Talk:Wave packet/Archive 1

Non-stoppable animated gifs
It is very difficult to concentrate on the text, which clearly needs a lot of concentration to read, while the wiggling gifs constantly distract. Can they be made startable/stoppable? — Preceding unsigned comment added by 2003:EE:C715:9800:3C22:B5E9:9FEE:DFB7 (talk) 07:51, 6 March 2022 (UTC)

Wave packet image
To those who felt the wavepacket representation image was confusing.I believe it is useful to have a graphical representation of a wave packet on this page. Any suggestions on how to improve the image to your tastes? Note that the "axis labels" of a wave packet are completely arbitrary. It depends on the representation one is working in, the point is the shape of the graph not the labels.

For now I'm restoring the image, so the site will be more clear. Let me know how to improve it, or upload a better one rather than just deleting content.


 * My main objection to the image is the totally ambiguous arrows. If the horizontal axes is position, then I would gather that the Position arrow is indicating the line (whereas its not clearly pointing to it), and then the momentum arrow is trying to indicate that the momentum is related to the width of the wave packet.  The caption still does nothing for me, because it does not explain what the arrows are indicating, or that we are in position-space.   I'll add to the caption, but it would be ever so much more clear if the image itself was made more clear.  Laura Scudder 20:30, 17 Mar 2005 (UTC)


 * If you read the text the ambiguity in the position is supposed to demonstrate the uncertainty principle.  What I was trying to illustrate is that the "position" is somewhat fuzzy, I'm not sure how else to draw it.   I suppose I could remove the arrows?  Not sure that this would be more effective though.   Some sort of gaussian distribution overlay may work better?  I'll have to explore some possibilities. John187 16:33, 18 Mar 2005 (UTC)


 * I understand the idea of not labelling the graph, since a wave packet looks identical in position-space and momentum-space... however, having a graph without any label on the x- and y-axis is really bad form. It's confusing to specialists and non-specialists alike, since it doesn't really represent anything (are you plotting amplitude vs. time (amplitude of what?), or time vs. distance, or length vs. mass ... or what?). Why not just have two pictures: one for position-space, one for momentum-space, and let the reader see how similar they look. And you don't need to label "position" if the x-axis is already "position"... it should be clear to the reader that the position is ill-defined, since a range of positions is included in the packet. --132.206.205.100 14:07, 15 Jun 2005 (UTC)


 * Simply removing the arrows would be worse without axes labels. Then it's just a meaningless graph.  The optimum would be axes labels and, as suggested, plots in both position and momentum space to convey the relationship.  And if there must be arrows in addition to axes labels, I think instead of momentum and position, it'd be preferable to be clearer and choose for instance &delta;p, &delta;x.  I would just do this myself, but I currently lack ability to generate acceptable quality pngs. --Laura Scudder | Talk 16:19, 15 Jun 2005 (UTC)

This discussion of wave packets seems almost completely based on their interpretation in quantum mechanics, and contains little discussion of generic properties of wave packets. The simplest example that I can think of is the complete lack of information on the group velocity, a concept that is both a general property of a wave packet and is a part of nearly every subject that works with waves and wave packets. The current organizational structure makes it difficult to include such information.

Complete Rubbish Without Math
If this article doesn't include math, or links to a more mathematical article then it is complete rubbish. Metaphysics should be in a wholly separate article and not included in wave packets. These are not only quantum phenomena, they are mathematical outcomes of waves. Please for the sake of wikipedia clean this article up. I don't want to insult the gracious individuals who have made this article but it is incomplete and misleading. More math please, less junk philosophy!


 * Wikipedia is produced by volunteers, and I personally don't understand the math you describe. Are you volunteering to provide the improvement you describe? If so, let's see what you had in mind. If not, do you really think your attitude is what will get people to happily pitch in and fix it? Art LaPella 01:33, 14 November 2005 (UTC)

I would almost think about agreeing if there was something on wikipedia that made math vaguely accessible. To be honest I find most math self indulgent jargon, overcomplex, poorly expressed and almost all of it could be represented in an accessible form using plain English. There are plenty of intelligent people who would love to know more about such topics, but the barrier to entry with this hieroglyphic math is just too high. Get rid of the symbols, please, and use words. I don't care if mathematicians claim to have ownership of this domain; dispense with the arrogance, dispence with the jargon, keep the symbolic codes for private fraternals, and make the effort to explain these concepts in plain, philosophical language. —Preceding unsigned comment added by 83.208.165.249 (talk) 21:00, 10 October 2010 (UTC)

Wavepackets and metaphysics
It seems rather odd that an article on wavepackets is about 50% on metaphysics...the article itself reads almost like an opinion piece. I'm going to put an NPOV tag on it; this page also needs some cleanup I think. --HappyCamper 12:39, 12 January 2006 (UTC)
 * I don't understand the science very well and you may be right about too much metaphysics, but I do have an opinion on unexplained tags. What opinion is over-represented or under-represented? I didn't notice any typical cleanup problems like spelling or grammar. Art LaPella 21:42, 12 January 2006 (UTC)


 * For example, for me, the first sentence is far too loaded: "The wave packet is one of the most widely misunderstood and misused concepts in physics." Also, at minimum, it should mention some of the influences that de Broglie had on this concept. This name is currently absent from the article!


 * Well, I suppose it would be unfair of me to add those tags, but for some of the articles that I found, I added these two, and after 9 months everything was fixed up. You can remove them if you like, as I probably won't get a chance to add more mathematical detail to this anytime soon. :-) --HappyCamper 22:04, 12 January 2006 (UTC)
 * Better. Art LaPella 02:13, 13 January 2006 (UTC)


 * As far as I'm concerned the metaphysics section could be cut back. I haven't done it myself yet because I'm scared of only leaving some disconnected nonsense and making the situation worse.  &mdash; Laura Scudder &#9742; 23:55, 17 February 2006 (UTC)

I have to say, I'd rather not see any reference to metaphysical concepts on this page. Consider it a personal bias if you will, but I think a subject that is fundamentally scientific in nature is serriously comprimised when metaphysics are included. I propose a rewrite, of a few of the paragraphs at least, it tends to read like a 10th grade science essay rather than an encyclopedia article. For instance, the first sentence should be eliminated altogether, it does't really add anything but ambiguity to the article. But thats just my $0.02. Davepetr 20:11, 17 February 2006 (UTC)

10th grade may be aiming a bit high... if it's possible to write an article in a simpler form that's equally accurate, let it be so. As far as the metaphysics section goes - a lot of that is just really bad logic, and I'm going to weed some of it now. Pjrich 06:10, 1 March 2006 (UTC)

Mathematics Section
I am starting to put in a mathematics sections and some relevent diagrams. It is going to take place over the next couple of weeks. If able, feel free to make corrections or additions. --Nkrupans 06:13, 20 January 2006 (UTC)

Sorry for editing the front page. My comment should be placed here:

Let me add a little comment without which above-mentioned derivation would be hard to comprehend by most people myself included. Not everybody has simply patience to solve puzzles each time he or she tries to learn anything about quantum mechanics. This differential equation has a simple and useful solution in the form of something close to Maxwell distribution:


 * $$ A(k) = A_{0}\exp[-\frac{1}{\sqrt{2\pi}}\left(k - k_{0}\right)] $$

where Ao and ko are constants. greg_park_avenue

Not Rubbish, but needs guidance
Okay... Wikipedia is a fascinating eperiment, but if anything it is an attempt to harness to energy of lots of people to explain a lot of things in a short amount of time. I am only one person, and I happen to be a physicist, and I don't have the time to go around rewriting every basic physics article that I find in serious need. In the spirit of the grand experiment what I can do is offer some guidance and maybe release some of that vast reservior of energy. So, here goes:

A wave packet per se is just a superposition of individual wave solutions that produces a localized packet by operation of superposition of the individual components arranged so that it reinforces at the center and interferes increasingly towards the extremes. The article is more or less correct in conveying this despite that it goes astray in the details. This can be fixed. However... we need to reel in the diversion about collapsing wave packets. One of the major intrigues of quantum mechanics is the so-called collapse of the wave function -- not the collapse of the wave packet. Well, maybe we can think about collapsing wave packets too, but that is not what you are thinking about when you are learning quantum mechanics and contemplating its postulates. Please go check this out by reading up on basic quantum mechanics and then see if you agree that collapsing wave packets is not what you want to convey as an example of what happens when you measure something. A wave function can be a superposition of states each of which is an eigenfunction of some operator with associated eigenvalues and these functions need not be waves as you usually think of them and need not be assembled into a packet. A measurement does produce only one eigenvalue as the result and so we describe the result by saying that the superposition collapsed to just one state -- in which it can remain for awhile. In this context we are talking about the state of a system of some kind and, in principle, a system can exist in a superposition of states distributed in a way that maps the probabilities for it to "collapse" into one of them upon measurement. The superposition of states is analagous to the superposition of things in a wave packet -- but is a much more general concept. Bound states, for example, are not visualized the same way as propagating packets.

Contrast this with the simple idea of a wave packet. A wave packet is a superposition of -- well, waves! Okay, waves can be solutions to quantum mechanical problems, but you don't need to invoke quantum mechanics as such to have waves or wave packets. It is also true that we worry about how to more realistically describe how particles propagate and since waves correspond to (mostly) freely propagating particles the packet becomes a model for this to some extent. In that context we get some real insights and it is those insights that I think the article should focus on.

My recommendation is that one or two contributors go and grab an elementary quantum mechanics textbook or find some online course material and then find within that some discussion of wave packets and see how they get introduced and for what purpose. Study this deeply and put it into your own words -- but be careful to stay on topic before you suggest the possible connection to other ideas. Maybe even before indulging the world of the quantum, just pick up a math physics book or find some online course material and see if they describe wave packets there.

Can you produce an article that answers basic questions like: what is compulsory about the shape of the packet or distribution of the coefficients as a function of the wave number k? Is there any arbitrariness in the shape? What keeps the packet from falling apart as it propagates? What happens if the wave number k is not proportional to angular frequency (i.e. becomes dispersive) and when does this happen in practice? Can someone upon reading the article grasp at least one example of Heisenberg's Uncertainty principle applied to wave packets?

Sorry for resorting to pedantry, but writing a really good article is real work no matter whether the topic is basic or advanced. Maybe another expert can drop in and spend more time -- maybe even someone with particular expertise appropriate to this topic. My goal was to point out that whoever is working on this needs to do more study -- but should be allowed to do so and I am confident as much help as necessary will show up. If they are good readers they might not need much help and experts can just tidy things up at the end. --scanyon 09:48, 8 June 2007 (UTC)


 * Rather than ask amateurs to study your subject until they know it as well as you do (their time might be better spent repairing millions of simple problems summarized at Community Portal), you might choose one or two interesting physics articles that you find in serious need, and rewrite with references (so we know whom to believe). Then someone like me can clean up the typos and such. Art LaPella 22:02, 8 June 2007 (UTC)


 * I am not being patronizing when I suggest that I am confident that whoever made major contributions to this article demonstrates enough competency to correct the problems with just some guidance and study. Also please accept that I am personally already engaged in a major revision to a math article which touches upon some things I have an interest in and is rated by the powers that be as having high priority.  I need to finish an existing commitment using available time before becoming too involved in another one.  This is kind of fun, but its also very time consuming.  Then also believe that you do not have to master physics to do an article on wave packets.  You do need to do some reading and take some guidance.  Its not really an advanced topic.  Finally, though, in rummaging through Wikipedia-physics I have seen plenty of physics expertise at work which exceeds my own talents so I know its available and I am not at all trying to be patronizing by declaring that I "don't have time" to do this as if I could save you but I'm too busy to care.  The reality is that I care, but genuinely don't have the time that I think this deserves.  Besides, in my general experience, the time you invest in figuring out what went wrong here will strengthen you and educate you at the same time.  If you poked your nose in this in the first place you must have a desire to master something and this is your chance.  There is no more powerful tool for achieving mastry than struggling to explain something clearly.  Fortunately for you guys, the web is full of wonderful free course material and all you need is time and curiousity to find it.  Of course, there is also still the old fashioned way -- go find a book or two in a library or bookstore. You are on the right track, just follow the leads that I and other like me provide and I think it will turn out fine. Leave me occassional notes on my talk page and I will see what I can do to help as you go forward. --scanyon 06:36, 9 June 2007 (UTC)


 * Looking through page history, it appears that "whoever made major contributions to this article" (my career is financial) and in particular to the section you disagree with, is User:John187 and perhaps User:Nkrupans. Those editors have made only 2 edits each to Wikipedia in 2007, and no edits to this article, so neither is likely to be reading this. Art LaPella 23:09, 9 June 2007 (UTC)

Hmmm... well, what is your feeling about just deleting the digression on "collapse of packets" for now and seeing who becomes unhappy about it? I am quite new to the Wikipedia process, so I am hesitant to delete other people's work without discussing it with them. In the customary scheme of things, though, deleting and reverting are the most direct ways of contesting correctness of material and the worst that can happen is you upset someone enough that they come back and make a case for themselves. If that is the way the system works then so be it. On the other hand, it would be nice, at times, if a pool of experts could be distinctly queried from time to time in an effort to recruit help on a specific matter. Maybe there is such a mechanism and if so you could explain it to me.

And by the way, the math as shown is okay except for the last step, but is not presented in as general or explanatory a way as I think it should be. I can take care of this for you later, but there are worse problems. There are, as I hinted in the first set of comments, lots of things that can be said about the parameters used to describe a specific wave packet and its behavior in a set of circumstances. To the uninitiated, the central questions that take thought to answer are: just how does a wave packet come about and what factors determine its shape -- or equivalently, the distribution of k values? Is a propagating electron reasonably described by a wave packet (this *is* a question with quantum theory implications) as we would expect a light pulse to be? The mere fact that I can build a mathematical structure does not mean it is a valid description of what happens in nature or by engineering. Where do we see (or think we see) wave packets? If we set about to make our own (is this pulse shaping?) when and how do we do that and for what purpose? For someone like me, the math is no problem. I can also discuss some of the motivation in general terms, but the best expert would be someone who literally deals with wave packets -- maybe an optical physicist or engineer -- maybe someone who does interferometry for a living. Again, I can bore you with the details of the math later on. Its the motivational material that takes more scholarly research to do it justice. Building a bibliography will take a little time.--scanyon 04:49, 13 June 2007 (UTC)


 * I have 8895 total edits on Wikipedia. You have done more than enough to justify "deleting the digression" (I'm speaking procedurally, not commenting on the scientific merits). This is especially true because the text I think you want to delete doesn't have any supporting references in the text, and may therefore "be removed" (Verification). Anybody who doesn't like the text being deleted now, should have shown up last week. But if you want to discuss this with a pool of experts, try Wikipedia talk:WikiProject Physics. Art LaPella 07:45, 13 June 2007 (UTC)

Okay, I spent some time looking into this in order to more clearly and succinctly state the problem and contemplate how deleting material can be justified. Please look this over and then confirm your agreement about what should be deleted at this time.

The article is about wave packets. As a mathematical construct, wave packets do not require appeal to quantum mechanics in order to be defined. It so happens that in a quantum mechanical context solutions to Schrodinger's equation can be waves as they are normally conceived and in all cases superpositions (linear combinations) of solutions are solutions in the aggregate to the Schrodinger's equation. Since wave packets are, by definition, superpostions of waves that result in a localized wave form, it is possible to construct a wave packet of solutions to some quantum problems. This results in a way to demonstrate Heisenberg's Uncertainly principle concretely, but something similar can be demonstrated with respect to wave packets even without resorting to a quantum scenario. Now, in a qauntum context, wave packets are a way to construct a localized solution for free particles consistent with some particle characteristics in cases of moving particles. Light is known to be quantized into photons and it becomes possible to model light and electrons alike using packets formed from solutions to the free-particle situation -- and they move as they should in a vacuum and display dispersion in media as the wave packet model would predict. Now stop for a second. We got to quantum mechanics becuase we wanted to show an important example of something. Quantum mechanics was not needed to define wave packets, but wave packets play a role in quantum mechanics -- of course so do differential equations, Hilbert spaces and lots of other things. That does not mean in a basic article about wave packets we need to explore all those topics in depth.

In sections 3, 4, 5 and 6 the article takes off into the quantum world with a vengeance. Some statements made are related to the main topic, some statememnts are poorly made or made without adequate context. There is a preoccupation with "collapse of the wave packet." There is, indeed, a central postulate in quantum theory about what happens when you take a measurement of a system in a state that is a superposition of basic solutions (eigenstates or eigenfunctions of the measurement operator) and that postulate is often described as "collapse of the wave function." Okay, a wave packet is a wave function of a particular sort and in the quantum context we could impose a measurement on it and precipitate a collapse. Packets are complicated things in some contexts, however, and there is even a notion of reviving collapsed wave packets. Does this create enough relevance to justify section 4 in particular? I don't think so -- but that is just my opinion.

There is more. If you google long enough you will discover that some authors are sloppy about using "packet" terminology. I did not do a scholarly survey, but it was clear that its fairly common for authors to lapse into calling ANY superpostion of states a packet -- but this is patently inappropriate unless you are referring to a resultant wave form which demonstrates a localization. I suspect that whoever wrote sections 3, 4, and maybe 5 encountered this and failed to comprehend the history and the distinctions that would have indentified this as a departure from well defined ideas.

There is even more. In googling I discovered that the current Wikipedia article on wave packets is often quoted as authority in the context of discussing 'collapse of the wave packet" -- so we are contributing to a problem instead of contributing to clarity! Enough already.  I am going to delete sections 3, 4, 5 and 6.  Let the chips fall where they may.--scanyon 07:57, 14 June 2007 (UTC)

(addendum) In the cold light of morning I have edited out a reference above to an uncited article which I suggested was a case where a professional author referred to something as a wave packet that was not. Although the english in that paper was not at full fluency, I reread it this morning and, despite some conceptual conundrums that haunt all experiments of this type, he was correct as far as he went. Fortunately I did not malign him by directly citing the article. Still, the abuse sometimes happens -- as I think it did in the article -- and the key point is that collapse of the wave function is the general issue and wave packets are just a possible case. In the case of wave packets per se, the collapses most notoriously explored often involve a collapse of the indeterminancy of some property of the particle -- like spin or polarization. The more general case of collapse is what is fundamental, but there is fascination with packets that shows up often owing to a mystical attraction to quantum mechanics. Quantum is truly counterintuitive, but why and how need to be explored elsewhere -- in which case the ability to make reference to a compact description of wave packets per se will prove helpful.

Lastly -- with sections 3-6 gone what I think should ultimately happen is that appropriate mention should be made of the fact that wave packet formalizations are important in quantum mechanics because they provide a way of modeling localization of particles. I just don't think a full-blown exploration of some otherwise interesting aspects of quantum theory are appropriate here and I more stongly believe that distracting readers into thinking that wave packets are a qauntum phenomenon per se is a disservice. --scanyon 16:23, 14 June 2007 (UTC)


 * As best I understand the above, sometimes it says that the deleted sections were irrelevant to an article about wave packets, and sometimes it says that the deleted sections were largely wrong. If they are right but irrelevant to this article, I hope you have considered the alternative of merging any useful information with an existing article such as Wave function collapse, or starting a new article with it. Art LaPella 02:28, 15 June 2007 (UTC)

Hmm.. I suspect what you are perceiving is a residual discomfort on my part for having deleting the material. What is true is that some of the material deleted was correct as far as it went, but was poorly connected to the main topic. My gratuitous comments about some of it needing to be better clarified by establishing a firmer context is true, but probably superfluous at this point. My afterthought was meant to move somewhat in the direction you are suggesting. In any case, I think the deletion was appropriate and that moving that way requires a fresh start. I strongly agree that subsequent attempts to redeem some parts of the deleted material need to include references to available related articles and Wave function collapse would be among them. It is certainly true that wave packets are utilized in quantum theory, but if you look at any textbook or monograph on quantum -- starting with Dirac's own "Principles of Quantum Mechanics" you will find that it is not an extensively considered topic. Accordingly, the article should focus on explaining wave packets per se well and then providing a link-rich, but succinct discourse that connects the explanation to its applications in quantum theory. Within the context of quantum theory it might be appropriate to have a subarticle on quantum wave packets. I will try to do something about cleaning up the math within a week and will do so in a way that sets the stage for what we agree should follow. As some folks in previous discussion have complained, the topic is essentially a mathematical one and perhaps one reason it got off on a tangent was that the needed mathematical depth left it otherwise empty ... but that's just a theory on my part. --scanyon 18:14, 16 June 2007 (UTC)

Question about image
Is the following appropriate on this page? Oleg Alexandrov (talk) 07:20, 27 November 2007 (UTC)



Animating dispersive propagation
It would be nice if some wizard, able to produce animations like the one illustrating nondispersive propagation, could produce a similar animation for the (dispersive) solution of the Schroedinger equation I added (unfortunately I do not possess that ability). It would be important because wave packet dispersion has played an important role in rejecting Schroedinger's early interpretation of the wavefunction. An animation of
 * $$|u(x,t)| = \frac{1}{(1+4t^2)^{1/4}}e^{\frac{-x^2+2k_0xt}{1+4t^2}}$$

(even though this is not a solution of the Schroedinger equation) would be particularly clarifying when compared with the nondispersive example.WMdeMuynck (talk) 13:13, 20 January 2009 (UTC)

Gaussian wavepackets from Schrödinger equation article
See here.-- F = q(E + v × B) 11:38, 11 January 2012 (UTC)

Same as "packet train"?
Is this the same as a "packet train"? 31.143.63.175 (talk) 18:43, 28 March 2012 (UTC)

Wave packet of an electron
I was looking for a formula of the wave packet of an electron but got confused by this article. In "Basic behaviors of wave packets" there is a "final" formula of U(x,t) which suggest that the envelop has a fixed width of t = 1/c = 3.3 ns. But in "Gaussian wavepackets in quantum mechanics" I see a different formule, without sine/cosine, and a variable packet-width. And what is t (time from when?), what is r?. And a is defined as "width of the wavepacket" while later on the "width of the Gaussian" is expressed in a... And why does the e-term not influence the packet width? I think this needs more explanation to be useful. DParlevliet (talk) 21:54, 20 February 2014 (UTC)
 * I took your questions into consideration in a minor edit of the relevant sections, but, frankly, I am not sure you should be here if you have no background in wave propagation or quantum mechanics. Most of your questions on the "final" formula apparently pertain to the wrong formula, the nondispersive wave example! The wavefunction (electron wave) would be represented in the dispersive wave exemplified at the end of section 2. The probability density of a free electron in one dimension, localized at x=0 at t=0, spreads in time as the sqrt of the denominator of the Gaussian (exponential), as stated.
 * In the next section, 3, this very example, without fussing the normalization, is generalized by a more realistic  3-dim position vector r  and arbitrary width a, which was =1/2 in section 2, and essentially the same formula, mutatis mutandis, is adduced in the end. What determines the width is again the sqrt of the denominator over a, and when a=1/2 it essentially coincides with the spreading of the previous section. For simplicity, it has vanishing momentum, in contrast to the uniformly moving example of the previous section, so it stays put at the origin, and spreads--melts away.  I think all this is self evident, but one must have some familiarity with wave phenomena. Cuzkatzimhut (talk) 16:40, 21 February 2014 (UTC)
 * Probably you are right. But on the other hand Wikipedia is not intended for specialists (see WP:TECHNICAL). Those don't read Wikipedia, they have their own study books. I can understand that calculating cannot be done without specialised math, but the final formula should also be rewritten in a form which is understandable for readers with average math knowledge, perhaps for with examples. Perhaps I will add some, to what I understand, so if its wrong, please changed it to be right. DParlevliet (talk) 08:44, 25 February 2014 (UTC)
 * The cos+isin I added, and then deleted the isin because in the study book "Quantum Mechanics", from Richard Fitzpatrick, I found that "a general one-dimensional real wave function can be represented as the real part of a complex wave function... for ease of notation the 'take the real part' aspect is usually omitted" (page 17). Is that wrong? According the formula the width start with square root a, isn't it? And what is t=0? Can you choose this origin yourself? Can you give examples, for instance with an electron? DParlevliet (talk) 08:31, 26 February 2014 (UTC)
 * It is OK to focus on the real part of a complex exponential, but one must say so---one cannot write that a complex number equals its real part!! I see no harm in your original expansion that I reverted to, and the injunction to focus and plot the real part as of now. The origin of time is arbitrary, and yes, you can choose it yourself: if you just call the width at your time origin √a, then, at nonzero time, the width is what is given. The wave given might as well describe a free, stationary electron. If you want to translate it with uniform drift motion, you generalize the 1d formula of the previous section instead. Thanks for your alert on the missing sqrt sign. Fixed now. Cuzkatzimhut (talk) 11:40, 26 February 2014 (UTC)

What is the formula for the propability in x (distance)? DParlevliet (talk) 16:52, 26 February 2014 (UTC)
 * Well, as the article gives, an electron spread out in a Gaussian around the origin with width √a when you start your stopwatch, will have the spacial probability density in distance  r from the origin,
 * $$P(r)= |\Psi|^2= \left( {a \over \sqrt{a^2+(\hbar t/m)^2} }\right)^3  ~ e^{-r^2 a \over a^2 + (\hbar t/m)^2} ~,$$
 * that is, it will still be centered at the origin, but now spread to a fuzzball with this ever-increasing width, as provided. In infinite time, it will have spread away all over the place, still centered at the origin. That's just quantum mechanics, not something odd about wave propagation.  If you want it to be moving, uniformly (freely), you just translate the origin by k0t in the direction of your choice.   Cuzkatzimhut (talk) 19:50, 26 February 2014 (UTC)
 * Thanks. I thought to read somewhere that the width in x was not depending on t, so that is wrong. DParlevliet (talk) 20:05, 26 February 2014 (UTC)
 * Absolutely not! The phenomenon described is the notorious "spreading of wavepackets" in quantum mechanics. I took off the gratuitous/mysterious [m] in the formula for the time-evolved width of the probability density, though. Of course, at all times, the expectation =0 and independent of t.   Cuzkatzimhut (talk) 20:32, 26 February 2014 (UTC)
 * So in what units is the width expressed, meters of second? Or can it be one of them, if "a" is chosen to be time or distance.DParlevliet (talk) 20:58, 26 February 2014 (UTC)
 * Units of length, of course, meters if you wished. √a  was defined as a distance, and so is  ħt /m√a, when you parse out its units. In this section, all constants have been kept, that is, one is working in conventional unscaled units. Cuzkatzimhut (talk) 21:12, 26 February 2014 (UTC)

Please look to my example, which seems to be wrong. If one starts with a larger a, the width increase with time is smaller.DParlevliet (talk) 09:28, 27 February 2014 (UTC)
 * Yes, it's wrong and I took it out... Your electron is not stationary, so with zero energy, merely diffusing at the origin, if it were ejected from an atom. And I have no clue what your units of the spread ended up in, but they look wrong ---do you appreciate the m in the denominator of the evolute is the mass of the electron and not a meter? I think you might read up on wavepacket spreading from a decent book, like Pauli's, which I adduced in the refs. The principle you tried to illustrate, however, is sound: a less localized electron would indeed spread more slowly, exactly by the argument of the previous paragraph. Perhaps you could apply yourself to an article in atomic physics? Cuzkatzimhut (talk) 11:47, 27 February 2014 (UTC)
 * I think that if a formula is given, it also should be explained what it means. I calculated ħt /m√a = (1.05E-34 * t) / (9.1E-31 * 1E-10) = 1.1E6 * t in meters. So after 1 ms, the width would be 1 km. So what is wrong? DParlevliet (talk) 13:14, 27 February 2014 (UTC)
 * OK, so your m was meters and your t was in seconds. What makes you think the formula is wrong? Have you studied quantum mechanics? Indeed, as per your calculation, starting with a width of an Angstrom, you puff up your probability fuzzball in one picosecond to 10000 Angstroms. In infinite time, you have spread all over the universe, assuming you were initially localized in a notional box of one Angstrom and were left alone (zero potentials and interactions) for ever and ever in that location. Note a coherent state in a harmonic oscillator potential does not diffuse like this, in fact, it does not spread at all! The formula is right, and should be left alone, and a good quantum mechanics course teaches you how to apply it. I have a terrible feeling you are misusing this editorial Talk page as a physics newsgroup/forum, instead . Please leave the article alone and seek discussion with physicists in a different venue. This should terminate our exchanges. Cuzkatzimhut (talk) 14:40, 27 February 2014 (UTC)
 * Alright, no need to get irritated. We just have to wait for an editor who knows how to apply the formula. DParlevliet (talk) 18:26, 27 February 2014 (UTC)
 * The formula is fine, and applicable in dozens of QM contexts, like propagation in dielectrics. Wavepacket diffusion is the cornerstone of QM. A free electron, or a free photon, for that matter, in macroscopic excursions in vacuum has its wavefunction completely delocalized, and described by (un-normalizable!) plane waves. I suspect  what you are seeking might well not have a place in this article. Cuzkatzimhut (talk) 19:28, 27 February 2014 (UTC)
 * The article mentions "the wave packet ... is interpreted as a "probability wave", describing the probability that a particle or particles in a particular state will be measured to have a given position and momentum", while several pictures show a "Gaussian state moving in one dimension ... in free space." So I suppose the wave packet of an electron moving in free space also falls within this article, and is much used, so interesting. You will be right that this formula is not adequate (although it does not mention why), but I hope there will be an editor which can give the right formula for that situation. DParlevliet (talk) 21:01, 27 February 2014 (UTC)
 * I found that my calculation was right, although the result is still strange for me. DParlevliet (talk) 14:01, 3 March 2014 (UTC)
 * Of course your plugin evaluation of the formula is right. It reminds you that after picoseconds, the QM description of macroscopic electrons is usually less important, due to delocalization. This is merely the strange glory of Quantum Mechanics. People describe macroscopic electrons, as in the "older" TV tubes, with completely delocalized plane waves, just like distant photons. Below picosecond times, however, as in diffraction experiments, such formulas are crucial, as illustrated in interference experiments in Tomonaga's classic Quantum Mechanics, VII Ch 8,  §61. Your illustration edit looks useful.  Cuzkatzimhut (talk) 14:59, 3 March 2014 (UTC)
 * Regarding your dispute with Mattedia, the formula in the article, is does that depends on the speed of the electron? Or is there is different formula. DParlevliet (talk) 15:56, 3 March 2014 (UTC)
 * Well, the two formulas are related. He is really talking about the u(x,t) of the previous section, 2, with the dimensionful constants in and generalized to 3D.  Setting ko =0  there, you get the analog of the section 3 cases. The "relativistic mass" stunt is pointless, and relativistic mass is deprecated by Einstein, Okun, Wheeler, etc... If the system went very fast, the time dilation, so the transformation of the colocated t, would slow down the spreading, all the way to zero for ko ~ c.
 * But pulling this type of stunt and implying a formal "expansion" shortcut to a relativistic limit not treated here (since the KG eqn is left out, and implying solutions of unwritten equations whose dispersiveness has not been discussed, and whose connection to the Schr eqn is fractured), by stopping time and setting it to 0 is hardly meaningful, don't you think? In any case, a diverging relativistic mass in lieu of the time dilation is bizarre to start with...Cuzkatzimhut (talk) 16:14, 3 March 2014 (UTC)
 * As you know I am no QM scientist, but I try to understand what the formula means, how it looks like. If the electron is at rest, it has no wave, so then you cannot call it a "wave" packet. DParlevliet (talk) 21:56, 3 March 2014 (UTC)

You seem to be interested in this enough to warrant reading a short quantum mechanics introduction, though... Strictly speaking, you are right that the spreading Gaussian is not wavy, but it is a solution to a wave eqn (the Schr eqn), and its moving cousin u in the previous section has waves (in fact, almost magically, it melts away to a plane wave for large t), so by convention people would call it a wave. In addition, these things are often solved by use of Fourier analysis, which resolves everything in waves. Cuzkatzimhut (talk) 23:37, 3 March 2014 (UTC)
 * I did read a lot about QM, but explanations about these waves are very scare. Could you enter the formula for a mass particle with speed (or show where it is in the article)? DParlevliet (talk) 08:01, 4 March 2014 (UTC)

? the 2nd formula from the bottom, section 2, u(x,t). it is only in one dimension, and with m and ħ set equal to one (nondimensionalization) for simplicity, so then ko ~ v, so it is simple and retains all the qualitative features of the general case. (I moved the moving figure to that section where it belongs. It is not wavy, as it is the probability density in x space, and not the wavy wavefunction itself.) If you look at the center of the travelling wavepacket wavefunction and monitor its spacetime development, you can see the monochromatic wave emerging, as in Tomonaga's treatment. The phase, exp(i ...), piece of it, can be rewritten as
 * $$e^{\frac{1}{2it} \frac{(x-k_0 t)^2}{1+4t^2}} e^{\frac{ix^2}{2t}} $$

and, at the center of the spreading wave (enforcing the Gaussian "constraint"), it morphs to exp(ikox/2), where ko/2 is the phase velocity, not the group velocity in the Gaussian envelope. This is a messy "physicist proof" and should not be taken seriously if one is not used to proper handling of it, which is the reason it would be a bad idea to insinuate it into the article at all. Cuzkatzimhut (talk) 11:29, 4 March 2014 (UTC)
 * Yes, your edit is fine. I thought it was clear at the beginning of the section that "just centered at the origin" is talking about a nonmoving particle sitting at the origin with ko=0, and rapidly melting away, probability-wise.... Cuzkatzimhut (talk) 17:07, 4 March 2014 (UTC)
 * You are right, but most readers skip the math and jump to the final formula, so it is always good to repeat there all suppositions made during the math. About the moving photon, I suppose you mean the formula in chapter "Dispersive"? Is it possible to also write this with h/t/m/ etc. like the formula in "Gaussian wavepackets ...", so actual value could be filled in? If this is not possible without becoming messy, could you write as an example what the formula would be in the case an electron is emitted by an atom? DParlevliet (talk) 11:09, 5 March 2014 (UTC)

I'm sorry, this conversation is going nowhere, and I am wrapping it up. The example of the stationary electron is adequate; just translate the origin with the speed (group velocity of your choice) as you could have noticed form the "Dispersive" 1D subsection above: the spreading rate will not be affected by the motion--that is the whole point in skipping it in the next section. The exact units are in formula (41.32) of Tomonaga Quantum Mechanics, VII, North Holland, 1966, Ch 8,   § 41. Please do not modify the page. It does not need an example of free streaming semiclassical electrons described by wavepackets, an inferior description of them, as I demonstrated. Your present example reminds the reader when to stop thinking about wavepackets in a system (unless you want to do interference, as in T  § 61). You could give it a break and try to use good books better. Cuzkatzimhut (talk) 15:41, 5 March 2014 (UTC)

Avoid Edit War on Wavepacket limits
I already left a message to User talk:Mattedia on the unwarranted counter-reverts. Let us reach a consensus on this page first, instead of peremptory off-mainstream and unsound edits. The section involved describes a stationary, non-translating Gaussian wave packet at the origin, which arises as a solution to the NR Schroedinger equation. Relativizing a stationary bump by switching off its time dependence on the grounds of time dilation, arrived at by a spurious limit, readily seen in the Schr eqn solved, and its behavior in that limit, hardly means anything useful to the reader. If there were another article with relativistic dispersive eqns (which is what this section is all about), it might be useful to link its conclusions here. But, as it stands, the peremptory edits spatchcocked in are worse than meaningless. Moreover, introducing interactions really muddies the waters, as dispersion is suppressed even for coherent states of the NR quantum harmonic oscillator, where it has nothing to do with time dilations. Please brush up on Tomonaga, QM vII. If you wished to write a coherent, responsible remark on the nondispersive features of the KG, you could somehow insert it in the previous section, where the wavepacket is actually translating, or after the Airy wavetrain section. It is a lot of work to do it right, and confusing the reader with garbled, ill-considered, unattested flights of fancy is not cool. Cuzkatzimhut (talk) 18:34, 2 March 2014 (UTC)

§--Farhan babra (talk) 15:02, 15 October 2014 (UTC)

Wave packet in Quantum Mechanics
I want to add this topic, in detailed form(not in mathematical sense). The outline of my plan of work will be as follows:

Free particle wave packet - plane wave solutions are not normalisable, general solution is a combination(superposition) of these solutions.

Fourier transform pair - wave functions in position and momentum basis.

particle to be localise in small region of space => some distribution in k-space and vice-versa.

Evolution of wave packet with time. Group and phase velocities.

Gaussian wave packet. Definition, evolution, properties and uncertainty relations.

Spreading of a wave packet.

Note that this layout is flexible as of now, and thus subsequent addition of new subtopics may be possible depending upon the appropriate contents.

--Farhan babra (talk) 15:05, 15 October 2014 (UTC) — Preceding unsigned comment added by Farhan babra (talk • contribs) 14:59, 15 October 2014 (UTC)


 * All items you touch upon are already covered, addressed, or, wikilinked in the extant article. Make sure you propose your modifications here, in the talk page, before you actualize them in the article. The goal of the article is to be concise in its coverage and explanation, and to send the reader off to relevant articles, if there is a moot point to be explained elsewhere. Make sure this doesn't turn into a lengthy remedial QM course to laymen! Cuzkatzimhut (talk) 14:26, 16 October 2014 (UTC)


 * I think Cuzkatzimhut gave good advice and would like to add that there there is a need for lengthy remedial QM courses for everybody from laymen to undergraduate physics majors on Wikiversity and Wikipedia. I work on the latter, and have put a few "essays" on Quantum mechanics.  I prefer Wikiversity because I would like to see an evolution from Wikipedia as the world's biggest encyclopedia to Wikipedia/Wikiversity/Wikibooks as the world's largest encyclopedia/bookstore (with everything open source).  I just posted an article on the v: Casimir effect because I could see no good place for it in Wikipedia's article on it. --guyvan52 (talk) 13:03, 17 October 2014 (UTC)


 * I dont think so, wavepackets in context of quantum mechanics is not properly explained or illustrated anywhere in wikipedia. And I will try to be concise and relevant. The initial possible addition/modification will be first discussed here and then be finalised to put the proposed content/section in the article.

--Farhan babra (talk) 13:35, 17 October 2014 (UTC)


 * Sounds like a good plan. We need more quality writing on this subject. --guyvan52 (talk) 15:00, 17 October 2014 (UTC)


 * I decided to delete the section - Gaussian wave packets in Quantum Mechanics at first because its written very compactly. So I suppose one hardly bother to refer that section. --Farhan babra (talk) 13:59, 19 October 2014 (UTC)
 * No, don't deleted a whole section without having something new. It is Wiki rule to improve, not to delete.DParlevliet (talk) 13:12, 20 October 2014 (UTC)
 * Please desist! Compactness is a virtue, not a deficiency! I keep on insisting that a reader not proficient in quantum mechanics should not be here, but evidently you do not hear that. If you can find the suitable places in wikipedia that refer to the elliptical spots, please wikilink them first, before burdening the article with more baggage. An experienced, intelligent, well-meaning reader should be able to skim through the article quickly to get the message. The specific section you have taken a mysterious dislike to, in fact, summarized the issues quickly, inviting the motivated reader who wishes to know more to read on. Please, first do no harm by proposing small substitutions/improvements, first, to reassure the rest of us you are not plunging into a major rewrite. I suggest that you shape your improvements in your sandbox of your personal page, as per wikipedia policy, then invite people to make comments on it and approve it, and leave the actual article alone for now! Something tells me you really are thinking of a quantum mechanics tutorial, of which there are several, and you just found yourself in the wrong article.Cuzkatzimhut (talk) 13:47, 20 October 2014 (UTC)
 * Let me be the third editor to say the same thing. Please begin an article on your user page and leave a message on my personal talk page at User_talk:Guy_vandegrift.  I will help you find a place for it.--guyvan52 (talk) 15:23, 20 October 2014 (UTC)


 * Speaking of edits to this section, am I the only one bothered by the following equation?
 * $$ \psi(\mathbf{r}) = e^{-\mathbf{r}\cdot\mathbf{r}/ 2a}~ ,$$
 * I would have mentioned that this holds only at t=0, or even perhaps used the unorthodox notation:
 * $$ \psi(\mathbf{r},t=0) = e^{-\mathbf{r}\cdot\mathbf{r}/ 2a}~ ,$$
 * This is just a suggestion.  I never make edits that are not absolutely necessary.--guyvan52 (talk) 15:35, 20 October 2014 (UTC)


 * Yes, I agree. I would also be concise. I was just suggesting. But if someone is profound in quantum mechanics, then he using this as a reference is rather peculiar. As a student or others not so good in the subject will very likely to refer wiki in this regard. So a proper description and elaboration will be more helpful. If one is able to explain the reasonable amount of information in minimal words, then its ok but for the very sake compactness, skipping even the crucial ideas won't serve the purpose. So, following what you said, I am not disliking the section, rather I want it to be more descriptive on physical grounds, not mathematically rigorous- one can do calculations on his own. I am sure it isn,t a tutorial, one can go through the books or notes. But my idea is to put a physical meaning and attributes to wave packets in quantum mechanics. And, I will take your suggestion, instead of deleting I will try to elaborate it appropriately and add some new materials which I already discussed in as compact form as possible but without destroying its very purpose and essence. --Farhan babra (talk) 08:53, 21 October 2014 (UTC)

According the Copenhagen convention there is no physical explanation, anyway not in our known classical physics. It only acts according formula in our world. So don't be surpised if edits are refused because of being personal philosophy(so no science). If you want to try, don't change the section but add a new one and present it here or in the sandbox first. DParlevliet (talk) 13:14, 21 October 2014 (UTC)


 * The undiscussed tutorial inserted appears misguided and unhelpful. The Schroedinger equation was already given in the preceding section, without the spurious "benefit" of ħ   and m, which are not really used here, but only provide dimensions, which are self-evident from nondimensionalization. Worse yet, the discussion is in only one dimension, which has already been completed above, instead of fleshing out the 3D case which follows. I think the insert sections completely disorient the reader getting the basic idea, and drag him into a substandard tutorial of QM, rambling over issues unrelated to the article.
 * In any case, the trashing of the reference tree is unacceptable, and references should be left alone. I cannot imagine why the vanity insert of somebody's lecture notes clarifies wave packets more.
 * The gratuitous mumbling on potentials when free propagation is discussed is misguided. Already, the function of the oscillator potential in stabilizing Schroedinger wave packets (coherent states) has been alluded in a footnote. Maybe salutary work could be done in that article. I believe the inserts I've seen so far do not belong---and besides they should have been proposed on this page first. I do not revert them, to prevent edit sparring, but the author should remove them himself. Cuzkatzimhut (talk) 19:07, 23 October 2014 (UTC)
 * As a lay man I found the additions quite useful and understand it better now. I hope more of the article can be made more understandable. If I may quote wiki rules: "A Wikipedia article should not be presented on the assumption that the reader is well versed in the topic's field" and "Texts should be written for everyday readers, not just for academics". Wikipedia is to learn, by as many readers as possible. It is not a reference for science, who will not use Wiki anyway. They have their own books. DParlevliet (talk) 19:49, 23 October 2014 (UTC)

So, then, free particle was a waste of time? Cuzkatzimhut (talk) 20:10, 23 October 2014 (UTC)
 * Perhaps, because they handle a wave packet but it still lacks the explanation Farhan added. I don't say this is the optimal of definite edit, but it concerns the wave packet and is a good explanation for the target group of Wikipedia, so it should be somewhere in this article. DParlevliet (talk) 07:17, 24 October 2014 (UTC)


 * First of all, I have yet to do some more editing. A line or a equation used again is not a mistake, rather a convenience. As DParlevliet mentioned, wikipedia is meant for providing information, which I tried give in simpler words keeping in mind the general audience not expert in quantum mechanics. I did not used any lecture notes except for some mathematical expression, it reflects what I grasped and understand about it. Experts can always refer the books. Furthermore, explaining the problem in one dimensional is more helpful and provides better physical insight. Generalisation is easy and can be found in texts. And I had not trashed any references. I just wanted to add a reference and it somehow happened, issue  which I was unable to solve, sorry for that.  --Farhan babra (talk) 13:15, 24 October 2014 (UTC) — Preceding unsigned comment added by 158.144.57.11 (talk) 11:45, 24 October 2014 (UTC)


 * OK, if someone finds it useful, I took the Solomonic decision to keep it there, against my better judgment, of course, and put it in a collapsible box ("Background") so it does not destroy the continuity between the two sections that the entire article relies on, and lower the article's rating to even lower than a C. This way, the novice reader who is too reluctant to look up the elementary basics of the Schroedinger equation in the appropriate wikis can have a quick refresher in situ, and be reminded that plane waves are not normalizable. But please experiment with references in your sandbox and not on the article page.Cuzkatzimhut (talk) 14:28, 24 October 2014 (UTC)
 * I have restored to the Farhan version. Cuzkatzimhut is not the owner nor the one who can make Solomonic decisions. Perhaps I restored too much, but then don't do too much changes in a short time. Remember again what I told about wikirules. There are no wikirules against repeated information. Whatever is useful for the understanding of the topic is welcome here and must not be hided in collapsible box. If there is mayor critic against Frahan edits it must be voted here, not decide bij one.DParlevliet (talk) 20:03, 24 October 2014 (UTC)

I am sorry you have to resort to peremptory rollbacks as a battle of wills. I tried to accommodate everyone, and I do not believe I effaced any information by putting it in a collapsible box. I believe your disruptive edits is of the kind already demonstrated in your banishment records. Please discuss the point first. That was the very consensus of these tutorial remarks, tofirst appear here, or in the author's sandbox, to be discussed and corrected (language and logic), instead of throwing them in the middle of an oft-read article while mucking up its references and leaving the site, expecting someone else to fix it. The information is there, but not forced on the reader with a minimum of experience with wave propagation and QM. Evidently, you decided that references you retrashed are not useful to the public. Cuzkatzimhut (talk) 20:06, 24 October 2014 (UTC)
 * Valuable information for the core public of Wikipedia should not be hidden in a collapsible box. You also did not discuss your changes here or in your sandbox. If you want to improve language and logic that is no problem, but don't hide it. Keep to Wiki rules. DParlevliet (talk) 09:12, 25 October 2014 (UTC)
 * You appear to be echoing back to me my points, without regard to the essential facts underlying them---tendentious lawyering? You and the author appear to be the only ones arguing for the "value" of the review information. Hideboxes make such supplementary information available to the unprepared reader, while sparing the mainstream reader. It is alarming you are misreading the "Wiki rules" this way, but not surprising, given a somewhat problematic record, there. Small edits do not belong in anyone's sandbox.
 * However, as per consensus, Farhan was supposed to provide his major inserts here or in his sandbox, first; that was the point of the prefatory discussion! But he did not, and he failed to move them when asked, making a mockery of the whole earlier discussion. Do you not understand the difference? I did not delete them, but put them in a hidebox to protect mainstream readers. Your impulsive and peremptory disruptive edits made the references unavailable to the reader, and, as I said, the insert tutorial subverted the crucial continuity between the two central sections, one of which explicates the other one preceding it. Quantum mechanics related pages in WP have a loaded history of flakey, inexpert content piled up on them by droves of passers by, exhilarated by the apparent lack of quality safeguards, and can degrade very quickly (see the deserved rating of this one) if left to the disruptors. As a result, perfectly plausible pages turn into pshycho-physico-babble funhouse mirrors overnight. As a start, we can 'now vet future large inserts here, before they reach the general reader.  There is no other reason for not vetting inserts here except peremptory efforts to bypass editorial review. Please do not repeat any stuff about inexpert readers coming to this page to learn basic QM. There are plenty of tutorials on QM at every level in WP. It's just not OK to come here perceiving a lack of maintenance and patrol presence and try one's chance of replicating other articles where intrusion might be less straightforward.  Cuzkatzimhut (talk) 11:07, 25 October 2014 (UTC)


 * I suggest the author(s) of the hidden material create a new article called Wavepackets in quantum mechanics. Leave a note on my talk page and I will make suggestions (I have 21 refereed publications and know how to write). What you wrote and what was hidden into a box was clear and certainly needed on Wikipedia.  But each 'pedia article has its own flavor and focus.  Putting two focuses into the same article makes it difficult for both intended audiences.  --guyvan52 (talk) 13:48, 25 October 2014 (UTC)
 * Outstanding proposal. This was my intent with the Hidebox. By the way, please do brush up on hide boxes, everyone. They are used to spectacular effect in the uncertainty principle, etc. They are not a mark of marginalization, as unreasonably implied. There are also a plethora of tutorials on QM all over WP, but scattered all over the place, and I did not presume to pick the optimal one, where basic solution techniques of QM are discussed. Ideally, people are supposed to teach QM out of WP... but this here is a special topic, intended to instruct readers on dispersiveness versus nondispersiveness, and the crucial features of diffusion of free systems, with the paradoxical aside of the Airy wave train on the side. I wish the project luck, and I would recommend the guidance on Draft, which should have been utilized in the first place. Cuzkatzimhut (talk) 14:26, 25 October 2014 (UTC)

On Wikiversity I use the quick 'n dirty and. Also, I saw something many years ago about non-dispersive wavepackets in AJP, but never read the article. I suppose it was about the Airy function solution. I don't understand why these solutions are  unnormalizable. I am not asking you to explain this in the talk page, but it's something you might want to put into the article. And yes, professors use Wikipedia. The ones that do just don't crow about it the way the ones who don't do.--guyvan52 (talk) 15:01, 25 October 2014 (UTC)
 * Yes you probably saw the classic AJP of Berry and Balazs. Thanks for letting me off the hook of the explanation, here, more or less straightforward; but sticking in more technical stuff in this article would make me guilty of what is being charged above, namely making the article too unappealing to the novice... The primary AJP source is adequate, I think, and well worth going to, for the motivated reader. In phase space, however, that statement is manifest, given the effective variable being just one combination, and the integral being double (over phase space), so one of the two integrations will yield infinity. Cuzkatzimhut (talk) 15:58, 25 October 2014 (UTC) Accordingly, I added the "half-line proof" which makes wavefunction non-normalizability an explicit consequence of the Wigner function's dependence on its variables. This is not overtly technical---it just invites the knowledgeable reader to see the obvious, while sparing the novice.Cuzkatzimhut (talk) 11:16, 26 October 2014 (UTC)

Would this link violate copyright laws?
Since this article compare/contrasts dispersive and non-dispersive wavepackets, this link might be interesting. I wrote it for AJP and put a copy of it on my website, and AJP has never complained. I know Wikipedia has higher standards ...would it be "legal" to put a link into this article? --guyvan52 (talk) 19:36, 24 October 2014 (UTC)
 * One could certainly adduce that publication's ref; and then allow a link to your online fair use copy. I believe you have the right to post such on your own (author's) website, and then link it from WP. If there were an unlikely breach of copyright, it would be by your site, and certainly not by WP! In my experience, countless authors have posted e-prints of their published work on Researchgate. Cuzkatzimhut (talk) 19:49, 24 October 2014 (UTC)
 * "Review" is just fine. I was trying to be minimal and restore the missing articles (syntax) and upend the inverted logic; so I did not fuss the wording too much. The page gets over 200 hits a day, so restoring the references yanked out of the readers' reach was a priority, rather than subtler wordsmithing... Thanks. Cuzkatzimhut (talk) 00:04, 25 October 2014 (UTC)

Deletion of misplaced tutorial
A QM tutorial has recently been inserted against consensus and good advice. The section is problematic for several reasons
 * It doesn't belong here. This is not a textbook
 * It is written by a new user who is clearly using a Wikipedia article to explain to himself/herself unfamiliar terms. This observation is based on errors in the tutorial a new-comer is likely to make. (And NO, I'm not going into detail.)
 * It is poorly written. Among other problems, there are inline equations in LaTeX, and there is the conversational we, which is badly suited for a Wikipedia article.
 * There are no references. This, by definition, especially taken together with the above observations, qualifies this as original research.

Having said that, let me make it clear that I am a proponent of having detailed explanatory articles with proof outlines. This may mean that I stretch the rules in favor of my viewpoint, but I never break the rules (severely). Most importantly, I don't go against consensus. The way that the tutorial has evolved is unacceptable. The fact that one user has expressed appreciation of it is one more reason to delete it, since it isn't entirely correct, not referenced, original research, etc. YohanN7 (talk) 01:14, 26 October 2014 (UTC)

The deleted material for reference:

In quantum mechanics, when a particle is not subjected to any external force (i.e., it exists in a region of constant potential, which can always be chosen to be zero, because one has the freedom of assigning zero potential to  all points in space), the free Schrödinger equation describing it is given by
 * $$ - \frac{\hbar^2}{2m} \nabla^2 \ \psi(\mathbf{r}, t) = i\hbar\frac{\partial}{\partial t} \psi (\mathbf{r}, t) ~,$$

which, in one dimension, has the general solution of the form
 * $$ \psi(x,t) = A e^{i(kx-\omega t)}~.$$

This represents plane waves. These plane wave solutions are not normalisable because the resulting probability density is uniform everywhere. The Hamiltonian commutes with the momentum operator $$ \hat{p} $$, whose eigenfunctions are e^{ikx}. There exist two methods of constructing normalisable solutions (wave functions). First, one may restrict the wave functions within a region (see Particle in a box).

Another way is to take linear combinations of plane waves with  several  k (momenta), called wave packets,
 * $$  \psi(x,0) = \frac{1}{\sqrt{2\pi}} \int^{\,\infty}_{-\infty}   \phi(k) ~ e^{ikx} \,dk $$.

where  $$  \phi(k) \,$$ is the wave function of the same particle in the momentum basis. The $$ \psi(x) \,$$ can be normalised by choosing appropriate $$  \phi(k)$$s. $$ \phi(k) \,$$ and $$  \psi(x) \,$$  constitute a fourier transform pair for t=0 (we are given the wave packet at some t, which can be taken to be zero, and asked to determine the wave function at a later time). One may check that both are normalised by the application of Plancherel theorem.

If we require to localise the particle in some finite region, we need some of the coefficients $$  \phi(k) \,$$ to be non-zero, i.e, a superposition of waves with different k's. Therefore, the more precise the location of particle, the more plane waves needed with a wide range of momenta, the less precise the momentum of the particle---the Heisenberg Uncertainty inequality.

Evolution of wave packet

Each energy eigen­function evolves by acquiring a phase factor, where $$ \omega(k) =\frac{\hbar^2k^2}{2m} \, $$ corresponds to the energy eigenvalue associated with k.  $$  \omega(k)\, $$  can have a more complex form if the particle encounters a different potential, potential barrier, for example. Ascribing this time factor to each constituent eigenfunction, yields the travelling wave packet.
 * $$  \psi(x,t) = \frac{1}{\sqrt{2\pi}} \int^{\,\infty}_{-\infty}   \phi(k) ~ e^{i(kx-\omega(k)t)} \,dk $$.

Suppose the wave packet is peaked around at some particular k; expanding $$ \omega(k)\, $$ around its vicinity  and retaining up to two terms, yields a moving wave packet (the envelope of plane waves) with a group velocity
 * $$v_g = \frac{\partial \omega}{\partial k}\,$$.

By contrast, each plane wave travels with the corresponding phase velocity $$v_p = \frac{\omega}{k}\,$$. The group velocity, i.e., the apparent velocity of wave packet with which it travels, is twice the phase velocity, the velocity of the constituents.

YohanN7 (talk) 01:18, 26 October 2014 (UTC)

"Reinsertion" of misplaced tutorial
I moved it to Quantum_mechanics under the title Wavepackets and uncertainty in quantum mechanics. The author is free to move it again, but the "plethora" of tutorials on the 'pedia, is so disperse and scattered in style that it is almost useless. Let's set learning free by assembling and organizing the best of these tutorials on the 'versity. If and when the tutorial is polished, we can link to it from this Wave packet article.--guyvan52 (talk) 11:47, 26 October 2014 (UTC)

Claims for which I have no reference (or proof)
The article says
 * The wave-like and particle-like characteristics never manifest themselves at the same time (i.e. in the same experiment).

This is, AFAIK true, but I can't think of a reference that says it verbatim. (I put the statement there so blame me )
 * These may be murky waters, cf the Afshar experiment, where they fuss it... Cuzkatzimhut (talk) 11:04, 27 October 2014 (UTC)
 * Acceptable formulation now? YohanN7 (talk) 12:47, 27 October 2014 (UTC)
 * I'm not sure it is important, but I would personally avoid implicit judgements, so, then, say something like: "(... experiment——see however the Afshar experiment and lively discussion of it.)" It is all a parenthetical footnote... If the reader were interested, she/he could pursue the links... Cuzkatzimhut (talk) 13:45, 27 October 2014 (UTC)
 * Hm. There were no intentional implicit judgments, but I reformulated according to your suggestion. (Perhaps the Afshar experiment should be considered unverified. I'm certainly not up to date.) YohanN7 (talk) 14:21, 27 October 2014 (UTC)

More interesting : There is no such thing as a particle "perfectly" localized in both position space and momentum space. By "perfectly", I mean probability zero (not small) to find the particle having momentum or position outside a finite interval. This is trivial for analytic functions (for if $Ψ = 0$ in an interval, then $Ψ ≡ 0$). For $C^{k}$ functions it is probably easy to show (some derivative will yield delta functions (or worse) implying an infinite spread in the other domain). But for smooth functions, I don't know how to show this, but I'm sure I have read the statement somewhere. The crux of the matter is probably to prove that a smooth bump function (of any sort) has infinite extent in the Fourier transformed domain.

This last issue may be worthy of inclusion in the article because nine out of ten introductory QM texts give the impression that "perfect" localization in both domains is possible (which is wrong, provided I'm right here). YohanN7 (talk) 01:25, 27 October 2014 (UTC)

"The initial uncertainty ΔxΔp = ħ/2 has now increased by a factor of ħt/ma."
In the section Wave packet, it is written: "The initial uncertainty ΔxΔp = ħ/2 has now increased by a factor of ħt/ma." Should this not be "a factor of $$e^{\hbar t/ma}$$"? Sławomir Biały (talk) 21:51, 13 December 2016 (UTC)
 * I'm not sure what you might have in mind... The comparison is between the t=0 width of the gaussian, versus the very large t width of the same gaussian, bigger than the original by a factor of ħt/ma, as also exhibited in the square root of the preceding paragraph. Cuzkatzimhut (talk) 22:35, 13 December 2016 (UTC)
 * Sorry, I interpreted it as something intended as an exact formula (in particular, one that cannot hold when t=0!), when it is actually intended asymptotically. I will add a small clarification.   Sławomir Biały  (talk) 23:34, 13 December 2016 (UTC)
 * That's fine. The exact formula is in the previous paragraph, but in this (last) one, one just hacks it without calculation... Cuzkatzimhut (talk) 23:40, 13 December 2016 (UTC)
 * Yes, I see that now. Thanks,  Sławomir Biały  (talk) 23:42, 13 December 2016 (UTC)

Units
h and m are set to 1 in the free propagator equation, but not in the Gaussian Wave packet part. Maybe someone who understands the equations better than me can made that uniform? -Charlie- (talk) 18:08, 3 September 2017 (UTC)


 * Why, on earth? to deprive the reader of the basic ability of switching back and forth, effortlessly and naturally, from standard to natural units of the problem? Units are taken out to simplify, when superfluous, and put back in, selectively, to guide reinstatement, or simplify a limit, if needed. Why is it needed, here? What does one gain by uniformization of units, and why would there unique, unambiguous reinstatement not be natural? Cuzkatzimhut (talk) 19:40, 3 September 2017 (UTC)


 * Sorry, it looked for me that the unit switch was done unintended here. What about a short comment when units switch in the calculation? -Charlie- (talk) 08:54, 4 September 2017 (UTC)

Images appear to be out of place.
The images are mis-positioned. The image of quantum tunneling is out of place here.

I will make some edits to fix what I see is broken. I'll live this for discussion. Johnjbarton (talk) 03:35, 28 May 2023 (UTC)


 * Johnjbarton (talk) 02:52, 23 June 2023 (UTC)