Talk:Wave function/Archive 2

Archive 1

Archive 2

Archive 3

Archive 4

Archive 5

Moved

I've moved the page, per the above discussion and the request at WP:RM. It seems that Wavefunction collapse should move as well, huh? I don't see any reason to go through a five-day procedure for that; I'll just move it. -GTBacchus^(talk) 00:12, 21 February 2007 (UTC)

Ok, that's done, and Normalisable wave function as well. Cheers. -GTBacchus^(talk) 00:31, 21 February 2007 (UTC)

I think we should think again about this. "Wavefunction collapse" always refers to the quantum case, so we can't just carry over the terminology from "wavefunction" which includes the non-quantum usage. --Michael C. Price ^talk 23:50, 4 November 2007 (UTC)

I am glad you moved it. No less an authority as the on-line OED redirects wavefunction to wave function. For those of us who poo-poo "authority" let's at least follow the logic of grammar and present day English usage. Wavefunction would be following the germanic eigenfunktion where one can join two nouns and keep them in equal importance. In English wavefunction would mean that wave is a prefix and function is the root. Prefixes are meant to modify or adjust the stem (according to the OED), much as adjectives do to nouns. A wavefunction would be akin to a wavy function, I suppose. Using one noun to modify another noun usually has a genetive meaning. Thus "wave function" is akin to a "function of a wave", mathematical I suppose. It gets trickier with the ablative meaning in the OED's example "speedboat ride", "ride on a speedboat". Laburke (talk) 03:05, 30 October 2011 (UTC)

Diagrams for interpretation

The article really needs more diagrams relating mathematics to physics... the wavefunction can be visualized. I produced a couple for the wavefunction in one dimension and for one particle, hopefully it makes interpretation clear(er). Maschen (talk) 16:04, 23 August 2011 (UTC)

The positioning of the images need improvement, although I'm not sure how to do that. Good images, though. -- cheers, Michael C. Price ^talk 18:53, 23 August 2011 (UTC)

Thanks for feedback, but how do you propose to adjust the images? The page to image syntax is in the article Wikipedia:Picture tutorial. Maschen (talk) 23:31, 23 August 2011 (UTC)

Reforming the article

As can be seen above this article has had a lot of problems and a negative history (I know - I have complained just above), which shouldn't be the case for a topic like this. To set the article streight the following should be resolved.

1. A lot of mathematics is repeating thoughout the article in a way that doesn't help, especially on normalization, where there is an entire article on Normalizable wave functions, so there is repetition with another article as well. Some notation for probability is non-standard. It best to state space over which the wavefunction is defined and the probability integrals for finite volumes of space, since normalization is then just the integral over the full space, equal to 1. The normalzation condition only needs to be stated once. Also, first there was not eneogh explaination as to what wavefunctions are (in QM), now there are repetions of the vector formalism at the beggining then end of the article.

2. Also, since wavefunction spans a number of contexts shouldn't a disambiguation page be created? Before this is done, the current page should be moved to a new name titled Wavefunction (quantum mechanics), then the other applications of wavefunctions (such as PDE solutions as stated above) can be developed into new artciles. Then the disambiguation page can be created.

Maschen (talk) 10:03, 24 August 2011 (UTC)

Problems with recent edits

Here are a few of many problems I have with the recent edits to the article:

• "Note that the wavefunction describes a system of particles in a quantum state, it does not "describe the behaviour quantum state" itself, which is defined by quantum numbers." <-- I have no idea what this means
• "Simple examlpes of wave functions are common quantum mechanics problems; the particle in a box, which corresponds to wavefunctions for standing waves at various vibration modes, and the free particle (or a particle in an infinitley large box), correspoding to a wave function for a travelling wave (in this case sinusoidal)." <-- Even ignoring the typos, and the lack of clear explanation, this is wrong. The wave functions for a particle in a box are any continuously-differentiable function inside the box which is zero at the edges of the box. The stationary states are standing waves, not the wavefunctions. Likewise, the free particle wavefunctions are any normalizable continuously-differentiable function of space, not just traveling waves.
• "By the uncertainty principle, the momentum uncertainty is less than the position uncertainty (momentum is known to a higher degree of accuracy than position)." <-- Huh????
• "In all cases, the wave function provides a complete description of the associated physical system - it contains information about the system to be extracted by operators." <-- A typical reader will not understand the phrase "to be extracted by operators"
• "Note that ψ is not a function of any of the quantum numbers because they are not continuously variable, they are only integer parameters to label a specific wavefunction for a quantum state defined by the required quantum numbers." <-- You seem to misunderstand quantum numbers. It is perfectly possible to have an electron in, say, a superposition of 1s and 2s states in a hydrogen atom. The spatial quantum numbers are optional labels and do not need to be mentioned in the definition at all. The spin quantum numbers should be inside the parentheses, because ψ is a function of them. ψ is a function simultaneously of continuous spatial variables and discrete spin variables, and the normalization condition involves its integral over continuous variables AND sum over discrete variable.
• "A wave function is either a complex vector with finitely many components or countably infinitely many components." <-- Doesn't a free particle has uncountably infinitely many components??
• "The modern usage of the term wave function extends to a complex vector or function, i.e. an element in a complex Hilbert space." <-- A key point is that function can be viewed as a type of vector, because the set of all possible functions is an infinite-dimensional vector space. This might be the hardest and most important mathematical aspect of introductory quantum mechanics courses. This sentence not only fails to explain this, it doesn't even get it right. ("vector OR function"??)
• Hydrogen atom example:

• Again, this is an article about wavefunctions, not stationary states. This is written as if they were the same thing.
• Formula for the Bohr radius is incorrect by a factor of two, and should not be written in Gaussian units without saying so. People usually assume SI.
• What is the reader supposed to learn about wavefunctions by reading this example? I can't think of anything. They'll just see some formulas, but have no idea why the formulas are true or what the formulas mean or why they should even care.

• "Below the basis vectors are unit vectors, which are completley arbitary but non-equal, non-zero, and dimensionless." <-- This is wrong. You can't pick three vectors all in the same plane and expect them to be a basis for position space. Everything in this section is so much more complicated by the decision to include both rectangular and polar and cylindrical coordinates all at once. Why make things so complicated?? Why not just use x,y,z?? With statements like "and X is some dimensionless factor, possibly dependant on any of the coordinates $\scriptstyle {r_{1},r_{2},r_{3}}\,\!$", only very mind-reading readers will understand that this is referring to the "sin θ" factors of polar-coordinate integrations and so on. The mechanics of doing integrals in spherical and other coordinates is the subject of other articles on wikipedia; for this physics article, we can just write the integral in normal notation.

Well that's just a few to start. I wish people would not edit extensively articles on subjects they don't understand very well. :-( --Steve (talk) 15:04, 28 August 2011 (UTC)

The next worse things are the diagrams I added - arn't they ?...

This isn't a vain self-obsessed attempt to add material to an article so my name takes up the edit history (i'm not in for credit), I actually want to help draw images relating maths to physics as (if only slightly) clearly as possible. More were planned for three dimensions and multiple particles but when I try fixing up maths it always ends up over-complicated, so the diagrams will lead to the wrong ideas...

I should add though about the quantum numbers, that in the definition section the numbers n₁, n₂ etc are any quantum numbers, no specific one is the principle, spatial or the spin quantum number. The n is misleading.

Maschen (talk) 23:51, 28 August 2011 (UTC)

About motivations: I have no doubt that you have been working hard and altruistically, don't get me wrong :-)

About quantum numbers: In a math course, you might define a function $f_{1}(x)=5x+11$ and another function $f_{2}(x)=16x-12$. The "1" and "2" are just labels. The teacher would say "These are examples of linear functions. In general a linear function would be $f(x)=ax+b$". The teacher would not say "In general a linear function would be $f_{i}(x)=ax+b$ where i is the number that labels the function." The quantum numbers are the same sort of thing, they are just labels. For example, the 100 orbital of the hydrogen atom is the function $\psi (r,\theta ,\phi )=e^{-r/a}$. Some people might choose to use $\psi _{100}$ as the label for this particular $\psi$. Other people might just call it $\psi$, as in "Let $\psi$ be the 1s wavefunction of a hydrogen-atom electron...". Or they might call it $\psi '$ if they previously used the letter $\psi$ for something else. Or they might use a different letter, $\phi$. Anyway, the spatial quantum numbers are just labels. They do not belong in the general abstract definition. In the general definition you can use $\psi$ with no subscripts, and people will understand that it's also OK to give wavefunctions other variable names, like $\psi _{100}$ or $\phi$ or whatever. I'm mainly talking about spatial quantum numbers but this is true for spin too: Spin belongs as an argument inside the parentheses, along with the coordinates, not as a subscript. There should be no subscripts in the general definition.

About diagrams: I like them, but I'm not sure they all belong in this article, because wavefunctions are not the same thing as stationary states. It seems to me that the solutions to the time-independent Schrodinger equation are part of the topic Schrodinger equation and stationary state but not really an important part of the topic of wavefunction. The wavefunction is the mathematical apparatus that is used for describing any state, whether or not that state happens to be an energy eigenstate. Of course it should be mentioned in the article that a wavefunction might be a stationary state, and some examples can be given of wavefunctions that happen to be stationary states, but the properties of stationary states should not be the dominant theme of this article, and examples of stationary states should certainly not be the only examples of wavefunctions in the article. There should be non-stationary-state examples too so that readers don't get the wrong idea that wavefunction is another word for stationary state. In my own illustrations of wavefunctions ([1] [2] [3]), you'll see that I have always put both stationary and non-stationary states for exactly that reason. :-) --Steve (talk) 03:54, 29 August 2011 (UTC)

Ok - thanks for constructive feedback, as always. So some next changes to make are then:

1. Remove subscripts in definition (and explanatory context related to them), insert the spin number s since its the fundamental property of the particle/s, which would appear as

$\psi =\psi \left(\mathbf {r} _{1},\mathbf {r} _{2}\cdots \mathbf {r} _{N},\mathbf {S} _{1},\mathbf {S} _{2}\cdots \mathbf {S} _{N},t\right)\,\!$

linking to spin (physics) article and possibly stating

"where

$S={\sqrt {s\left(s+1\right)}}\hbar ,\quad s=n/2,\quad n\in \mathbf {N} \,\!$

and N = field of natural numbers".

2. The curvature image doesn't really belong to the article I suppose. Either eliminate or move the curvature images to the Schrödinger equation article, to the Versions,Time-independent equation section where the wavefunction is described in 1 dimension and is time-independant.

The other images are not just standing waves as stationary states though, there are travelling wavepackets. There's nothing exactly wrong with that since as you pointed out ψ can be any continuously differentiable functions, and travelling waves are just a specail case. As I said more images were planned; for stationary and non-stationary states, so if I was to add more stationary-only states wouldn't become the theme.

Also isn't continuously differentiable one criterion for a function to be continuous? Continuity was stated in the Born interpretation section.

Maschen (talk) 07:59, 29 August 2011 (UTC)

About spin: You do not understand how to incorporate spin into a wavefunction. Therefore I suggest that you don't write about it! (This is true for other topics too! Do not write about things you don't understand well! Please please please!!) You can just give the definition for spin-0 particles, which you seem to at least slightly understand. Maybe someone else will incorporate spin later. This is the approach in most introductory textbooks anyway: Usually only spin-0 is discussed in the first introduction.

About the Schrodinger equation article: The article right now is a general discussion of the fundamentals of this broad and important topic. It doesn't get bogged down in details like "How can I solve the 1-dimensional time-independent Schrodinger equation in my head?" I do not suggest that you edit the article to bog it down with these details! I suggest you leave it alone. :-/ --Steve (talk) 13:16, 29 August 2011 (UTC)

Fair enough - I’ll just delete my curvature image, the subscripts in the definition and not touch anything after...

Before I zip it and leave you all in physics/maths tranquility: I didn't say it before either - i've seen your images before you linked to them, they were better than mine since they were animations including real and complex parts of the wavefunction.

Maschen (talk) 17:06, 29 August 2011 (UTC)

That's very kind of you to say! I had a go at rewriting, you are more than welcome to complain if I deleted something useful and good, or wrote something bad, or whatever. Maybe it was accidental, or maybe I had a reason and I can explain and discuss. --Steve (talk) 19:46, 3 September 2011 (UTC)

Not at all. Most of you here seem to be postgraduates, post-doctarates or beyond, so I respect your positions, and it must be a P ! A ! I ! N ! to have early undergraduates (only just about to start 2nd year) get this all wrong... many of you would be better at writing this topic than me.

The only slight objection is; (I know its only notation), but why use subscripts instead of brackets for the probability notation? Perhaps its easier to read? And for "all space" in the integrals why not give a symbol - at the top of my head why not \mathcal{R} (caligraphic R), to take repetition of words out and for ease of writing? Well it doesn't really matter anyway...

When I first came to the article I never intended to touch any of the maths and only meant to add diagrams, but the probability formalism seemed (to me at least) a little ropey so I found myself re-writing half the article wherever probability came in, and over-generalized. Sheesh its guilt to leave all the work to postgrads or beyond to tidy up the mess, I really need to use talk pages BEFORE editing anything...

Maschen (talk) 21:35, 3 September 2011 (UTC)

No worries. :-) About subscripts: I dunno, that's the way it was last month. I guess parentheses can be functions or multiplication, whereas a letter with a subscript is expected to be just a number, which is what it is. Other notations are also OK, don't get me wrong. About "all space", I was trying to minimize the amount of time it takes for an average reader to take in the equation. But I know it's non-standard. I think $\mathbb {R}$ is the most common symbol for 1-dimension, or $\mathbb {R} ^{n}$ for n dimensions. I'll put that in... --Steve (talk) 02:32, 4 September 2011 (UTC)

Awesome! Thats tons of hard work you've done, i'll reward you when I find out how. Cheers - Maschen (talk) 10:07, 4 September 2011 (UTC)

Hydrogen Density Plots

It might be an idea to make the image File:Hydrogen Density Plots.png bigger to about 450 × 450 px²? so the quantum numbers can be seen while reading the article instead of having to click on it each time. Thats how it is in the Quantum state article. Maschen (talk) 10:34, 4 September 2011 (UTC)

I thought I would just do it, clearing any odd spelling typos while at it - this much editing will not harm. Maschen (talk) 10:34, 4 September 2011 (UTC)

Ontology of the Wave Function needs its own article.

It's an incredibly interesting subject with major thinkers struggling to find answers for it and we only have a petty paragraph. Also it can not be claimed that not a lot can be written because that small paragraph cites several big thinkers who have public texts. --62.1.29.164 (talk) 12:59, 4 September 2011 (UTC)

A number of articles seem to be possible from the term Wavefunction. If a new article on this is to be written it will need

its own name - presumably and without shock Wavefunction (ontology). Then this article may need to be called something else more specific, related to the wavefunction in QM. On top is are the issues above for other applications of wavefunction, which may have their own names. I did move the page at one time from Wavefunction to Wavefunction (quantum mechanics) but an admin moved it back. Thoughts on creating a disambiguation page assocaited with the term wavefunction and all the names of the articles need to be considered before creating the new article, and to be sure they don't mutually overlap too much with each other or with other QM articles.

I don't suppose that helps much, chances are you knew this already... Maschen (talk) 21:03, 4 September 2011 (UTC)

The ontology of the wave function already has its own article. The article is: Interpretations of quantum mechanics. I just added better links so that readers will not miss it. :-) --Steve (talk) 19:10, 5 September 2011 (UTC)

Forget it... Maschen (talk) 00:06, 6 September 2011 (UTC)

Formalism section?

The formalism section now seems redundant, since much of the vector formalism there has been incorperated into the wavefunctions in vector form section. Though I would rather not delete the formalism section, since its someone else's work and there are some extra bits which may be of use. I'll leave it to anyone to decide for now... F=q(E+v^B) (talk) 22:31, 15 October 2011 (UTC) — Preceding unsigned comment added by F=q(E+v^B) (talk • contribs)

Diagrams again...

I have returned for a while, having read up more about wavefunctions. I will not write content, but re-incarnate some diagrams from months ago...

Maschen (talk) 22:30, 26 October 2011 (UTC)

For the last diagram it might be worth adding to the caption that the two particles recoil off each other with 100% certainty, or words to that effect. -- cheers, Michael C. Price ^talk 06:44, 30 October 2011 (UTC)

Thanks, first feedback already positve, i'll add the caption and lauch the diagrams onto the main article. If someone dislikes then they'll delete - and (should) explain here... Maschen (talk) 01:54, 2 November 2011 (UTC)

I deleted the images there.. they in the article now and have stood the test of time in not being deleted, so I take it people think theyr'e fine.

Maschen (talk) 16:33, 21 November 2011 (UTC)

Historical context

Hi...

All: I added a brief (and rather vague even after all that time and energy...) historical outline of the introduction of the wavefunction in the 1920-30s, named all those physicists responsible and dated events, added referances. Furthermore added extra context in the definitions and re-headed the section, and tweaked a other few bits and pieces. Hope its better...

Sbyrnes321: Looking back through the edit history makes me realize you wrote the historical note (and much of the 1st 1/2 of the article) so I do apologize for that. I know its very l8 to say that now after the elaped time of a couple of weeks. Do you think this history expansion is fine? or anything on the article for that matter......

...all comments are very welcome...and feel free to complain about mistakes...

F=q(E+v^B) (talk) 02:42, 21 November 2011 (UTC)

Not at all, as far as I can tell. Maschen (talk) 16:33, 21 November 2011 (UTC)

Further modifications

• Re-wrote a few bits here and there so those sentences/phrases make sense.
• Re-grouped all the introductory stuff together, so that the formalism follows after. It still doesn't seem coherent enough.
  • All mathematical intro stuff has been placed under a new heading Mathematical introduction - so that a layman will know what's coming before rather than after.
• Re-titled the section Wavefunctions in vector form to Wavefunctions as vector spaces and removed the dablink:

" "Wavefunctions in vector form" are not and must never be confused with "Wave vectors", a different physical quantity and concept.",

- there is no need if the title it worded properly.

• I don't really know why I added the headings to the history section. They don't add anything so i'll remove them. This is a setion which should be written in continuous prose, not chopped into subsections.
• Also added more referances

On this page I mainly talk to myself, but indicated the reason for change anyway. --F=q(E+v^B) (talk) 14:55, 11 December 2011 (UTC)

The linear algebra has become very intricate and technical - but nessersary. None of it should be deleted, but its making the article very long. Since most readers will probably read through the introductory stuff, then look further down the page for more info, the show/hide table code can be used so readers not interested in the maths can skip it and carry on (or click "hide" if it too blinding), interested readers can click "show" and read on. Also the article will reduce in length so the scroll bar is not so awkward to use. It would propose its reasonable to add show/hide table code for each subsection of mathematical detial the linear alg. formalism. It'll also frame the maths and make it look better (perhaps).

-- F = q(E + v × B) 12:05, 15 December 2011 (UTC)

I reverted the last change becuase it was pointless. Its better to expose maths so its clearer what the subsections are about, then let readers click hide when it gets too much for them. --Maschen (talk) 16:54, 15 December 2011 (UTC)

Update ratings

The article is no longer "start class" - it has reached a formiddable standard of quality from what it once was (only just a few months ago), but hasn't been nominated for a "good/top quality article" or whatever. So, I thought it was appropriate to update to at least a B class.--Maschen (talk) 14:32, 6 January 2012 (UTC)

Definitley. =) -- F = q(E + v × B) 07:52, 21 January 2012 (UTC)

Transfer content

I propose moving the normalization formulae to Normalizable wave function - it’s more relevant there than here. Here readers can just read the definitions of the wavefunctions, and see how to normalize it separately on the main article. These formulae are a little detracting, and the main article is too small anyway. I would also propose logically tabulating the wavefunctions in this article and associated probability integrals, to take out the repetition of "the probability of...", "the normalization condition reads". It’s pretty tiresome and boring for a reader to read these repeated lines (almost word for word) when everything can be laid in front concisely in the form of a table. -- F = q(E + v × B) 07:52, 21 January 2012 (UTC)

Forget this, i'd be inclined to merge Normalizable wave function into here, if anything... see also here.-- F = q(E + v × B) 15:30, 21 January 2012 (UTC)

Disambiguation page (again)

I think the disam. page for wave function should be created and move this current title to Wave function (quantum mechanics). There are many things which could refer to "wavefunction" - so on the disam. page the following can be listed:

Solutions to the linear wave equation

• Mechanical wave function: longintudinal and transverse displacements s and y respectivley
• Sound wave functions: fluctuating displacements and pressures s and p respectivley
• Electromagnetic wave function: oscillating electric and magnetic fields E and B respectively

Solutions to non-linear wave equations

• Soliton wave function, solution to the Korteweg–de Vries equation

Quantum mechanics

• Quantum mechanical matter wave function (link this article): Ψ
• Wavefunction collapse
• Wavefunction renormalization
• Normalizable wave function

(these would obviously be sections). Yes/no? The trouble is I can't find pages directly leading to the mathematical treatment of these functions (except for EM and Soliton waves), just the articles for each type of wave... -- F = q(E + v × B) 08:36, 21 January 2012 (UTC)

• Oppose. Just because something is a function describing a wave does not mean it goes by the name "wave function". I have never heard anyone use the specific phrase "wave function" to refer to, for example, the electric field as a function of space and/or time. The quantum-mechanical things are subtopics or aspects of the QM matter wavefunction, not descriptions of something different also called the wavefunction. So they shouldn't be disambiguation links, they should be "main article" links within this article. On the other hand, I am not opposed to putting a note at the top of this article saying, "For functions related to waves, see Wave (disambiguation)." --Steve (talk) 14:50, 21 January 2012 (UTC)

Fair enough - forget it. In my experience though, I have come across this terminology, sometimes. Its pretty rare though. One example source is Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, ISBN 0 7167 8964 7.-- F = q(E + v × B) 15:31, 21 January 2012 (UTC)

I know I am very pernickety about things... it has already been discussed before, though it really does make the article easier to read for non-experts - change the $\mathbb {R} ^{3}\,\!$ back to "all space"? Its tedious so I'll actually do the editing (if in others are in favour), but again just a suggestion. BTW the easy way of doing such things is to use the find/replace tool on notepad/Microsoft word (for those that didn't already know).-- F = q(E + v × B) 15:43, 21 January 2012 (UTC)

Merger proposal

I propose merging Normalizable wave function into this article. A normalizable wave function is a fairly trivial case that is already covered here. RockMagnetist (talk) 16:31, 21 January 2012 (UTC)

So far so good, thats 3 (including me) for a merge.-- F = q(E + v × B) 17:07, 21 January 2012 (UTC)

MERGE ABOUT TO BEGIN - DON'T EDIT YET-- F = q(E + v × B) 17:24, 21 January 2012 (UTC)

Done. Opinions?-- F = q(E + v × B) 18:09, 21 January 2012 (UTC)

One problem: Help:Merging says the following:

... note the merger (including the page names) in the edit summaries. (This step is required in order to conform with Wikipedia's licensing requirements. Do not omit it nor omit the page name.)

Perhaps you could redo the edit with the proper edit summary? RockMagnetist (talk) 18:22, 21 January 2012 (UTC)

F###... Sorry - from now on I should read and not overlook these rules inside out. =( -- F = q(E + v × B) 18:48, 21 January 2012 (UTC)

I think i'll just change R³ to "all space" actually, in one edit. It looks better also, and its not like anyone will debate over such a simple matter. Before that, a few minor edits will be needed for consistency with the rest of the article.-- F = q(E + v × B) 19:01, 21 January 2012 (UTC)

Rockmagnetist: Sorry to undo your change - but I already added the banner before, it was identical. Apparently they're supposed to be at the top? That’s how it looks on most talk pages anyway...-- F = q(E + v × B) 21:22, 21 January 2012 (UTC)

Archive 1

Archive 2

Archive 3

Archive 4

Archive 5

so the wave function...

Is it the probability of a particle hitting a space. Like in the double slit experiment is it the probability of the electron hitting any of the points on the electron buildup? — Preceding unsigned comment added by 86.185.99.78 (talk) 18:18, 11 December 2012 (UTC)

The second paragraph says:

"Although ψ is a complex number, |ψ|² is real, and corresponds to the probability density of finding a particle in a given place at a given time, if the particle's position is measured."

which is how the wavefunction is interpreted. Strictly the probabilities are not "at" any points, but within an interval of points (see the integrals). For the double-slit experiment in QM, you could see Schrödinger equation (Particles as waves). Does that help? Maschen (talk) 18:27, 11 December 2012 (UTC)

Yes thank you! :) — Preceding unsigned comment added by 86.185.99.78 (talk) 18:41, 11 December 2012 (UTC)

article contains no mention of wavefunction's importance to charge conservation

This article makes no mention of the deep connection between the gauge invariance of the wavefunction and conservation of electric charge. It would seem to me to be important to note that connection in this article. On a lesser note, don't particles without electric charge have wavefunctions? If so, then how does that work to conserve charge?

BrianStanton61 (talk) 22:42, 17 December 2012 (UTC)

I haven’t come across this particular point in the literature to understand in detail. Are you referring to this? Do you have any secondary sources (textbooks, papers, research monographs etc) that could cite this point your making in the article? Thanks and welcome, M∧Ŝc²ħεИ_τlk 22:58, 17 December 2012 (UTC)

The source I was referring specifically to was the Charge conservation article. It states: This (i.e. changes in the phase of a wavefunction being unobservable) is the ultimate theoretical origin of charge conservation.

67.81.217.212 (talk) 23:18, 17 December 2012 (UTC)

Wikipedia articles don't count as sources or references; it's still necessary to provide sources as said to back up the statement. Of course you can link to the article. M∧Ŝc²ħεИ_τlk 23:52, 17 December 2012 (UTC)

It is surely worth mentioning somewhere in this article that the wave function of a charged particle is not gauge-invariant. It is important in practice, and it's also important conceptually to understand that a wavefunction is at least partly an arbitrary convention (even beyond the famous global phase factor).

But I don't think the connection to charge conservation needs to be mentioned in this article. The "main plot" is the (Noether's theorem) relationship between gauge invariance and charge conservation. Yes, changing the wavefunction happens to be one aspect of changing the electromagnetic gauge. But I don't think that adds up to a particularly direct connection between wavefunctions in general and charge conservation.

(I am not particularly knowledgeable about this and could be wrong.) --Steve (talk) 03:42, 18 December 2012 (UTC)

In light of this thread, would it be worth mentioning the Aharonov-Bohm effect? M∧Ŝc²ħεИ_τlk 11:16, 31 March 2013 (UTC)

Recent edits (late March → early April 2013)

Much of the vector spaces section has been almost completely trimmed/rewritten. The silly mix of inline LaTeX and HTML maths will be fixed later, no time right now. M∧Ŝc²ħεИ_τlk 12:28, 5 April 2013 (UTC)

Wave function as probability of feeling a sensation

On April 18, I changed the words "…the probability density of finding a particle…" to "…the probability density of sensing a particle…." This was reverted by Maschen with his personal comment, "'sensing'? 'finding' was clearer."

Another member of the priesthood, 52 years ago, had made the following statement about wave functions: "…the wave function permits one to foretell with what probabilities the object will make one or another impression on us if we let it interact with us either directly or indirectly." Also, in discussing the communicability of a wave function, he wrote, "If someone else somehow determines the wave function of a system, he can tell me about it and, according to the theory, the probabilities for the possible different impressions (or 'sensations') will be equally large…." (Eugene P. Wigner, "Remarks on the Mind-Body Question," The Scientist Speculates, I. J. Good, ed., pp. 284-302, Heinemann, London)

Professor Wigner clearly asserted that a wave function tells "one" the probability that "one" will see a flash, or a dark spot on a photographic plate, or some other sensation. This is of considerable importance because it relates a wave function's probability to "one's" subjective and inter-subjective private sensations or impressions, not to external, public, objective particles that can be found. The "one" who is mentioned by Professor Wigner is the observing subject in whose nervous system the sensations are felt.Lestrade (talk) 12:50, 19 April 2013 (UTC)Lestrade

"Sensing" has a connotation of human sensory perceptions like hear/see/taste/touch/feel... "Finding the particle" seems far clearer, more scientific and less philosophical, and direct, to me.

But if you feel strongly about this - then add it - I will not revert next time. If someone comes to this page in the future to post a section heading "what does "sensing" have to do with it?", be sure to explain to them.

P.S. In case you say it - I would never disparage/disregard Wigner, a crucial contributor to quantum theory. M∧Ŝc²ħεИ_τlk 17:26, 19 April 2013 (UTC)

Wigner seems to have been dragging science into the realm of the subjective. This is totally unacceptable to any scientist. Science’s supreme goal is to be totally objective. It wants to describe all observed objects without regard to the condition of the observing subject. But this may not be fruitful. It may be impossible to have an observed object without an observing subject. Einstein took the relativity of the object to the subject into account. Quantum mechanics seriously considers the role of the observer. Wigner’s definition of the wave function with regard to the probability of subjective sensation may not seem "far clearer, more scientific and less philosophical, and direct," but it might be a necessary part of the definition of "wave function."Lestrade (talk) 20:56, 19 April 2013 (UTC)Lestrade

Real valued wave function proposed by de Broglie

It should be mentioned in the article that a real valued wave function has been formulated by Louis de Broglie.--188.26.22.131 (talk) 12:44, 2 August 2013 (UTC)

You need a reliable secondary source for this before mentioning it in the article. M∧Ŝc²ħεИ_τlk 07:03, 27 October 2013 (UTC)

Wave function #Requirements

First of all, “continuously differentiable in the sense of distributions” is an oxymoron. The continuous differentiability (C¹) means that the derivative is a continuous function. Not more and not less. If one uses generalized function spaces for such definitions, such as D′, then a requirement to be differentiable certain number of times is meaningless. Any D′ “function”, by construction of the differentiation operator, is differentiable any finite number of times.

Second, if one source on the wave function formalism requires that it must be continuous, then it does not mean yet that the continuity as an absolute requirement represents a mainstream view. Incnis Mrsi (talk) 12:32, 10 January 2014 (UTC)

The line:

"It must everywhere be a continuous function, and continuously differentiable (in the sense of distributions, for potentials that are not functions but are distributions, such as the dirac delta function). "

seems to suggest delta potentials, but I agree it is badly worded to be useless (even if it was technically correct). (I didn't write it by the way, if that helps). The original line read:

"It must everywhere be a continuous function, and continuously differentiable (at least up to all possible first order derivatives)."

I may be able to add more sources to the section later, the Eisberg and Resnick book is a respected one, but agreeably not enough. M∧Ŝc²ħεИ_τlk 22:42, 10 January 2014 (UTC)

So, did you agree with letting W^2,1W^1,2 in? Incnis Mrsi (talk) 21:31, 11 January 2014 (UTC)

Yes, thanks for that. M∧Ŝc²ħεИ_τlk 22:09, 11 January 2014 (UTC)

Compress

Currently the article makes laborious reading in introducing wavefunctions as functions, then vectors, then several sections on wavefunctions as functions, then another several sections on wavefunctions as vectors, concluded by ontology. That's fine, but the article could be streamlined by explaining the two side by side, making more room for sections on examples and other interesting facts. I'll try, anyone is free to revert. M∧Ŝc²ħεИ_τlk 17:07, 11 January 2014 (UTC)

What facts about vector space do you deem very relevant? BTW, look at my recent change in quantum state: the article contains a stuff that this article misses, and it is relevant here more than there. Incnis Mrsi (talk) 21:31, 11 January 2014 (UTC)

For this article: linear combinations, the inner product (should be obvious why), normalization (again obvious), and completeness condition (useful for representing states and operators in an orthogonal basis), for the discrete and continuous bases, which should be orthonormal.

The edits to streamline were not finished, something else came up. I'll carry on now. M∧Ŝc²ħεИ_τlk 22:09, 11 January 2014 (UTC)

It hasn't worked... The article is top-heavy with abstractions... But rather than reverting, could we manually move the sections back (move the Dirac notation after the function space examples, then the Dirac notations for the individual examples like "one spin-0 particle in 1d", "one spin-0 particle in 3d", "one spin particle in 3d" etc...)? M∧Ŝc²ħεИ_τlk 01:11, 12 January 2014 (UTC)

I'll move things back. M∧Ŝc²ħεИ_τlk 12:41, 12 January 2014 (UTC)

units of psi

From the lede:

The SI units for ψ depend on the system. For one particle in three dimensions, its units are m^–3/2. These unusual units are required so that an integral of |ψ|² over a region of three-dimensional space is a unitless probability (i.e., the probability that the particle is in that region).

Question: Why isn't the ψ unit trivially L^–3/2? To integrate the square of this with L³ (volume) over all space, to get unitless P = 1?

What you just said is exactly the case for one particle in 3d, which the example already is. What would the units be for two particles (with positions r₁ and r₂) in n (= 1, 2, 3...) dimensions? We need:

$\int d^{n}\mathbf {r} _{1}d^{n}\mathbf {r} _{2}\left|\psi (\mathbf {r} _{1},\mathbf {r} _{2})\right|^{2}=1$

so for this case ψ has dimensions of L^–2n/2. In general for N particles (with positions r₁, r₂, ..., r_N) in n dimensions we have:

$\int d^{n}\mathbf {r} _{1}d^{n}\mathbf {r} _{2}\ldots d^{n}\mathbf {r} _{N}\left|\psi (\mathbf {r} _{1},\mathbf {r} _{2},\ldots \mathbf {r} _{N})\right|^{2}=1$

so ψ has dimensions of L^–Nn/2. In momentum space the dimensions would be p^–Nn/2 or (MLT^–1)^–Nn/2 = (M^–1L^–1T)^Nn/2. The results would be the same for particles with or without spin because the summations are over the dimensionless spin quantum number.

The units in the lead have been queried before. I think it's time we add a specialized section on this somewhere. The alternative would be to add the units after each normalization condition, although IMO it would be better to describe the units collectively. M∧Ŝc²ħεИ_τlk 10:19, 24 November 2013 (UTC)

You are using m for "metre" which even though linked is bad because so easily misunderstood to mean mass. This is a dimensional argument. For that we use "L" or length which cannot be misunderstood. Insisting on meters is like insisting that the dimensions of m in E =mc^2 is "kg". That's just bad writing. Fix it please. SBHarris 20:17, 24 November 2013 (UTC)

(edit conflict)Yes, m is for the SI unit metre (as the sentence "For one particle in three dimensions, its SI units are m^–3/2." in the lead says), while L is for the dimension of length. Standard symbols for two different things. The reason for not discussing dimensions (in the context of units) and using associated symbols in the lead is to prevent any confusion/conflation with dimensions in the context of space. I'll try and clarify again. M∧Ŝc²ħεИ_τlk 20:27, 24 November 2013 (UTC)

I restored your [length]^–3/2 in the lead and units section. Better? It should get the point across to the reader. Apologies for being slow on that particular uptake. M∧Ŝc²ħεИ_τlk 20:32, 24 November 2013 (UTC)

Thank you! SBHarris 01:11, 25 November 2013 (UTC)

Warning: idiot’s dashes in the thread (11 items). Also, several idiot’s dashes in the article as of now: unfortunately, I did not notice those then edited. Incnis Mrsi (talk) 21:31, 11 January 2014 (UTC)

If you're referring to my edits which used the minus sign the "special characters" -> "symbols" character map atop the edit window, I'm not insulted and will not argue. M∧Ŝc²ħεИ_τlk 22:09, 11 January 2014 (UTC)

Firefox has a nice feature: it can highlight all occurrences of a search term on the page. Can your browser do it? Enter the “–” symbol into the search term bar and look where does it appear in the article and here, on the talk page. Incnis Mrsi (talk) 08:36, 12 January 2014 (UTC)

Maschen deems I’m a great expert in wiki typography and waited for me to have abominations in the article slaughtered. Thanks, I fixed 8 instances of an idiot’s dash. Do not paste the code from this section to articles. Incnis Mrsi (talk) 16:48, 12 January 2014 (UTC)

Uncountably infinite dimensional Hilbert spaces

This section has problems; at the very least its title.

"Uncountable many components" is not the same thing as "Uncountably infinite dimensional Hilbert spaces". The Hilbert spaces involved here are countably infinite dimensional. YohanN7 (talk) 08:18, 12 January 2014 (UTC)

I agree: somebody learned poorly that the cardinality of the set of xes in ψ(x) counts dimensions only in the discrete case (i.e. “function” spaces built atop of an atomic measure). Paradoxically, the probability amplitude article now explains it better than this one. Incnis Mrsi (talk) 08:36, 12 January 2014 (UTC)

Fortunately, Maschen reduced it by now, AFAIK. Incnis Mrsi (talk) 16:48, 12 January 2014 (UTC)

Pauli

Pauli and the Pauli equation belong in the history section. YohanN7 (talk) 08:21, 12 January 2014 (UTC)

What makes Pauli equation less relevant than Schrödinger equation? Incnis Mrsi (talk) 08:36, 12 January 2014 (UTC)

Not sure I understand what you mean. Let me rephrase; Pauli and the Pauli equation are missing from the history section.YohanN7 (talk) 11:28, 12 January 2014 (UTC)

Obviously, and anyone could add him. I'll do it now. M∧Ŝc²ħεИ_τlk 12:41, 12 January 2014 (UTC)

I briefly mentioned it. Incnis Mrsi (talk) 16:48, 12 January 2014 (UTC)

The "off-topic" section...

...is inside here...

Two-state Simple examples can be found from a two-state quantum system, two energy eigenstates: $|\psi \rangle =\psi _{1}|E_{1}\rangle +\psi _{2}|E_{2}\rangle .$ where the basis vectors are the energy eigenstates $|E_{1}\rangle$ for the first energy level, $|E_{2}\rangle$ for the second energy level. (Common alternative notations are simply |1⟩ for and |2⟩, respectively). Another example is two spin states of a spin-½ particle, neglecting spatial degrees of freedom: $|\psi \rangle =c_{1}|\uparrow _{z}\rangle +c_{2}|\downarrow _{z}\rangle ={\begin{bmatrix}c_{1}\\c_{2}\end{bmatrix}}={\begin{bmatrix}\langle \uparrow _{z}|\psi \rangle \\\langle \downarrow _{z}|\psi \rangle \end{bmatrix}},$ where the basis vectors are the spin states $|\uparrow _{z}\rangle$ for "spin up" or sz = +1/2, $|\downarrow _{z}\rangle$ for "spin down" or sz = −1/2, (Common alternative notations include |+⟩ for and |−⟩, respectively). In these examples, the particle is not in any one definite or preferred state, but in a superposition of them: in both states at the same time.

Just moved it to here after this edit so it is not sat in the article in the way of anything. M∧Ŝc²ħεИ_τlk 17:46, 12 January 2014 (UTC)

Non-interacting particles

The multi-particle free wave functions are usually considered to be tensor products, each particle living in a private Hilbert space, and the total wave function residing in the tensor product of these. The way of writing it as an ordinary product is just a matter or notation. Interpreting it as an ordinary product is wrong, because the "natural guess" as to what a multi-particle free wave function ought to be is a sum (by linearity of the governing equation) if it is supposed to be residing in a "single" Hilbert space. (Naturally, (at least some, haven't investigated in detail) operators are (sums of) tensor products too with the identity operator in appropriate places.) YohanN7 (talk) 21:04, 8 March 2014 (UTC)

Yes I'm aware of that, but the section is not about abstract vectors and tensor products only wavefunctions (The symmetrized/antisymmetrized tensor products are linked to through Identical particles#Wavefunction representation). M∧Ŝc²ħεИ_τlk 22:18, 8 March 2014 (UTC)

I am, of course, aware of that you are aware of that, but it still deserves mention in the article because it's borderline whether the average reader (taking QM101) is aware. The literature usually (very vaguely) introduces the tensor product notation (without much (if any) elaboration on tensor products) and then immediately scraps it. One sentence would suffice. I'll try to think of something. Cheers! YohanN7 (talk) 22:27, 8 March 2014 (UTC)

The edits are an improvement, but it is still problematical. The wave functions are state vectors, and that passage (state vectors and Dirac notation) is irrelevant. I'll sleep on the matter and see if I can reformulate it tomorrow. Nightie! YohanN7 (talk) 01:17, 9 March 2014 (UTC)

---

The section is blatantly wrong not for any of YohanN7’s reasons, but because it conflates concepts of interaction and quantum entanglement. We know that an interaction usually leads to entanglement, but it is egregiously wrong to assert that an entangled system of two particles always owes its entanglement to their interaction. Should I clarify the distinction with examples or you already understood me? Incnis Mrsi (talk) 13:55, 9 March 2014 (UTC)

No, please do not do that because no such assertion is made. Also, the quantum entanglement article is confused as to what the definition of entanglement is. On the on hand (consider a pair of identical particles), it's entangled if the state of one particle can't be described without mention of the other; Quantum entanglement is a physical phenomenon that occurs when pairs or groups of particles are generated or interact in ways such that the quantum state of each particle cannot be described independently – instead, a quantum state may be given for the system as a whole., (first sentence in lead). Using this definition we do not have entanglement here. On the other hand, the state of the two-particle system cannot be factorized (either for bosons or fermions) and using the alternative definition (given in quantum entanglement#concept); If the quantum state of a pair of particles is in a definite superposition, and that superposition cannot be factored out into the product of two states (one for each particle), then that pair is entangled, we do have entanglement in the present case.

Thus, let us keep entanglement at arms-length distance here. YohanN7 (talk) 23:25, 9 March 2014 (UTC)

The phrase in question is

For N non-interacting distinguishable particles, which do not interact mutually and move independently, the wavefunction can be separated into a product of separate wavefunctions for each particle…

I do not know what the 3 books referenced below this text say about interactions actually. But any person with a basic notion of quantum information knows that a state of the form |ψ₁⟩⊗|ψ₂⟩ is called non-entangled. Not any state of a composite system of two non-interacting constituents is non-entangled. An entangled but (currently) non-interacting composite system can be produced by a decay, for example. In short, the quoted text is false. Also, I am not interested in what wiki cranks and waste-makers wrote about the quantum entanglement; I just expected that user:YohanN7 might be familiar with the concept. Incnis Mrsi (talk) 18:07, 10 March 2014 (UTC)

Seems like over-explanation. I reduced it. M∧Ŝc²ħεИ_τlk 18:20, 10 March 2014 (UTC)

Oops, I might have been jumping the gun above. I don't know much about this. Is Ψ₁(x₁, ...)Ψ₂(x₂, ...) + Ψ₁(x₂, ...)Ψ₂(x₁, ...) entangled? Does not Ψ₁(x₁, ...) describe one of the involved particles? If so, according to one of the definitions above, there is no entanglement, but according to the other definition (the one also given by Incnis) there is entanglement. I'm too confused at the moment to contribute with anything useful. YohanN7 (talk) 20:54, 10 March 2014 (UTC)

An interesting question that I initially ignored. Classification of Ψ₁ ⊗ Ψ₂ + Ψ₂ ⊗ Ψ₁ depends on whether the particles are identical. If they are not, then the state is entangled. If they are identical bosons, then the state is not. If they are identical fermions, then such state is not possible. Note that the definition of indiscernibility is tricky: you must ensure that all freedoms are accounted for before symmetrizing or antisymmetrizing the tensor square space. Incnis Mrsi (talk) 10:43, 15 March 2014 (UTC)

Thank you for an enlightening answer. YohanN7 (talk) 20:51, 15 March 2014 (UTC)

---

The second paragraph (idential bosons) is, at best, confusing. YohanN7 (talk) 23:25, 9 March 2014 (UTC)

What is confusing?

In response to both of you - the section was originally to sharpen up the previous statement of separability for non-interacting particles, for the cases of any number of distinguishable particles, bosons, and fermions. To have general formulae kept to a minimum of mathematics, and which almost always appear in QM books (hence the citations...). No entanglement. No tensor products or linear algebra, which are linked to Identical particles#Wavefunction representation and Bra–ket notation#Composite bras and kets, and could be described later in the sections which at least attempt to deal with linear algebra in abstract generality.

Would you give an example where particles are entangled and not interacting? Even so this is not the article on quantum entanglement. M∧Ŝc²ħεИ_τlk 17:47, 10 March 2014 (UTC)

It says that the Ψ_i, 1 ≤ i ≤ N all have distinct quantum numbers. Moreover, N_i, 1 ≤ i ≤ N is the number of particles with the quantum number of Ψ_i. This means that either some of the Ψ_i have the same quantum numbers or that there are fewer than N Ψ_i in case N_i > 1 for some i. (Also that reciprocal of the square root had me confused for a while, more immediate would be to say square root of the reciprocal.) YohanN7 (talk) 20:39, 10 March 2014 (UTC)

It's taking up more space than contributing useful content. So it has been temporarily deleted.

Here it is for now:

Non-interacting particles The idea of "non-interacting" is hypothetical, since a system of real particles always interact. For N non-interacting distinguishable particles, the wavefunction can be separated into a product of separate wavefunctions for each particle: $\Psi (\mathbf {r} _{1},\mathbf {r} _{2},\cdots ,\mathbf {r} _{N},t)=\prod _{i=1}^{N}\psi _{i}(\mathbf {r} _{i},t)=\psi _{1}(\mathbf {r} _{1},t)\psi _{2}(\mathbf {r} _{2},t)\cdots \psi _{N}(\mathbf {r} _{N},t)\,.$ This separation of variables is a simple method for solving partial differential equations like the Schrödinger equation. If the particles interact, then the wavefunction cannot be separated into the separate wavefunctions of the particles. For N non-interacting identical bosons, the wavefunction is a sum over permutations (indicated by σ) of one-particle wavefunctions ψ1, ψ2, ..., ψN (each of these has a different set of quantum numbers): $\Psi (\mathbf {r} _{1},\mathbf {r} _{2},\cdots ,\mathbf {r} _{N})={\sqrt {\frac {N_{1}!N_{2}!\cdots N_{N}!}{N!}}}\sum _{\sigma (i_{1},i_{2},\ldots ,i_{N})}\psi _{i_{1}}(\mathbf {r} _{1})\psi _{i_{2}}(\mathbf {r} _{2})\cdots \psi _{i_{N}}(\mathbf {r} _{N})\,,$ where N1 is the number of particles all with the same quantum numbers of ψ1 (the number is called the multiplicity), etc. and the sum is taken over all the permutations of i1, i2, ..., iN which lead to different terms. The number of terms is N! / N1!N2!...NN!, so the reciprocal of the square root of this is the normalization constant. The equivalent for N non-interacting identical fermions is (again each of ψ1, ψ2, ..., ψN has a different set of quantum numbers): $\Psi (\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N})={\frac {1}{\sqrt {N!}}}\sum _{\sigma (i_{1},i_{2},\ldots ,i_{N})}\mathrm {sgn} \sigma \left(i_{1},i_{2},\ldots ,i_{N}\right)\psi _{i_{1}}(\mathbf {r} _{1})\psi _{i_{2}}(\mathbf {r} _{2})\cdots \psi _{i_{N}}(\mathbf {r} _{N})\,,$ where "sgn" is the sign of the permutation: −1 for an odd number of permutations and +1 for an even number of permutations. This expression is the Slater determinant. Note for fermions the multiplicities of all the one-particle wavefunctions can only be 1, by the antisymmetric property. The number of terms in the sum is N!, so the reciprocal of the square root of this is the normalization constant. In these equations, the wavefunctions are complex scalars multiplied according to ordinary complex number arithmetic. Using the formalism of quantum state vectors in Bra–ket notation, the formulae are similar but the ordering of single particle states |ψ1⟩, |ψ2⟩, etc. in each term is important as they are tensor products - a non-commutative operation unlike ordinary complex number multiplication. The single particle states each live in a Hilbert space, and tensor products of the single particle states give new states which live in the corresponding tensor products of the Hilbert spaces (see Identical particles#Wavefunction representation, and below). The tensor product nature of multi-particle states is usually not explicit in the notation the literature (or here). References

M∧Ŝc²ħεИ_τlk 21:05, 10 March 2014 (UTC)

P(Ψ₂→Ψ₁)

This piece of Copenhagen currently resides in the “position space” section, that is misleading. The formula for the transition probability is correct, but it describes the situation where Ψ₁ is an eigenvector of the measurement. This depends on observable and doesn’t depend on the presentation (the position space or whichever). Interpretation of |Ψ(x)| as the probability density corresponds to a measurement of the position operator when in position space (it would be another coordinate operator in other presentations). These are two different scenarios, that usually are incompatble: in the non-discrete case the position operator has a continuous spectrum only. It is sad that my work on probability amplitude did not result in significant improvements of QM literacy here. Incnis Mrsi (talk) 10:43, 15 March 2014 (UTC)

Better?

The new section Inner product should not be a subsection of “position space” section, but I don't have time a t m. YohanN7 (talk) 12:34, 15 March 2014 (UTC)

The additional material on the inner product was placed in that section for concreteness. The other inner products for more complicated systems are given later. Please keep the inner product section there at least for now.

What's missing from this article is a section on how the wavefunction is used to calculate the possible results of observables (eigenvalues of operators), as well as expectation values. Perhaps a general statement of the formula

$P(\Psi _{2}\rightarrow \Psi _{1})=|\langle \Psi _{1},\Psi _{2}\rangle |^{2}$

could go in a section like this somewhere? M∧Ŝc²ħεИ_τlk 13:23, 15 March 2014 (UTC)

I thought to contain all the inner products and terminology together in its own section. M∧Ŝc²ħεИ_τlk 13:42, 15 March 2014 (UTC)

Yup, structurally, that looks good. With the inner product, we have an Inner product space. Then appeal to completeness of the set of eigenfunctions of Hermitean operators and (what more?) conclude that we have a Hilbert space of solutions to the Schrödinger equation, i.e. an inner product space with a complete metric. Just thinking out loud. Not much is needed here, a sentence or two. YohanN7 (talk) 14:48, 15 March 2014 (UTC)

Prior to yesterday, the reason for including superposition followed by the inner product in the first section (1d one particle states position representation) was to provide the very basic features of an inner product space in a concrete setting. This flow is a bit disjointed now, which is why I'm not sure if it was an improvement. M∧Ŝc²ħεИ_τlk 08:58, 16 March 2014 (UTC)

Ironically, the [complete] inner product space argument discredits all C¹ and Sobolev space stuff presented above in the article. By the way I do not see where this completeness has actually any use. Although one can’t rely, without completeness, on the spectral theorem for an arbitrary self-ajoint operator, it will likely hold for practically important operators. In any case, the article must clarify why the metric completeness in one part of the article does not contradict to differentiability restrictions in another. Possibly, we should hint that the concepts of a wave function and of an abstract state vector have certain subtle differences. Namely, one gets a wave function from the Schrödinger equation, but operates with state vectors doing various measurements. Incnis Mrsi (talk) 09:20, 16 March 2014 (UTC)