because the "e" in eV stands for "electron charge with unit": 1 Coulomb * 1 Volt = 1 Joule -> 1e * 1V= 1.602 ×10E−19 C * 1 V = 1.602 ×10E−19 J , so there is only a (quite explicit) numeric factor, the electron charge in Coulombs, not a constant with units. If you use eV for momentum, mass etc then yes, you have implicit, hidden factors and, more important, hidden units, but that's not the case for energy. Aside from this, my main point was that the corrections were replacing Joules with Hertz without adjusting the numerical value, which is incorrect no matter what! -- Sergio Ballestrero 10:32, 1 July 2007 (UTC)
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- Sergio, I agree with your main point, but only partially with with your first point. If one goes from joule to any other "energy unit", one has to multiply with one or more natural constants. This is also true for the eV, as you say yourself: you multiplied with the charge e of the electron. I call e a natural constant with SI unit coulomb. However, I grant you that the eV is special among the "energy units" in the sense that it has the dimension of energy (as does the hartree), while most of the other "energy units" do not even have the dimension energy (for instance Hz has the the dimension 1/second). --P.wormer 13:24, 1 July 2007 (UTC)
[edit] accessible discussion?
I saw the following sentences in this article:
- Therefore, an electron in a certain n-sphere had to be within a certain range from the nucleus depending upon its energy. This restricts its location. Also, the number of places an electron can be is also called "the number of cells in its phase space". The Uncertainty Principle set a lower limit to how finely one can chop up classical phase space, so the number of places that an electron can be in its orbital becomes finite. An electron's location in an atom is defined to be in its orbital, but stops at the nucleus and before the next n-sphere orbital begins.
Is this understandable for the proverbial "average reader"? I, for one, don't know what an n-sphere is, and why the phase space [for one particle a 6-dimensional space of points (q,p)] enters here.--P.wormer 14:53, 24 July 2007 (UTC)
- Valid criticism to be sure. Can somebody put in explanations and links to make this stuff more accessible? Otherwise it can only be read by the people who already know about it. P0M (talk) 18:07, 19 December 2007 (UTC)
[edit] Citation
I'm new on here, but I found a source to the Heisenberg qoute in 3.3 Uncertainty Principle in the Intro. to QM but am not sure how to go about adding it, if someone else will or give me more specific instructions that would help.
The html is http://www.aip.org/history/heisenberg/voice1.htm, and they have it cited as "From "The Development of the Uncertainty Principle", an audiotape produced by Spring Green Multimedia in the UniConcept Scientist Tapes series, © 1974."
Thanks RangerA 04:55, 31 October 2007 (UTC)
- Done. The original also cleared up some misreading in the quotation. Thanks. P0M (talk) 07:40, 20 December 2007 (UTC)
[edit] Absorption?
Where would I find an explanation of the quantum-mechanical understanding of absorption, for example, the absorption of a photon by an atom? Thinking of a photon as a particle, I can imagine it "hitting" an atom and transferring energy, but as a big wave, I don't see it. —Ben FrantzDale 04:07, 6 November 2007 (UTC)
- See Absorption cross section? 155.212.242.34 (talk) 13:45, 19 November 2007 (UTC)
I'll have to look around for some of my old notes to find a good book to direct you to, but I think I can explain the basics.
First, nobody ever actually sees either a photon or an electron, so what is said depends on observing what can be observed and then building a "model" that one hopes will not soon suffer the fate of all analogies.
The history of quantum mechanics is closely tied to the efforts to explain black lines in the spectrums of various sources of radiation. Research into that question led to the idea of electrons being able to be in orbitals only at specific distances from the nucleus of an atom. The emission of a photon was attributed to the dropping of an electron from a higher orbital to a lower orbital. Since the orbits were restricted to certain distances, the energies put into various photons were limited by the energy differences between orbitals. Available orbitals to move electrons into favored the blacking out (absorption) of certain characteristic parts of the spectrum when light passed through a gas or a plasma. So the picture that one forms mentally is of an electron dropping down a notch and, coincident with its assuming a lower energy position, the emission of a packet that carries that energy away. Or, an encounter between an electron and a matching photon can result in the electron jumping to a higher orbit. Think of an inflatable dome building. When the ceiling lowers, air is blown out a doorway. When air is blown in from outside the ceiling goes back up again. But to made a better analogy, air would have to move in and out of the building in "chunks" of a certain size. When a photon encounters an atom it can do either of two things. It can be reflected in a perfectly elastic collision, or, it can cease to exist coincident with an electron of that atom rising to a higher orbit. The probability wave that is involved with the propagation of light across space is theoretically infinite in extent. Where on the surface of that probability wave the photon will be manifest by blackening a photographic emulsion is a matter of probability, but the blackening of the photographic emulsion by that one photon occurs in a very small volume of that emulsion, the volume of the new molecule formed through the agency of the energy donated by the "sacrifice" of the photon. I guess that's one reason that one speaks of the "collapse" of the probability wave. One might make the analogy of a large soap bubble that encounters the tip of a leaf. When the bubble is punctured maybe it will shrink toward the sharp tip of the leaf and the soapy water will all end up on the tip of the leaf.
Our ideas of absorption come from things like paper towels picking up water. The absorption of a photon by an atom is more of a phase change kind of thing. Where does the work I do carrying bricks up a scaffolding? Besides moderately contributing to global warming I also make a brick building. If I am later under the building when it collapses I will be certain that the energy did not disappear. I will personally get part of it back -- at a much higher speed than I put the energy out. That's not a perfect analogy, of course. P0M (talk) 08:50, 20 December 2007 (UTC)
[edit] Missing book title
Midway through the article there is an author and a page number, but no book name. "Aitchison" is probably I.J.R. Aitchison. That author has two books currently available:
Gauge Theories in Particle Physics and Relativistic Quantum Mechanics
Can anybody help track down the book and revise the in-line citation? Thanks. P0M (talk) 18:04, 19 December 2007 (UTC)
- When I went to patch the hole in explanation of the reason for the matrix math I found the citation -- to an article, not to a book.
- I hope I've made the new part (starting with "In approaching the problem that Bohr gave him") clear enough. Without doing an actual lab experiment and doing everything in the real world the rationale for the math is very abstract.
- Parts of the explication farther down may need to be tweaked a little due to what I've added. P0M (talk) 18:45, 20 December 2007 (UTC)
[edit] Writing needs to be improved
I'm moving on through the article now that I've finally figured out something to say about the matrix math rationale.
The first paragraph of the section called "Uncertainty principle" is terribly muddy. Does anyone understand what the original writer of this part was trying to say well enough to make to both clear and correct? P0M (talk) 18:55, 20 December 2007 (UTC)
[edit] A questionable assertion
The text currently claims that:
- A wave is also a moving stream of particles.
It is not clear whether the writer was confusing light "waves" with water waves, talking about light "waves", or talking about water molecules.
A "photon" is a shorthand way to refer to something that is not a classical particle and is not a classical wave. It is an individually created disturbance that has wave characteristics and particle characteristics. The wave that is named in the definition of a photon is not "a moving stream of particles."
If the writer was talking about water waves, the statement is misleading because the water involved in a wave doesn't move across the surface of the body of water that is disturbed by waves. The molecules in a wave are moved up and down. The particles do not "stream" anywhere.
If one were forced to make a physical analogy, the "particle part" of a light wave could be imagined to be a surfer riding a wave in toward the beach. But there isn't any reason to claim that this surfer is different in kind from the wave that it rides. Analogies are risky, but the passage of a photon would be more like a water wave that moves smoothly across the water surface but, upon hitting the beach, delivers all of its force at a very small volume of space.
Does anyone follow what the original writer was trying to say well enough to clarify this point? P0M (talk) 19:28, 24 December 2007 (UTC)
- Someone else marked the above-mentioned section for clarification, and, seeing no way to clarify it I have removed it. The idea of reinterpreting a particle-like description in wave terms and thereby rationalizing the uncertainty factor is interesting but highly technical. If it is going to be in the article it needs to be put in a very cogent way. Otherwise it is a hindrance to the reader who is not already very familiar with the subject. P0M (talk) 23:05, 31 December 2007 (UTC)
[edit] added tables
The actual development of matrix mechanics has been made the subject of mystery and even mystification (or maybe it's obfuscation). I've at least found outline information on what went into Heisenberg's original calculations and have added two charts that show the kind of information he was working with and, also, give an indication of how suggestive of matrix math this information becomes when so organized.
Does anyone have access to the actual charts or data sets? P0M (talk) 23:10, 31 December 2007 (UTC)
[edit] Dogmatic assertion
Someone has tagged a statement for lack of a citation:
It is not possible to use quantum entanglement for information transfer since it would violate Special Relativity. (citation needed)
The statement is problematical for another reason: it uses theory to attempt to predetermine an empirical observation. At most, one could argue that information transfer using quantum entanglement would be inconsistent with Special Relativity. That's why Einstein et al. thought that quantum mechanics could not be a valid theory, which brings the argument full circle.
There is at least one serious proposal that would have to be given "equal time" if a cleaned up version of the quoted assertion is to be retained.P0M (talk) 17:20, 12 January 2008 (UTC)
See, e.g., http://www.analogsf.com/0612/altview.shtml P0M (talk) 16:57, 13 January 2008 (UTC)
[edit] Transwiki?
I think that this article should be moved to Wikibooks... any suggestions? CJ Miller. (That's my name.Don't wear it out.) 01:33, 9 February 2008 (UTC)
[edit] Is this sentence right?
In the following passage (Section "The Pauli exclusion principle", 3rd paragraph):
According to Schrödinger's equation, there are three quantum states of the electron, but if two electrons can be in the same orbital, there has to be another quantum number
I was trying to follow the text up to here and found this sentence dubious (note that I am a complete layman on QM). Shouldn't it be:
According to Schrödinger's equation, there are three quantum numbers of the electron, but if two electrons can be in the same orbital, there has to be another quantum number
190.1.52.62 (talk) 19:30, 25 March 2008 (UTC)
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- It is badly written, unfortunately. Besides that, that section is lacking in citations. I think it is one of those statements that can be defended if you can figure out for sure what the writer was trying to say in the first place.
- Anyway, thanks for pointing this problem out. I'll have a look at what Pauli had to say on the subject and see whether or not I can make it clearer and peg it to Pauling's The Nature of the Chemical Bond. One good thing is that Pauling is an extremely careful and clear writer. (Why is it that the really great physicists are generally clearer writers than the people who are not such great physicists but try to "popularize" their work?) P0M (talk) 02:31, 27 March 2008 (UTC)
190.1.52.62 (talk) 19:36, 28 March 2008 (UTC) Thanks for addressing my question!
[edit] testing
I've read the entire article, and I was wondering if someone could prepare a test for me. I'm home schooled. 68.143.88.2 (talk) 16:22, 16 May 2008 (UTC)
- Supposing that I were even willing to to do so, before I could make a test I would have to know what level you are operating at and what your goals are. Right now, I'm more interested in the whole issue of testing since we have a national program that is allegedly forcing primary and secondary schools to "teach to the test."
- Testing serves at least two legitimate purposes. The purposes are different, so the tests may be different too. One purpose is to guide learning. Weekly quizzes in a language class are very important because they keep students aware of whether they are actually learning things. Quizzes or exams at longer intervals give a pretty good indication of whether the material is really sticking with the student. Another purposes is to certify competence to other people. If somebody has taken French in college for four years and has gotten all A's, the bank that is thinking of hiring this individual to work in a branch office somewhere in France depends on the college to give a good representation of that student's actual level of competence. The bank will probably make further evaluations after the have chosen the individual as a candidate for the job, but an individual who had C's and D's all the way through his or her education in French would not be a candidate to begin with.
- Then there is the question of "inside" and "outside" knowledge. I entered the physics department of a major university with a poor math background. Partly that was due to coming from a small town that did not offer courses in spherical geometry or trigonometry, and partly that was due to my own limitations. Our school divided the year into trimesters. During the first trimester we did mechanics (f=ma, etc., but using calculus to derive formulas rather than memorizing formulas), and I did not do very well. During the second trimester I got an A -- not because my math had improved but because the subject was electricity and I had already spent years doing things like making crystal radios, wiring the light dimmers for the spot lights on the stage of my high school, and trying to understand how things like resistors really work so that I could understand why voltage drops more and more as resistances are added in series, but voltage rises as resistors are added in parallel to whatever resistors were already there. So if the book said the formula for resistors in series is R = Ra + Rb +..., I didn't have to memorize the formula because it was just a representation of what I knew as a fact of life. Our grad student lab instructor knew all the math, but he had never really worked with things like voltmeters, and sometimes he made some funny mistakes. ;-) The third trimester we did thermodynamics, and I was lost again.
- Being able to report on what some physicist says about some experiment is far different from having done the experiment yourself. Reporting on what Young said about the double slit experiment is far different from having reproduced the same experiment yourself. And just looking at the fringe pattern is different from working out the relationship between the frequency of the light coming into the apparatus, the width of the slits, the distance between the slits, and the pattern that emerges on the screen.
- If you are trying to do the physics major kind of physics, then there is not very much in the Introduction to Quantum Mechanics that can really help you. On top of that, much of the math is beyond the level of a student who has taken college calculus five days a week for an entire year.
- The math involved in relativity theory where things like time dilation are calculated is not difficult. The hard thing to do was to ask what would happen if the speed of light was taken as a constant. Imagine that you have a device on board a space ship that can measure the speed of light Light comes in at the nose of the ship and exits at the tail. You note the time a pulse comes in and the time when that pulse goes out, so you can compute the speed. What would it mean for things like the measure of time if you discovered that no matter how fast you flew toward a distant star, and no matter how fast you flew away from it, it took the same amount of time for a pulse of light to pass from one end of your ship to the other? In other words, what would it mean if the speed of light turns out to be a constant? That simple question fueled Einstein's thought.
- Another thing that happened around the turn of the 20th century was that scientists realized that you have to be very careful about how you define things, and then you have to be consistent, i.e., it will get you into trouble if you change definitions in the middle of things without realizing what you have just done. When some physicist asked, "How do we measure time?" s/he came up with a simple kind of clock. Instead of using a pendulum and the regularity of its swings to make a "tick," the new kind of clock made a pulse of light travel a long distance, hit a mirror, hit a detector adjacent to the place it left from, and then that incremented a counter and also sent off another pulse. Each trip was a "tick" in the new clock. One of the standard texts of relativity theory starts with this simple experiment and then asks what would happen if two spaceships passed each other in interstellar space.
- Now, going back to the Introduction to Quantum Mechanics, what are the insights involved that changed people from talking in terms of classical physics to realizing that they had to use the new physics? What did we learn about the fuzzy parts of our thinking? P0M (talk) 18:58, 16 May 2008 (UTC)
[edit] Easy to understand??
This is supposed to be a "a generally accessible introduction to the subject"? I guess it might be easier to understand than the (non-easy) Quantum Mechanics article, but this is by no means something that the layman could understand. I know this is a complex subject, and that explaining complex things simply is difficult. Still, I hope that this article could be made from "easier quantum mechanics" to "actually easy and fairly simple to understand quantum mechanics". Above all, remember that we don't need to go into that much detail, because this is the "easy" article :). Some suggestions:
- -The overview section is really long, maybe we could trim it down?
- One of the things that might make the article easier to understand is if we took all the history out of the "Overview" section and put it into a new "History" section that details all the, well, history of QM, like Einstein, Bohr, etc. (see below)
- Would it be possible to just put a few simple paragraphs explaining what quantum mechanics is, without going into a lot of detail just yet?
- -History
- I noticed that there is a TON of historical data floating randomly in this article. I don't know how important this is to quantum mechanics, and of course if it is an important point in the study of quantum mechanics to know who-did-what-when-and-why then of course these little snippets of history should be kept, but if this is not the case, then we have way too much history on our hands here. It seems like what's happening in this article is that instead of explaining QM in an easy-to-understand manner, we are writing about the major "players" in the development of QM. We are going through all these theories and concepts person-by-person, looking at "what Planck did" and then "what Bohr did" etc etc, and not at "what all of this actually is". The history, though an important part of QM as a whole, should not be the focus of the introductory article to the field. Thus, I think that the aforementioned "History" section might be a good idea. That way, we could have a brief discussion of what each genius did, and we can mention their experiments and everything, but we could also try to bring the focus away from the people and towards what QM actually is. Bottom line: what matters more, the who or the what?
Thanks for the time and effort, sorry if I have no idea what I am talking about...I don't know much about QM or even physics for that matter (hah! matter! no pun intended! but how amusing!). About the history, again, if its really important and integral, then I apologize, but it just seems to me, as someone who doesn't know much about the subject, that there is too history for an introductory article. Thanks, 71.178.238.238 (talk) 04:23, 19 May 2008 (UTC)
- You might want to look at what some other secondary sources have done with the subject. Brian Greene has written a couple of good books that include chapters on the subject.
- Just explaining what classical physics is can be a problem, yet it researches the stuff of everyday life. As soon as you hit Relativity theory, people start making simplistic interpretations that can only mislead others, but at least they are talking about effects that can be observed in things like global positioning satellites. Quantum mechanics deals with things that cannot be seen, and one of the really important lessons tied up with quantum mechanics is that it is essential to keep clear on what is observed vs. what is inferred.
- It would be relatively easy to say what quantum mechanics does, i.e., what it explains, what it helps us do, how it directs technological initiatives, etc. But then the question would be, "Well, how does quantum mechanics provide for some of the useful things in our life?" One kind of explanation is like saying that a large box with windows and wheels, you get it, turn a key, and move across the terrain. That kind of explanation would do just as well for Cinderella's magic carriage. The other kind of explanation gets into fuel, pistons, mechanical and electrical connections, tires, etc., and even explaining a relatively stripped-down mechanized vehicle more specifically than saying that you get it and it goes will turn out to involve a long chain of explanations.
- I started reading about quantum mechanics sometime before 1958, and my first impression of it was that it had something to do with an "uncertainty principle." For a long time I thought that it was possible that the things in the real world were all at actual points in space and time, and that our knowledge was just limited by the fact that we inevitably mess with these real locations when we try to measure them. It turns out that I had gotten it wrong.
- If you don't want people to get misled, then you have to be really careful. You have to include things to head people off from taking some things the wrong way. Furthermore, you can't jump into the middle of the story. You have to provide a road between the things that readers already understand and the new insights you want to teach. P0M (talk) 15:50, 19 May 2008 (UTC)
- I've just been through the article again. The intro section gives readers the basic "consumer's view of QM" (equivalent to: Car -- you get in, you get delivered, you get out). Everything else in the article has to do with quantum phenomena, so if the reader wants some flesh on the idea that quantum mechanics deals with how nature works on the very small scale and why that should matter to us then you must work through some concrete examples. But these examples also tell the story, in digestible quantities at each stage, of how the answers to problems that cropped up in the labs each contributed parts to the jigsaw puzzle that was eventually assembled to account for everything except for gravity.
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- I agree with both User:71.178.238.238's criticisms. The history is too much, and the overview needs to be simpler. What this introduction to QM needs to address is why QM is so much harder for people to understand than classical mechanics. What about an approach like Richard Feynman uses in his Lectures on Physics? He starts out by saying: "Things on a small scale behave like nothing you have any direct experience with. They don't behave like waves, they don't behave like particles, they don't behave like clouds, or billiard balls, or masses on springs. Even the experts don't understand it the way they would like, because all of our human intuition applies to large objects. But small objects just don't act the same way." Later he goes on to enumerate in simple language how small objects' behavior is different: wave - particle duality, inability to measure variables with arbitrary precision, probabilistic results of measurement, the ability of objects to be in different states simultaneously. These are the things beginners have trouble with in QM, and they should be faced head-on in this article. --ChetvornoTALK 22:59, 21 May 2008 (UTC)