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Solutions of SE

Has it been proved that analytical solutions to the time-independent schrodinger equations of molecular systems are impossible, or is it just the case that solutions have not been found. 212.140.167.98 13:29, 31 July 2007 (UTC)

An "...electronic term...makes analytical solution of the Schrödinger equation impossible for many-electron systems." in Quantum Chemistry in the 21st Century (IUPAC). David Spector (talk) 20:40, 26 November 2011 (UTC)

Solutions of SE

It is stated at the beginning of the article that the Schrodinger Eqn applies to macroscopic systems or even the whole universe. If that is so, can we please site an example? If not, that needs to be deleted.

The citation link is broken, but I think it should point to http://prola.aps.org/pdf/PR/v28/i6/p1049_1

 —Preceding unsigned comment added by 64.134.160.237 (talk) 00:49, 24 June 2010 (UTC)

I believe the theory is that these equations apply to all physical and chemical systems. However, solutions have only been found for a few cases simple enough to permit an analytical solution, such as the energy levels of the electron in a hydrogen atom. But you can't discard basic physics which has been verified again and again in simple cases, and has never been disproven, just because it is intractable in the general case. That is a sure path to ignorance. Besides, there are surely some closed-form solutions in macroscopic quantum phenomena such as superfluidity, superconductivity, or Bose–Einstein condensates such as clouds of laser-stopped photons. David Spector (talk) 20:50, 26 November 2011 (UTC)

Equation box

In the equation box in the opening paragraph, i'd like to change the text underneath from:

Two forms of the Schrödinger equation

to...

Two forms of the Schrödinger equation, the time-independent form above, and the time-dependent form below

...or something similar, unless anyone has any objections. Larryisgood (talk) 22:25, 20 March 2011 (UTC)

Time-dependent Schrödinger equation

${\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi ={\hat {H}}\Psi }$

Time-independent Schrödinger equation

${\displaystyle E\Psi ={\hat {H}}\Psi }$

<-- My suggestion. :-) --Steve (talk) 23:43, 20 March 2011 (UTC)

Looks good to me! Larryisgood (talk) 21:32, 21 March 2011 (UTC)

Introduction mistake

"For systems in a stationary state (i.e., where the Hamiltonian is explicitly dependent on time)" should be changed to "For systems in a stationary state (i.e., where the Hamiltonian is not explicitly dependent on time)" Wuliowng (talk) 15:20, 21 April 2011 (UTC)

Done. --Bduke (Discussion) 23:52, 21 April 2011 (UTC)

Wave function is not the probability amplitude

It is stated in multiple places that the wave function as opposed to the square of its magnitude is the probability amplitude. — Preceding unsigned comment added by Gradatmit (talk • contribs) 16:35, 15 June 2011 (UTC)

Yes, per probability amplitude: “In quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density.” So I think the article is right. At least, that is the usage of “probability amplitude” I am familiar with. — Tobias Bergemann (talk) 08:47, 16 June 2011 (UTC)

Schrödinger Equation for distributed quantum objects

Recently, the ststistical interpretation of the wave function has been questioned arXiv:1111.3328v1

MichaelEngels (talk) 17:07, 28 December 2011 (UTC)

I think this was always the case. IRWolfie- (talk) 13:38, 2 January 2012 (UTC)

"Short heuristic derivation"

It says "citation needed", so soon I intend to tweak it and add the referance QM, Eisberg & Resnick et al + the simalar derivation. Its in there, but might be a bit longer (also its called a "plausibility argument").-- F = q(E + v × B) 02:19, 22 December 2011 (UTC)

btw in the edit history I seem to have forgotten to write "clean up" after slight...-- F = q(E + v × B) 02:21, 22 December 2011 (UTC)

Before re-writing the derivation - I would propose blending togther the subsections in the Versions section with the corresponding subsections in the The Schrödinger equation section. It has already been pointed out that there is too much repetition above (a long time ago) and this hasn't really improved. For the size of the article, there isn't as much informaiton as there could be due to repeating the various forms of the schro eqn.

Would anyone favour/oppose such a change? I have requested here for further opinions.-- F = q(E + v × B) 23:00, 23 December 2011 (UTC)

I thought I would just do it. Please don't panic or worry or get angry - nothing has been thrown away (except the repetition of the same equation twice in some cases - which was the whole point). It has been blended into the extension. The apparent repetition is due to the different number of spatial dimensions, particles, and time-dependence.-- F = q(E + v × B) 16:22, 26 December 2011 (UTC)

Btw I removed a completley useless statement in the properties section (probability conservation + continuity equation):

"The reason that the Schrödinger equation admits a probability flux is because all the hopping is local and forward in time."

What does it mean? Also the relativity section has been blended into the properties section, since its motsly about how the schro eqn doesn;t transform under Galilean and Lorentz transformations etc.

-- F = q(E + v × B) 18:48, 26 December 2011 (UTC)

No value

This entry is of no value to a layman. It strikes me more as a vanity project/pissing contest for physicists. For non-physicists, it makes no meaningful effort to explain the concepts and is inpenetrable for the type of person who would look up a topic such as this on Misplaced Pages. I came here for more information after reading about the concept in a Hawkings book. If he can lower himself to explain the basics to the ignorant masses surely the Wiki community can at least make the effort. — Preceding unsigned comment added by 24.148.43.34 (talk) 08:17, 1 January 2012 (UTC)

Which entry do you mean? The derivation? The whole article? And in what way do you find it doesn't explain the principles? Please say, if you would like it to be improved. -- F = q(E + v × B) 16:09, 1 January 2012 (UTC)

Once again, I have requested further input from the wikiproject talk page.

PS - It was NOT a vain attemt to show off - irrelavent of what user:24.148.43.34 thinks. As explained above - the whole point of the re-write was to make the construction of the equation more rigourous, and create a systematic outline of the various forms of the schro eqn + properties... -- F = q(E + v × B) 16:25, 1 January 2012 (UTC)

The IP hasn't offered any contructive criticism so I would ignore the comment. IRWolfie- (talk) 13:36, 2 January 2012 (UTC)

Well... at least thanks for your opinion =) I look forward to those of the other editors... -- F = q(E + v × B) 14:55, 2 January 2012 (UTC)

I'm not an editor of this article, but I am a WP editor and am somewhat knowledgeable in physics. I happen to agree with the spirit of this comment. While it is true that some subjects (such as quantum mechanics) are specialized, and a more or less full understanding requires lots of mathematics, there is no reason that such difficult articles couldn't have two parts, one accessible to anyone, and one meant for a specialist in the particular field of study. There have been a few attempts to do this, such as Introduction to quantum mechanics for Quantum mechanics, but they're not really very successful, because they really don't reach "laypeople". For examples of what I mean, special relativity could be described in its first paragraph as "the strange way in which the laws of physics behave when things move very fast," and quantum mechanics could be described in its first paragraph as "the strange way in which the laws of physics behave when things are very small or very low in energy." To a physicist, "the strange way" is very unsatisfying. It is like a teaser for a movie that doesn't actually say what the movie is about. But to an "ordinary person" wanting a handle on a strange-sounding concept, such a superficial description might well be just the thing they need.

This article could certainly be presented in summary form using words understandable to any intelligent and educated person, such as someone who has read a book by Prof. Hawking. If all difficult articles were presented in two or three parts corresponding to levels of difficulty, WP would be helpful to anyone, and anyone could choose how deeply to study any particular topic. Such a change could make WP useful to a much wider range of people and for a wider range of purposes without compromising the coverage, content, and quality of WP. David Spector (talk) 21:15, 2 January 2012 (UTC)

Your absolutley right, and your feedback it accouted for at the wikiproject page - thanks. =) -- F = q(E + v × B) 22:05, 2 January 2012 (UTC)

Some general comments:

It seems to me like the "Special cases" section does very little to teach people about the Schrodinger equation. Instead, it mostly teaches people about the various forms that a Hamiltonian operator can take. (I could be wrong.) I suggest moving much of it to Hamiltonian (quantum mechanics). Dropping some of these examples will do a lot to make the article less intimidating-looking.

I'm nervous about the relativity section. The equation i hbar d psi /dt = H psi is true in relativistic quantum field theory, isn't it? When people say "S.E. doesn't apply for relativistic situations", they really mean "S.E. with the normal nonrelativistic single-particle Hamiltonian doesn't apply for relativistic situations." i.e., it's the Hamiltonian that's nonrelativistic not the "Schrodinger equation" itself as defined at the start of the article.

An area where the article could use expansion is discussing S.E. as a wave equation--how is it mathematically similar to other wave equations, how is it physically similar? For example, right now diffraction and interference are not even mentioned! And perhaps, how is it different from other wave equations--e.g. why is the Schrodinger wave equation a first-order differential equation, whereas other wave equations are second-order? How does wave-particle duality come about?

Another area for expansion is discussing the fundamental connection between energy and frequency more broadly... For starters, the fact that particle physicists don't even bother distinguishing between the two. (Natural units uses hbar=1.) --Steve (talk) 02:31, 3 January 2012 (UTC)

• Yes - I’m aware about the relativity point you mention:

${\displaystyle {\hat {H}}\Psi =i\hbar {\frac {\partial \Psi }{\partial t}}}$

in general

${\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V}$

for the Schrodinger Hamiltonian,

${\displaystyle {\hat {H}}=c{\boldsymbol {\alpha }}\cdot {\mathbf {\hat {p}}}+mc^{2}}$

for the Dirac Hamiltonian......would you propose to write that? its in the Dirac equation article anyway...

• Also all the different forms of the Hamiltonian are already in the article you link to (and more). Its just that in this article, people can expand on them, like the H and He atoms under the appropriate headings, or the free particle in one or three dimensions. Also it takes the pain out of repeatedly saying on the spot "for one particle in three dimensions, time-independent potential the schro eqn is" etc by systematically showing them in the current way - all different forms in one place. I really don't know about inclusion anymore, it was admitted at the wikiproject physics talk page that it was all too intimidating but someone proposed we keep it...
• I was thinking about mentioning diffraction + wave-particle duality..... then thought it may detract from the article as the schro eqn to the wavefunction...
• Also the schro eqn is a mixed order diff eqn, not just 1st order in time but 2nd order in space.
• Would we need to mention energy and frequency anymore than currently is? the de Broglie and Planck-Einstein relations are already there, would explaining it further here instead of the main article (de Broglie relations) not detract from this article? probably/not?
• And do we really need natural units? I wasn't part of the argument above, but I would rather it didn't happen again...

your opinions are much respected though =) , I’m simply tired... If you would like to go for what you say then please do!.. (btw - to get rid of the columns just look for "multicol", I used them to save room because there was so much empty space without them).

-- F = q(E + v × B) 03:04, 3 January 2012 (UTC)

I tried to clear up the problems with relativity, hopefully that’s clearer. And as gently as I could, some very simple mathematics was introduced to provide the reader the ultimate structure to the equation: Total Energy = Kinetic + Potential Energy = Hamiltonian, and in addition introduce the operators one step at a time, making them resemble the classical equations and more recognizable, in doing so illustrate the Hamiltonian is what makes the schro equn N/A to relativity. Who knows, maybe a step closer to success this time?..... =|

Btw, I forgot to mention previously that I do know what nat. units are, and that there are several systems for them, but can't remember every single convention inside-out or even use them (yet). The only normalizations to unity I can remember are ħ = c = 1 and sometimes G = k = 1 also, and can just about translate some simple quantum or relativistic equations in books written in nat. units to SI by dim. analysis... That’s just my background, I would rather not have nat. units but it’s up to others if they want them - I don't mind. It would definitely make it easier for non-experts, which is the current theme of improvement for this article (at the wikiproject talk page): making it simpler without dumbing it down...-- F = q(E + v × B) 11:45, 3 January 2012 (UTC)

The only aspect of natural units relevant here is that physicists sometimes use hbar=1, and they use it because of the Schrodinger equation (in my view anyway). I am suggesting to discuss why physicists sometimes use hbar=1...I am not advocating to write the S.E. using hbar=1 in the first place, for the reasons discussed above.

I don't think the wave-like behavior of particles should be extensively discussed at wavefunction. Particles don't behave like waves because they are mathematically described by a wavefunction and the first four letters of wavefunction is "wave". They behave like waves because they obey the S.E. which is a wave equation. Maybe people are already familiar with wave-particle duality--that particles behave like waves--but only the S.E. allows them to know exactly how the waves propagate and behave. Maybe File:Doubleslit.gif ?

I hadn't seen matter wave article before. Maybe you're right about energy-frequency discussion going there, hmm.

Another potential topic that could be briefly discussed is to link the T.I.S.E. to the mathematical concept of spectrum of an operator. And another is the link to Spectral lines, which is currently mentioned only in passing. I haven't thought these through, just ideas. :-) --Steve (talk) 13:27, 3 January 2012 (UTC)

These points are fair enough. Apologies for not understanding all of what you meant =(

• Your correct about the wave-particle duality due the SE, the wavefunction article is very long as it is and focuses more on the wavefunction as a mathematical construct. The SE lead section or a new section can explain about the duality and its connection to the SE, and have the main link to that article. That animation you mention would do well, provided its fine with the creator.
• Mention of natural units in the article without using them shouldn't be any harm, in fact it should benefit because the reader will know how physicists simplify the calculations, or not differ between E and ω or p and k allowing interchangable use etc.
• The extra links about spectra of operators can be added, and so can spectral lines, when the content is expanded/re-written in the TISE section - both are certainly relavant.

-- F = q(E + v × B) 14:23, 3 January 2012 (UTC)

Column format

I removed the column formatting in the body of the text, for two reasons:

1. Although I can't find any rule against it in the style guidelines, it is highly unusual – and generally WP pages should have a uniform appearance.
2. It probably makes the page difficult to read for people using PDA's.

However, column formatting is common for lists like those in /* See also */, so I applied it there. RockMagnetist (talk) 16:53, 3 January 2012 (UTC)

It doesn't matter, its sorted out now.

First - the examples will be sorted out once and for all, which I have done now. Then the new additions can be done. I think its resonable to keep all of the examples and versions, otherwise its unclear what the applications of the SE are. In principle they could be a list of links, but thats boring and its better not to remove them now that they're already there. It serves as a conveinent summary of where the TISE is exactly soluble. -- F = q(E + v × B) 17:23, 3 January 2012 (UTC)

Sorry about the Solution methods and See also sections, editing took so long that you had edited before I finished, and so your changes were effectively cancelled when I clicked save. I just restored your changes now.-- F = q(E + v × B) 17:31, 3 January 2012 (UTC)

The wave-particle duality section will soon/now be introduced.-- F = q(E + v × B) 17:50, 3 January 2012 (UTC)

I'm so sorry for cancelling your edits again (and wavelengths this time)... it took ages to create the new section so by the time I was finished you had done loads... when I clicked save I cancelled all edits again. I just undid mine and inserted the new section at the same time, so all your edits should be preserved.-- F = q(E + v × B) 21:04, 3 January 2012 (UTC)

Thank you. Sorry I'm causing you this trouble – I think your edits are more important so I'll get out of your way for a while. RockMagnetist (talk) 21:39, 3 January 2012 (UTC)

Absolutley not at all. =) I'm the one that takes too long. -- F = q(E + v × B) 21:44, 3 January 2012 (UTC)

Well, your edits are addressing content, and I consider that more important. On another note - could you please have a look at the indentation guidelines for talk pages? Indentation really helps to identify threads in a discussion. RockMagnetist (talk) 21:57, 3 January 2012 (UTC)

I looked at the guidelines, they're nitpicky but I can live with it. Apologies for that. Anyway, I'm pretty much finished for now. We'll see what more content to add soon from other editors, if any, unless there is something you would like to add now. =)-- F = q(E + v × B) 22:03, 3 January 2012 (UTC)

There might be. I followed the link to the schools Misplaced Pages selection for this article, and in many ways that version (dating back to 2008) was better than this one. I may incorporate some of that material. That link is also a reminder that someone thought this article should be understandable by children (or at least, high school students). A challenge! RockMagnetist (talk) 22:35, 3 January 2012 (UTC)

It probably is actually. The sections Energy Eigenstates, Real Eigenstates, Unitary Time Evolution, Correspondence principle, some bits from Relativity, Gaussian Wavepacket, Galilean Invariance, Free Propagator, Analytic Continuation to Diffusion, Operator Formalism (?) from then could be included here in the relavant sections, but the article may become too long. Most of the other material not mentioned is either already included or is irrelavent. I would favour inlclusion of these, provided its really mixed throughly in.-- F = q(E + v × B) 23:14, 3 January 2012 (UTC)

We could recover some content from here and re-write: -- F = q(E + v × B) 23:39, 3 January 2012 (UTC)

You beat me to it! That was confusing. I am now going to combine Galilean invariance with relativity. RockMagnetist (talk) 00:18, 4 January 2012 (UTC)

Sorry to confuse you - no more will be added now, theres plenty of recovered info.-- F = q(E + v × B) 00:34, 4 January 2012 (UTC)

Yes, the article is now badly in need of copy editing, particularly reduction of redundancy. RockMagnetist (talk) 00:43, 4 January 2012 (UTC)

The derivation has lost its rigour, this version assumes in the first place the wavefunction is complex, where before it was demonstrated clear eneogh that the wavefunction can be assumed real, but must be forced to be complex. Also the entire structure to the SE is now repeated in the wave-particule duality and induced repetition. I will not revert, but I do intend to change it carefully still keeping to the same lines as this one.-- F = q(E + v × B) 08:40, 4 January 2012 (UTC)

Since the purpose of the section was to provide a heuristic derivation of the equation, it doesn't need to be rigorous - although the shortcomings of the derivation should be made clear. It's probably good that you need to combine it with the wave-particle duality section because that will slim down the article - but sorry for causing you more work. There is also redundancy between this section and the introduction. RockMagnetist (talk) 17:03, 4 January 2012 (UTC)

Anent the earlier mentioned "making it understandable for (high) school students," of course that is impossible if whoever said that wanted the whole article to be that way. However, I think Schrödinger's way of talking about the equation generating a sort of catalog of expectations could be worked into something that anybody with some experience in plotting a graph might be able to understand. I'm thinking of something like a table of probabilities for the location of something at t = 1, 2, 3, ... seconds that solutions of the equation would generate. One would not have to understand the math to be able to understand that it produces, for one t, a series of probabilities that are spread out along the perimeter of the expanding wavefront.P0M (talk) 09:06, 4 January 2012 (UTC)

We can incorperate that soon, sounds like a good idea. Btw I'm giong to blend the original derivation into the waveparticle duality section. they're too alike. I still don't plan on restoring the previous version for now. -- F = q(E + v × B) 09:21, 4 January 2012 (UTC)

I agree that we can't expect the whole article to be understood by high school students, but they should get something from the lead and introduction. Your idea sounds intriguing, Patrick0Moran. RockMagnetist (talk) 17:05, 4 January 2012 (UTC)

hello... I had a go at clearing up much of the chaos I caused in the properties section and derivation. The energy eigenstates has blended in nicely. Galilean invariance has been reduced significantly, but will be generalized to 3d. The only difficult thing is the Gaussian wave packets and thereon. That idea above is the best I’ve ever come across for describing the SE, it will be included in the lead sections. well done for coming up with it P0M! =) -- F = q(E + v × B) 17:16, 4 January 2012 (UTC)

Let's slow down and think

I am concerned that this article is changing very rapidly, but not necessarily for the better. We are editing it without any clear plan to address the complaints of that anonymous reader. I have started a page that outlines the concepts that are appropriate for each level of understanding and added it to the TODO list. I think it would be a good idea to work on this list for a while before making any more major edits to the article. Unfortunately, I can't contribute much more for a while - I need to get back to my real job. RockMagnetist (talk) 18:28, 4 January 2012 (UTC)

Also, this being an encyclopedia, we need fewer derivations and more citations. RockMagnetist (talk) 18:59, 4 January 2012 (UTC)

The simplifications can be done as and when, there will be rapid changes due to the recovery of info. I'm aware that more citations and fewer derivations are needed, but derivations are not really a problem if we use show/hide boxes. Citations shouldn't be much of a problem, I have a few books that should help. Your list is definitely a logical one and will be implemented, but I also had a plan in my head: recover the lost info, clean up + find citations, make simpler, throw away the rest (of the added material). It would make sense to clear up the messy details and at the same time settle the content of your list into the lead of the article.

The non degenerate ground state section may as well be thrown away right now as its pretty much irrelevant, but the rest currently in the article can be chopped and changed without much of a problem. I haven’t forgot about that IP either. This time, if I add hidden boxes for all the current intricate derivations, please leave them there - they will help.-- F = q(E + v × B) 19:11, 4 January 2012 (UTC)

Your plan sounds pretty reasonable. Hidden boxes might help. I think I'll come back in a couple of weeks and see how it looks then.RockMagnetist (talk) 19:18, 4 January 2012 (UTC)

The first set has been added. The variational principle will also be thrown away - it doesn't add anything.-- F = q(E + v × B) 19:24, 4 January 2012 (UTC)

I thought you were going to use boxes for derivations. I strongly disagree with using boxes for whole subsections. Not only does that hide important parts of the article but makes them invisible in the TOC and unlinkable. RockMagnetist (talk) 19:40, 4 January 2012 (UTC)

In fact, on second thought I don't think boxes should be used for any part of the article. They simply hide material that should be either reduced or discarded. RockMagnetist (talk) 19:43, 4 January 2012 (UTC)

Do you plan to get rid of everything in the boxes, or just take the stuff out of them? There are some articles which use them for properties in the same way.-- F = q(E + v × B) 19:49, 4 January 2012 (UTC)

Maybe you didn't see this article: Variational method (it's exclusively about quantum, despite the title).

My two cents: Make better use of summary style, and don't be afraid to spin out new articles, Free particle in nonrelativistic quantum mechanics or whatever.

Also, here's a rule of thumb. If the endpoint of a discussion or example is a formula or equation, then probably that discussion or example should be deleted. The endpoint should always be a concept, in particular a concept that sheds light on the nature of the Schrodinger equation. It's OK to go "setup --> equation(s) with description --> concept --> new section". But it's usually a bad sign if you go "setup --> equation(s) with description --> new section". Usually that's a sign that you're getting too textbook-y and that you'll be boring and/or scaring readers and over-emphasizing inessential details. --Steve (talk) 19:51, 4 January 2012 (UTC)

Could be useful: File:Wave packet (dispersion).gif --Steve (talk) 19:54, 4 January 2012 (UTC)

I just took the stuff out. If you want to reinstate the boxes, I suggest soliciting more opinions first. RockMagnetist (talk) 20:00, 4 January 2012 (UTC)

Ok, Yes I have seen the Variational method article as well. As a change of plan, the new sections should just be added first, as proposed by Rockmagnetist and PoM, and any of you can delete/move the mathematical intricies at will. -- F = q(E + v × B) 20:04, 4 January 2012 (UTC)

By all means carry on with trimming the article. I think you were right to delete Variational principle. Another good candidate for deleting/moving is the material I just moved from Further properties to Solution methods. RockMagnetist (talk) 20:08, 4 January 2012 (UTC)

(pasted after edit conflict) I forgot to mention adding referances to as much as possible at the same time. Right now i'm in the process of that.-- F = q(E + v × B) 20:10, 4 January 2012 (UTC)

I found a great analogy between roller coaster on a track, and the SE (!) in The New Quantum Universe, T.Hey, P.Walters, Cambridge University Press, 2009, ISBN 978-0-521-56457-2. The particle is like the roller coaster, and subject to a gravitational potential, and there is an exchange between KE and PE, but E = KE + PE= constant, so it limits what the roller coaster can do. The track is the spatial constraint on the system - it can only move back and forth (the authors neglected heat loss + friction). Analogously, the coaster at the dips in the track are like particles in a potential well.

The same book goes on to explain the dualistic nature of Electrons and Neutrons, and applications to microscopy and diffraction respectivley, and later energy levels + stationary states.

Also there's always Feynman's lectures to the rescue (vol 3), which explains very qualtiativley about particles and waves diffracting in the form of a double-slit experiment and leads onto wave-particle duality, in doing so the associated discrete and continuous probabilities.

These would be the first steps for the new approachable sections. =) -- F = q(E + v × B) 20:37, 4 January 2012 (UTC)

The Talk:Schrödinger equation/Outline for different levels page looks fine to me. Thanks for the good work.P0M (talk) 21:21, 4 January 2012 (UTC)

I'm still in the process of adding refs, so everyone please don't edit right now! Btw RockMagnetist created that page, not me.-- F = q(E + v × B) 21:30, 4 January 2012 (UTC)

At least there are a sufficient number of references for what we have now.-- F = q(E + v × B) 21:45, 4 January 2012 (UTC)

Another analogy I came across was a couple of years ago at an open day to a uni, one post-grad was explaining how quantum tunnelling is possible. Throw a ball at a wall on the macroscopic scale - it will never penetrate it because it doesn't have enough energy. A particle "bouncing" in a finite constant-potential well can penetrate the barrier and pass through, if its energy is greater than the potential, though only by chance. The whole situation is described by solutions to the TISE for the situation, the chance of a particle doing something at some place is the wavefunction. Rather neat, isn't it? =) If only that person was writing this article (btw - his name was Steve!!!)... -- F = q(E + v × B) 22:15, 4 January 2012 (UTC)

Once again it took an embarrassing length of time to re-write the introduction, but at least its something......=( -- F = q(E + v × B) 22:41, 4 January 2012 (UTC)

I'm finished for now... I may not be able to continue for a day or so - due to uni... when I can, I will return and kill the chaos I caused. =( -- F = q(E + v × B) 01:25, 5 January 2012 (UTC)

I have added File:Wave packet (dispersion).gif as it is public domain, we should add the other animation File:Doubleslit.gif and File:Hydrogen Density Plots.png, if we get the permission.-- F = q(E + v × B) 01:43, 5 January 2012 (UTC)

Perhaps this one would be useful at some point?

I'm not sure what this file was originally used for, just noticed it a couple years ago.P0M (talk) 03:18, 5 January 2012 (UTC)

I'll return quickly for a while, I've seen that image at quantum tunnelling before, to how show particles can pass through a boundary (surley the orginal intension)? Its another good find, the article lacks any image so it should be added to the case of a particle bound by a constant potential, but we don't wan't to make the mistiake of too many repeated animations of a simalar nature... eventually. -- F = q(E + v × B) 09:43, 5 January 2012 (UTC).

A more approachable scene has been set at the beginning, more will be done, but maths towards the end can be substantially trimmed down once and for all. All of the mathematical properties of the Schrödinger equation can in principle form a new article Schrödinger equation (properties) - everything has been referenced.-- F = q(E + v × B) 09:54, 5 January 2012 (UTC)

To help implement PoM's ingeineous idea, here is an image (I drew) which could be added:

Thoughts? Referance: Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, ISBN 0 7167 8964 7

--Maschen (talk) 18:56, 5 January 2012 (UTC)

It might be usable in the body of the article, but it's a bit too complex for the lead. What do the circles represent? RockMagnetist (talk) 19:04, 5 January 2012 (UTC)

The circles are the particle - see also wavefunction for simalar diagrams I drew, if you wan't. The caption has been repaired. There was another I drew months ago for the TISE - summarizing Ψ curvature, and the quantities T, p, k etc:

and dumbly added it to the wavefunction article, but it was effectively rejected for both wavefunction and this article (Schrodinger equation) (see here). I can modify it to the TDSE, since it would work out the same, it was just that I tried to stick with the explaination provided by the source it came from: Quanta: A handbook of concepts, P.W. Atkins, Oxford University Press, 1974, ISBN 0-19-855493-1. Not so sure anyone will find this one helpful. I'm not forcing these into the article: they are for take or leave.--Maschen (talk) 20:02, 5 January 2012 (UTC)

To Maschen: The first of those two images, File:1d step pot sol TISE.svg, is inconsistent. You are showing the time-evolution of a moving particle's wavefunction and describing it with the time-independent Schrödinger equation which does not apply to such cases. JRSpriggs (talk) 05:41, 6 January 2012 (UTC)

It isn't possible to descibe a moving particle by the time-independent eqution? For 1-d

${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\Psi }{dx^{2}}}+V\Psi =E\Psi \Rightarrow {\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\Psi }{dx^{2}}}+(E-V)\Psi =0\,\!}$

where the solutions are in the form

${\displaystyle \psi _{1}=Ae^{ik_{1}x}+Be^{ik_{1}x},\quad k_{1}={\sqrt {2m(E-V)}}/\hbar ,\quad V=0}$

before the pot. step, and

${\displaystyle \psi _{2}=Ce^{ik_{2}x}+De^{ik_{2}x},\quad k_{2}={\sqrt {2m(E-V)}}/\hbar ,\quad 0<V<\infty }$

after the step, boundary conditions at the step leading to

${\displaystyle C={\frac {2k_{1}}{k_{1}+k_{2}}}A,\quad B={\frac {k_{1}-k_{2}}{k_{1}+k_{2}}}A}$

and linear combinations of waves for different k are the wavepackets that localize the particle? Am I wrong? The particle is initially free so has kinetic energy and velocity, passing through the barrier slows it down becuase some kinetic energy transforms to potential energy. This is in the sources, two more I didn't mention are Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10) 0 07 145546 9, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-873730 --Maschen (talk) 11:49, 6 January 2012 (UTC)

The TISE is for potential functions independent of time, and so the Hamiltonian is independent on time, does it mean the particle can't move anywhare? What about the harmonic oscilator (animation already included) - not every wavefunction solution for that system is a stationary state (and yes I do know what these are now). Again - i'm not forcing for inclusion, just a little confused... and why was there mention that it might be included by RockMagnetist - and that he didn't mention any error then?--Maschen (talk) 12:02, 6 January 2012 (UTC)

Btw I also forgot to mention (needless to say but still), there seems to be an enormous mass of mathematical detail towards the end which seems completley irrelavent to the article, move to the main article on wave packets? Also perhaps create an article entirley on the exact solutions: Schrödinger equation (exact solutions), and mention the methods used here? leaving all the general points on the TISE + TDSE? just a suggestion...--Maschen (talk) 12:15, 6 January 2012 (UTC)

You are correct, up to the superposition of plane waves to form a wave packet. The complication with wave packets is that they are not stationary states because of the localization of the probability amplitude near the particle as it moves. The probability density is therefore space and time dependent, since the localization of the particle changes position with time, if you get what I mean! Look at the animation in the article in the implication section: is the probability of finding a particle only a function of position? It clearly isn't. JRSpriggs above is correct - this situation can't be described by the TISE. You need only say instead: time evolution of the wave packet for a free particle incident on a potential step, as described by the solutions to the TDSE, and readers will have an idea of what the meaning of the image is. So kill the equation in the image - since that is the TISE, and fix the caption.-- F = q(E + v × B) 19:53, 6 January 2012 (UTC)

Just to let you know - wavepackets can be obtained from the Fourier transform of the product of a plane wave and a momentum space wavefunction:

${\displaystyle \psi (x,t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\phi (k)e^{i(px-Et)/\hbar }dk={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\phi (k)e^{ik(x-\hbar kt/2m)}dk\,\!}$

the k-space wavefunction is needed since the sum (integral) is over all k, i.e. a wavefunction in k-space, but the k-space and x-space wavefunctions are Fourier transforms of each other. =)-- F = q(E + v × B) 19:53, 6 January 2012 (UTC)

It may be an idea not to add the Fourier integral equation to the image - you might scare away readers and it may be detracting for some... A purley diagrammataic one should not. Although don't get the wrong end of the stick - the quality of the drawings are really good, provided you correct the one mentioned!!! The other one may well be usefull in describing wave propagation, but you'll have to re-draw it for the TDSE (if you want to/have time), since that image contains the energy eigenvalues and not derivatives. =)-- F = q(E + v × B) 20:13, 6 January 2012 (UTC)

I made the correction. Everyone happy now?--Maschen (talk) 20:27, 6 January 2012 (UTC)

I am sorry that I did not express myself quite correctly in my last comment. The point is not that the particle is moving, but rather that the probability density depends on time (since the wave packet is moving). This is only possible if the wave is a superposition of eigenfunctions for different energy eigenvalues. If the energy had a unique value, then combining the time-dependent equation (which is always true) with the time-independent equation (which holds for a definite energy) gives

${\displaystyle E\psi =i\hbar {\frac {\partial \psi }{\partial t}}\,}$

which implies that the modulus of the value of ψ at any point in space does not change over time. And thank you for fixing the image. JRSpriggs (talk) 02:09, 7 January 2012 (UTC)

I understand now. Thanks very much to both of you for explaining that to me. No problem about the image either.--Maschen (talk) 03:50, 7 January 2012 (UTC)

From above: ...how quantum tunnelling is possible. Throw a ball at a wall on the macroscopic scale - it will never penetrate it because it doesn't have enough energy. A particle "bouncing" in a finite constant-potential well can penetrate the barrier and pass through, if its energy is greater than the potential, though only by chance. I think this is erroneous. This difference between a macroscopic object and a particle (fermion) is not due to the energy difference, but to the difference in object structure. An object is a random collection of component particles, with random bonds holding them together. This is what gives it "solidity". If you send a series of particles toward a thin wall, many will tunnel through (as predicted by the square of the wave function). This is due to their QM nature. The particles in a solid object also have QM nature, but the entire object has a vanishingly small probability of tunneling due to the random bonds. Example: the large quantum object represented by superfluidic helium can easily pass through even thick walls, such as glass. This essentially happens because the bonds in the helium are coherent and do not interfere with its QM nature. The helium molecules can and do tunnel freely. I'd like to see the article reflect that difference between large objects that have random structure as compared with those that have ordered (coherent) structure, such as Cooper pairs, instead of talking about tunneling as due to kinetic and potential energy. David Spector (talk) 22:23, 15 January 2012 (UTC)

I presume it was just an analogy.--Maschen (talk) 14:26, 16 January 2012 (UTC)

Is energy the right emphasis?

Although I appreciate the effort that has gone into writing a new introduction, I don't think it is the right approach. By focussing on energy, constrained motion and the Hamiltonian, you are basically describing the similarities between quantum mechanics and the Hamiltonian formulation of classical mechanics. That is fine as far as it goes, but it would be better to describe the peculiarities of the quantum world and how the Schrödinger equation addresses them. Also, an introduction that emphasizes energy could just as well be for matrix mechanics as for the Schrödinger equation. A more appropriate introduction would emphasize the wave-like form of the equation RockMagnetist (talk) 17:52, 6 January 2012 (UTC)

Initially there was plenty of reason to emphasize energy, but now you mention it the title may as well be called "Energy conservation/Hamiltonian mechanics". =( So you say we should kick off with wave propagation? We should definitely keep the analogies at the beginning, but reduce the amount on energy thereafter and move it to (say) the derivation/assumptions section? -- F = q(E + v × B) 19:53, 6 January 2012 (UTC)

I should mention I may not be able to edit for long periods of time: preoccupied with assignments/work (like everyone else), so it may take a while... I'll see what I can do later this evening. Its pretty bad of me to chaos the article and not be able to fix everything thereafter. =(-- F = q(E + v × B) 20:13, 6 January 2012 (UTC)

I'm in the process of drawing images to illustrate wave propagation - it should be done for later... today or tommorow. Thanks for your feedback above also. --Maschen (talk) 20:29, 6 January 2012 (UTC)

Yes, I think a lot of the material on energy could be moved to the derivation/assumptions section. That would also reduce some redundancy. RockMagnetist (talk) 22:02, 6 January 2012 (UTC)

I'll try now. Also - to force-render the equations as png, just type \,\! at the end before the </math>, \displaystyle is a bit long and clumsy. -- F = q(E + v × B) 23:26, 6 January 2012 (UTC)

Well thats it from me for a while... I have other commitments for a few days...-- F = q(E + v × B) 03:45, 7 January 2012 (UTC)

Your efforts are much appreciated, by surley everyone on the talk page.--Maschen (talk) 03:50, 7 January 2012 (UTC)

Its a very stomach-wrenching thing to say, especially after all the work you have all put in, but its tempting to restore the current article to this version schools Misplaced Pages selection for this article, then from the edit history, add back useful content from this version into the will-be new version (removing the irrelevant/unsourced content at the same time), instead of the other way round which is the current scheme.....another suggestion. I'm not in any way planning to do that, I don't even plan to touch the article (except for adding imagaes or referances).--Maschen (talk) 04:01, 7 January 2012 (UTC)

Actually, just at a glance (all I have time for right now), it's starting to look promising. RockMagnetist (talk) 05:13, 7 January 2012 (UTC)

I have a few seconds to add some important mathematical information missing from the article.-- F = q(E + v × B) 23:52, 7 January 2012 (UTC)

Just to let everyone know - I did the image yesterday but for some really annoying reason it will not upload properly - the SVG rendering is always ruined. The actual drawing is very smooth, here it becomes jagged and distorted. This will not be added to the article untill find a way to repair it and if everyone agrees - right now I don't have time, just like everyone. I intend to add the other images soon - no-one is objecting to them.--Maschen (talk) 14:12, 8 January 2012 (UTC)

It may be that the cache is not updating properly. I have seen this problem many time. The image may even show up properly on English Misplaced Pages but appear to be the old image on Commons. Sometimes I access the Commons image using a different browser.

Are you using Inkscape? If so, be sure not to let it save the default "Inkscape SVG" image. It does certain things like messing up the direction of arrowheads or the position and orientation of blocks of text. There is a more comprehensive note on the problem on Commons somewhere.

To get feedback on where the problem is really showing up, save your SVG image to the desktop, open it using your browser, and if it isn't right there then it will surely be wrong after you upload it. If you upload it and it doesn't look right, first, be sure you are looking at the real SVG image and not a rendering of it produced by Misplaced Pages. If it looks wrong, then download that image (giving it another name so that it does not wipe out your original file. Check the new image by opening it in your browser.

BTW, the image I see on this page at about this point looks very good on my browser. P0M (talk) 16:35, 8 January 2012 (UTC)

Just a quick response (while researching online), I used Serif Draw Plus X4 . I'll see about fixing it later - the article is not dying for it. Are you saying the image on your browser is very smooth - absolutley no kinks/jolts in it at all? Nice sinusiodal-shaped waves? Thanks a lot for your help, best. --Maschen (talk) 17:35, 8 January 2012 (UTC)\

I just downloaded the SVG you posted and opened it in Inkscape. It looks fine there. I had Inkscape export a PNG image, and it had the same level of jagginess that I can observe on Commons -- at 300% enlargement. It appears identical to the rendered image I get on this page. If I hit the button to enlarge the display size three times I can see some jagginess, but nothing exceptional. If I go to Commons and ask for a 2000 px rendering, then it comes on my screen with no evident jagginess at all.

Whoever codes the software makes some decisions about what will look o.k. at different sizes. I suppose if I downloaded a 2000px PNG image and then "shrank" it I would see no jaggies. The small image that this page displays is perfectly o.k. as far as I am concerned. Somebody who has 20/5 vision may notice jaggies. But such people must accustom themselves to seeing the moons of Jupiter and the pimples on models at 20 feet. :-) I suspect that if you clear your cache, turn your computer off for 20 seconds, restart, etc., all the aesthetic issues will disappear. P0M (talk) 19:06, 8 January 2012 (UTC)

Hi, cheers. I tried it and it didn't work. Thanks again for your advice though... I intend to do a better one at some point anyway - one suggestive to three dimensional plane waves also, and it will overwrite this one.--Maschen (talk) 21:47, 8 January 2012 (UTC)

O.K. I've tried it on all three of the browsers that I keep around, and the images all look fine to me. The SVG definitely does not have any problem with it. I'm curious. What browser and what System are you using? (I'm on a Mac with Sys X & using Firefox, Safari, and Chrome.)P0M (talk) 23:11, 8 January 2012 (UTC)

Internet explorer 8, Windows 7. I never use other web browsers - especially Firefox, that simply doesn't work for anything when I use it (not just becuase of me!). I'm pretty sure the image has no intrinsic properties, it can only arise from how its coded for a given browser.--Maschen (talk) 23:21, 8 January 2012 (UTC)

I dug out Opera and Camino. Both showed the image with no problems. If IE is giving you trouble with this one image it may be giving problems with some other images too. Perhaps Microsoft would like to know that their browser will not read Misplaced Pages without problems. (Have you looked at other SVG images from Commons?) P0M (talk) 01:10, 9 January 2012 (UTC)

Hello - sorry for such a long response, it seems the problem was in serif symmetrizing the curvature nodes so the curve became twisted in places. This effect appeared since I opened it up in inkscape and found the nodes were symmetrized, and have made the correction there. I haven’t looked at other images from commons yet, will in a minute. In doing so I'll upload the new version - it should appear here. Again - sorry about the long wait, given that you are trying to help me its pretty rude of me not to respond at least within a day... just been busy is all...--Maschen (talk) 20:13, 11 January 2012 (UTC)

Generally - at a glance of the quantum mechanics images they look perfect in quality.--Maschen (talk) 20:18, 11 January 2012 (UTC)

No problem with the wait. I'm glad you found out what was causing a problem. I can see a difference between the current image and the one before, but on my system the improvement involves unwanted extensions of lines that are so short that I did not notice them until I looked at the SVG images in a form that covered my whole monitor screen. Thanks for the beautiful images. P0M (talk) 01:06, 16 January 2012 (UTC)

No problem + Cheers - i'll add them all now, including the diffraction animation you found above.--Maschen (talk) 14:26, 16 January 2012 (UTC)

Wave packets...

I'm going to move all the stuff on wave packets: Gaussian wave packets, Free propagator and most of Analytic continuity to diffusion to the main article. There is simply no need for them in this article, as it has been suggested a few times before. The first equation on diffusion can be incorporated into the Positive energy section to take the context further, it is relavent to some extent.-- F = q(E + v × B) 11:35, 11 January 2012 (UTC)

Xcellent. I would propose to delete the equation box at the very begining of the article - they are completley pointless as the same equations are given below with explaination (havn't done it yet). --Maschen (talk) 14:26, 16 January 2012 (UTC)

On second thought it may as well be removed. Anyone who is so desperate for such repetition can explain themselves here on the talk page.--Maschen (talk) 14:34, 16 January 2012 (UTC)

Rewrote sections 1 and 2 (now 1 and 2 and 3)

My main changes were: (1) Have the time-independent equation boxed near the top. Since the time-independent equation is legitimately and often called "the Schrödinger equation", I think it's a disservice to readers to bury it so far deep in the article. (2) Shorten the discussion of roller coasters...this aspect is nothing new compared to classical physics. (3) Shorten other parts of the Implications section thanks to a lot of "main article" links (e.g. double-slit experiment is a fine article, I don't think we need to repeat quite so much of it here). Also, reducing redundancy within the section. Please let me know any thoughts... :-) --Steve (talk) 22:40, 29 January 2012 (UTC)

I like your approach and agree with your opinion of the roller coaster analogy. RockMagnetist (talk) 23:01, 29 January 2012 (UTC)

I agree also, that user:F=q(E+v^B) is a shithead of an editor. Fortunatley we have you two and numerous others who have a brain!

1. Pusey, Matthew. "The quantum state cannot be interpreted statistically". arXiv.

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