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Wikipedia talk:List of pages whose correct title is not allowed by MediaWiki

Should the title of this page include the word(s) lower case or lowercase? Which does everyone prefer? -- Mattworld 03:35, 16 Nov 2003 (UTC)
 * *NSYNC has been added, which is incorrect because the first character is not a letter at all, and the first letter is supposed to be capitalized. Wikipedia does not allow the asterisks in article titles, however.  Perhaps this should be moved to  List of pages whose correct title is not allowed by the Wikipedia software or something less verbose.  If so, The '5' Royales should be added, because it should have double quotes, not apostrophes. I know there are more, but can't think of them right now... Tuf-Kat 04:38, Dec 8, 2003 (UTC)

 * Yes, SETI at home comes to mind, which it should be at SETI@home Or, similarly, something like C++ has to be at C Plus Plus. --Minesweeper 12:32, 10 Dec 2003 (UTC)
 * I'll make the move, and add Too $hort. Tuf-Kat 18:12, Dec 10, 2003 (UTC)

C++
I don't think C++ should be spelt c++. -- Sverdrup (talk) 11:56, 12 Dec 2003 (UTC)

 * No, it should be spelt C++, but we can't have +s in page titles at the moment, else things'll go screwy. (Apologies if you already knew that, and I'm answering your question out of context, but nobody else had replied.) - IMSoP 13:34, 12 Mar 2004 (UTC)

Name of page
MediaWiki confused me with a possible bug while I moved this page. What links here seems to have to some caching problems (though it's always hard to tell where those come from) despite my best efforts to reload the page. Please point out if I have missed any double-redirects or other problems in moving this page. --Ellmist 07:07, 21 Dec 2003 (UTC)

Spinal Tap
What's the Unicode for that? :) - Hephaestos 23:52, 8 Jan 2004 (UTC)

 * N, followed by U+308. Your browser displays this like -> 'n̈'. I dunno how that looks, mine just displays the combining diaeresis as a spacing one, instead. Morwen 23:54, Jan 8, 2004 (UTC)

 * This page specifies that it uses ISO-8859-1, not UTF-8, so it's not surprising that the character does not show up right to you or me (I'm using Safari and Firefox to check the rendered page). I would think that all output from wikimedia would be Unicode (UTF-8 or UTF-16).  I also tested using the actual combining diaresis mark ('?' and '?'). --Bill 16:46, 2004 Oct 13 (UTC)

 * No that doesn't quite explain it because my personal Esperanto sandbox renders as UTF-8 and the diaeresis mark still does not show up correctly... I briefly checked monobook.css for a font specification but only saw that it specified a san-serif font. The rendered page must be getting a font specification from somewhere to a specific font that does not have the diaeresis mark.  --Bill 17:18, 2004 Oct 13 (UTC)

 * Nope, it's just a browser bug! 'n̈' looks like an n with two dots on top just fine under Mozilla and Firefox, but breaks under IE; actually, IE displays the second character as a "default glyph"-type square, so there could be something font-ish involved as well. But it's definitely a browser issue, not a MediaWiki one. - IMSoP 19:49, 13 Oct 2004 (UTC)

Why these titles don't work
Are there technical reasons why these titles are not allowed? It seems rather bizarre that certain characters can't be in titles -- I'm sure there are plenty of valid reasons to automatically capitalize the first letter, but the rest of them... Would putting the + in article titles really break something, or is it just developmental intertia? Tuf-Kat 08:49, Mar 12, 2004 (UTC)

 * The short answer is "yes". The long answer varies depending what character is being disallowed, but here are a few reasons:
 * + is used in web addresses to represent a space (e.g. when you type more than one word into a search engine). Using it in article names would potentially make parts of the system see their name wrong.
 * @ also has a special meaning in URLs, as a way of adding a username and password, and would have even more drastic consequences.
 * [, ], {, }, |, and probably some of the others listed at Naming conventions (technical restrictions) have special meaning within Wikipedia's syntax, which are processed before the pagename is determined. (e.g. 2024 points at 2024, not a page called  2024 .
 * $, \, ", ` (and some others) have special meaning in other bits of the software, and allowing them would create potential security flaws which would take a lot of effort to insure against.
 * Really weird characters, like the Spinal Tap case above, can only be represented using Unicode, whereas the English Wikipedia just uses ISO 8859-1 (I think). I guess that one's partly just "developmental inertia", arguably.
 * HTH - IMSoP 13:34, 12 Mar 2004 (UTC)

 * Thanks. Tuf-Kat

Stopping forced capitalization (from Village Pump)
[moved from Village pump by IMSoP 23:38, 17 Mar 2004 (UTC)]

The Pillows (a Japanese rock band) should actually be "the pillows" -- I've never seen their name capitalized on any official source. It would be easy enough to change this in the text, and I was about to, but first I tried to move the article to "the pillows." However, Wikipedia forced the capitalization of the T: "The pillows." Is there a way to force Wikipedia to keep that T lowercase? Garrett Albright 04:44, 12 Mar 2004 (UTC)

 * Nope. There's a page on Wikipedia somewhere that lists the articles that can't have a proper name because of Wikipedia limitations.  Other things like the iMac and the iPod are also incorrect, but so is L'Arc~en~Ciel (can't use tildes).  RadicalBender 04:46, 12 Mar 2004 (UTC)

 * Yes it can. L'Arc~en~Ciel is entirely possible. It's currently under review for a move, as the destination (with ~s) already exists. - ^demon 18:25, 10 November 2005 (UTC)

 * Take a gander at List of pages whose correct title is not allowed by MediaWiki. --Minesweeper 04:53, Mar 12, 2004 (UTC)

Created template
I have created the template : {wrongtitle} I have put it on C Plus Plus and will let it bake a bit to see if people like it before moving it onto other articles on this page (and mentioning the template on the page itself). Feel free to edit it at Template:wrongtitle. --TreyHarris 16:52, 14 Jun 2004 (UTC)

 * It's possible to pass on parameters to the template and use them, e.g. would expand to something like:
 *  pH is the correct title, but can't be used to due to technical limitations. The text below uses the correct terminology.
 * -User:Docu

I have made this change and filled it in for the articles using the template. You can say to get: {wrongtitle|title=correctTitle} --TreyHarris 22:49, 11 Jul 2004 (UTC)

Janacek
What about Leoš Janácek, currently at Leos Janacek? Hyacinth 22:04, 25 Jul 2004 (UTC)

bash
The bash home page does not seem to be consistent on whether the correct usage is BASH, Bash, or bash. I think we can remove bash from the list since Bash is an accepted spelling. --Bill 16:15, 2004 Oct 13 (UTC)

A Proposal
This idea is in my head for some time: the real problem is the title shown in the article, not in the URL. The title is automatically set from the article name in the URL and the first letter is capitalized. While this is normally perfect and should not be changed due to incompatibilities, dangling links a.s.o, it could help if the title can be set or suppressed in the article contents. My proposal would be to be able to set the title in an article, so one could create an article named C plus plus and show a title C++. One could create a tag <tilte>C++</title> which suppresses the normal title generated and replaces this with the contents of the title tag. Or one could create a __NOTITLE__ magic (like ) and use =....= (first entry inside toc) as title. This would lead to

 * free setting of title inside article including small letter at the beginning, all characters
 * No change in links, since the URL, Linkrefs is unchanged. One links to C++ (C++) as usual an gets an article with C++ as title
 * All articles can show the correct title, if it is expressible in wikipedia syntax, even in foreign languages
 * only addition of NOTITLE and and correct heading in article required

The C++ article would start like this __NOTITLE__ =C++= ...

--Hubi 11:25, 25 Feb 2005 (UTC)

Thousands of articles
The "What links here" appears to have a limit of 500 articles. Any way to modify the function in case thousands of Wikipedia articles have the template?? Georgia guy 14:49, 23 Apr 2005 (UTC)
 * It's not a hard limit; you can get the next 500 by clicking on the text "next 500". I have added Category:Articles with unsupported titles to the template, so you can use that instead, if you prefer. (It's sorted alphabetically, which is nicer.) -- Beland 05:51, 13 December 2005 (UTC)


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Talk:C standard library

Remove drive-by tagging by three-edit one-day anon IP
— MaxEnt 23:46, 20 March 2014 (UTC)
 * "Three-edit one-day anon" isn't a valid reason (this could be anyone who edited from an uncommon location). But the fact that there's nothing clearly "misleading" and no clarification is. — Vano 00:53, 21 March 2014 (UTC)
 * Yeah, IP addresses are human too, but the hatnote wasn't appropriate; I'd like to know what looked so misleading? — Dsimic (talk | contribs) 19:00, 22 March 2014 (UTC)

About subarticles (e.g. C mathematical functions)
There are links to a site called cppreference.com (and almost all subarticles has those links) and it looks like WP:SOAP and I think (for example) the link to should be replaced with  or. What do you think? --betseg (talk) 21:13, 15 November 2015 (UTC)

 * I don't know the background but a quick look shows cppreference.com is pretty good. By the way, WP:SOAP is referring to advocacy (arguing, for example, in favor of a particular political outlook). I guess you are suggesting the website is being promoted (WP:PROMO). I don't know, but evidence for that would be if a small number of editors were focusing on adding that link. Issues such as whether a link is spam are normally discussed at WP:ELN whereas there might be more attention to which is the best C reference at WT:COMP, with a link to the discussion at Talk:C (programming language) and here. Johnuniq (talk) 22:11, 15 November 2015 (UTC)

 * cppreference.com is "at top" or "near top" of C function searches on google, just like Wikipedia, so they can't be all bad. I've used those links and their information is pretty good. I like how they state the section number for each C/C++ specification.  •  Sbmeirow  •  Talk  • 01:58, 16 November 2015 (UTC)
 * Or maybe the site is at top because Wikipedia has a lot of links to the site? That's how Google works. --betseg (talk) 05:32, 19 November 2015 (UTC)
 * The engine automatically adds rel="nofollow" to every external link to prevent search engines from considering them in ranking - specifically to discourage this kind of "promotion". Though I agree that a site shall be judged solely by its content rather than position in a search result. — Vano 10:50, 20 November 2015 (UTC)
 * I think  would be the perfect thing to use.--betseg (talk) 12:08, 22 November 2015 (UTC)

Linking libm on linux
None of the linux distributions, over the past 18 years, I have every used, have required me to link with libm. The claim that you *must* link with -lm seems dubious if not outright wrong. Can anyone confirm that this real today? — Preceding unsigned comment added by 188.176.25.249 (talk) 09:18, 12 February 2019 (UTC)

Reading the page, I believe it's only saying you need -lm if you actually use any math functions (sqrt, sin, etc.), which comports with my experience. SJFriedl (talk) 21:56, 6 June 2020 (UTC)

A Commons file used on this page or its Wikidata item has been nominated for speedy deletion
The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for speedy deletion: You can see the reason for deletion at the file description page linked above. —Community Tech bot (talk) 19:09, 18 August 2022 (UTC)
 * 1999 ISO C.pdf

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title: Talk:C standard library


text:
Baasha of Israel

Baasha (, Baʿšāʾ ) was, according to the Hebrew Bible, the third king of the northern Israelite Kingdom of Israel. He was the son of Ahijah of the Tribe of Issachar. Baasha's story is told in.

Reign
Baasha became king of Israel in the third year of Asa, king of Judah. William F. Albright has dated his reign to 900–877 BCE, while E. R. Thiele offers the dates 909–886 BCE. Baasha came to power by murdering the previous king, Nadab, at Gibbethon, followed by the entire House of Jeroboam (Nadab's father and predecessor). Baasha had previously been a captain in Nadab's own army. Like many military leaders, he appears to have risen from obscurity. The Jewish Encyclopedia suggests that because he came from the tribe of Issachar, "he may have represented a local faction".

Over the course of his 24-year reign, Baasha was at war with Asa, king of Judah. He allied Israel with Aram and endeavored to strangle Judah's trade by fortifying Ramah, a city five miles north of Jerusalem. King Asa of Judah then bribed King Ben-hadad of Syria to switch sides and attack Israel, prompting the loss of extensive territory in Dan and Naphtali northwest of the Sea of Galilee. Baasha was forced to withdraw from Ramah. Asa of Judah utilized the materials of the abandoned fort for the fortification of his own frontier towns, Geba and Mizpah.

Though Baasha remained in power for life, he was not without his opponents. The prophet Jehu, the son of Hanani, foretold the destruction of his dynasty, which came to pass with the assassination of Baasha's son Elah.

title: Baasha of Israel


text: Talk:Milton Berle

title: Talk:Milton Berle


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Matrix mechanics

Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum jumps supplanted the Bohr model's electron orbits. It did so by interpreting the physical properties of particles as matrices that evolve in time. It is equivalent to the Schrödinger wave formulation of quantum mechanics, as manifest in Dirac's bra–ket notation.

In some contrast to the wave formulation, it produces spectra of (mostly energy) operators by purely algebraic, ladder operator methods. Relying on these methods, Wolfgang Pauli derived the hydrogen atom spectrum in 1926, before the development of wave mechanics.

Development of matrix mechanics
In 1925, Werner Heisenberg, Max Born, and Pascual Jordan formulated the matrix mechanics representation of quantum mechanics.

Epiphany at Helgoland
In 1925 Werner Heisenberg was working in Göttingen on the problem of calculating the spectral lines of hydrogen. By May 1925 he began trying to describe atomic systems by observables only. On June 7, after weeks of failing to alleviate his hay fever with aspirin and cocaine, Heisenberg left for the pollen-free North Sea island of Helgoland. While there, in between climbing and memorizing poems from Goethe's West-östlicher Diwan, he continued to ponder the spectral issue and eventually realised that adopting non-commuting observables might solve the problem. He later wrote: "It was about three o' clock at night when the final result of the calculation lay before me. At first I was deeply shaken. I was so excited that I could not think of sleep. So I left the house and awaited the sunrise on the top of a rock."

The three fundamental papers
After Heisenberg returned to Göttingen, he showed Wolfgang Pauli his calculations, commenting at one point: "Everything is still vague and unclear to me, but it seems as if the electrons will no more move on orbits."

On July 9 Heisenberg gave the same paper of his calculations to Max Born, saying that "he had written a crazy paper and did not dare to send it in for publication, and that Born should read it and advise him" prior to publication. Heisenberg then departed for a while, leaving Born to analyse the paper.

In the paper, Heisenberg formulated quantum theory without sharp electron orbits. Hendrik Kramers had earlier calculated the relative intensities of spectral lines in the Sommerfeld model by interpreting the Fourier coefficients of the orbits as intensities. But his answer, like all other calculations in the old quantum theory, was only correct for large orbits.

Heisenberg, after a collaboration with Kramers, began to understand that the transition probabilities were not quite classical quantities, because the only frequencies that appear in the Fourier series should be the ones that are observed in quantum jumps, not the fictional ones that come from Fourier-analyzing sharp classical orbits. He replaced the classical Fourier series with a matrix of coefficients, a fuzzed-out quantum analog of the Fourier series. Classically, the Fourier coefficients give the intensity of the emitted radiation, so in quantum mechanics the magnitude of the matrix elements of the position operator were the intensity of radiation in the bright-line spectrum. The quantities in Heisenberg's formulation were the classical position and momentum, but now they were no longer sharply defined. Each quantity was represented by a collection of Fourier coefficients with two indices, corresponding to the initial and final states.

When Born read the paper, he recognized the formulation as one which could be transcribed and extended to the systematic language of matrices, which he had learned from his study under Jakob Rosanes at Breslau University. Born, with the help of his assistant and former student Pascual Jordan, began immediately to make the transcription and extension, and they submitted their results for publication; the paper was received for publication just 60 days after Heisenberg's paper.

A follow-on paper was submitted for publication before the end of the year by all three authors. (A brief review of Born's role in the development of the matrix mechanics formulation of quantum mechanics along with a discussion of the key formula involving the non-commutativity of the probability amplitudes can be found in an article by Jeremy Bernstein. A detailed historical and technical account can be found in Mehra and Rechenberg's book The Historical Development of Quantum Theory. Volume 3. The Formulation of Matrix Mechanics and Its Modifications 1925–1926. )

Up until this time, matrices were seldom used by physicists; they were considered to belong to the realm of pure mathematics. Gustav Mie had used them in a paper on electrodynamics in 1912 and Born had used them in his work on the lattices theory of crystals in 1921. While matrices were used in these cases, the algebra of matrices with their multiplication did not enter the picture as they did in the matrix formulation of quantum mechanics.

Born, however, had learned matrix algebra from Rosanes, as already noted, but Born had also learned Hilbert's theory of integral equations and quadratic forms for an infinite number of variables as was apparent from a citation by Born of Hilbert's work Grundzüge einer allgemeinen Theorie der Linearen Integralgleichungen published in 1912.

Jordan, too, was well equipped for the task. For a number of years, he had been an assistant to Richard Courant at Göttingen in the preparation of Courant and David Hilbert's book Methoden der mathematischen Physik I, which was published in 1924. This book, fortuitously, contained a great many of the mathematical tools necessary for the continued development of quantum mechanics.

In 1926, John von Neumann became assistant to David Hilbert, and he would coin the term Hilbert space to describe the algebra and analysis which were used in the development of quantum mechanics.

A linchpin contribution to this formulation was achieved in Dirac's reinterpretation/synthesis paper of 1925, which invented the language and framework usually employed today, in full display of the noncommutative structure of the entire construction.

Heisenberg's reasoning
Before matrix mechanics, the old quantum theory described the motion of a particle by a classical orbit, with well defined position and momentum X(t), P(t), with the restriction that the time integral over one period T of the momentum times the velocity must be a positive integer multiple of Planck's constant $$ \int_0^T P \; {dX \over dt} \; dt = \int_0^T P \; dX = n h .$$ While this restriction correctly selects orbits with more or less the right energy values En, the old quantum mechanical formalism did not describe time dependent processes, such as the emission or absorption of radiation.

When a classical particle is weakly coupled to a radiation field, so that the radiative damping can be neglected, it will emit radiation in a pattern that repeats itself every orbital period. The frequencies that make up the outgoing wave are then integer multiples of the orbital frequency, and this is a reflection of the fact that X(t) is periodic, so that its Fourier representation has frequencies 2πn/T only. $$X(t) = \sum_{n=-\infty}^\infty e^{2\pi i nt / T} X_n.$$ The coefficients Xn are complex numbers. The ones with negative frequencies must be the complex conjugates of the ones with positive frequencies, so that X(t) will always be real, $$ X_n = X_{-n}^* .$$

A quantum mechanical particle, on the other hand, cannot emit radiation continuously; it can only emit photons. Assuming that the quantum particle started in orbit number n, emitted a photon, then ended up in orbit number m, the energy of the photon is $E_{n}−E_{m}$, which means that its frequency is $(E_{n} − E_{m})/h$.

For large n and m, but with n−m relatively small, these are the classical frequencies by Bohr's correspondence principle $$ E_n-E_m \approx h(n-m)/T.$$ In the formula above, T is the classical period of either orbit n or orbit m, since the difference between them is higher order in h. But for n and m small, or if n − m is large, the frequencies are not integer multiples of any single frequency.

Since the frequencies that the particle emits are the same as the frequencies in the Fourier description of its motion, this suggests that something in the time-dependent description of the particle is oscillating with frequency $(E_{n}−E_{m})/h$. Heisenberg called this quantity Xnm, and demanded that it should reduce to the classical Fourier coefficients in the classical limit. For large values of n, m but with n − m relatively small, Xnm is the $(n−m)$th Fourier coefficient of the classical motion at orbit n. Since Xnm has opposite frequency to Xmn, the condition that X is real becomes $$X_{nm} = X_{mn}^*.$$

By definition, Xnm only has the frequency $(E_{n}−E_{m})/h$, so its time evolution is simple: $$ X_{nm}(t) = e^{2\pi i(E_n - E_m)t / h} X_{nm}(0) = e^{i(E_n - E_m)t / \hbar} X_{nm}(0) .$$ This is the original form of Heisenberg's equation of motion.

Given two arrays Xnm and Pnm describing two physical quantities, Heisenberg could form a new array of the same type by combining the terms XnkPkm, which also oscillate with the right frequency. Since the Fourier coefficients of the product of two quantities is the convolution of the Fourier coefficients of each one separately, the correspondence with Fourier series allowed Heisenberg to deduce the rule by which the arrays should be multiplied, $$(XP)_{mn} = \sum_{k=0}^\infty X_{mk} P_{kn}.$$

Born pointed out that this is the law of matrix multiplication, so that the position, the momentum, the energy, all the observable quantities in the theory, are interpreted as matrices. Under this multiplication rule, the product depends on the order: XP is different from PX.

The X matrix is a complete description of the motion of a quantum mechanical particle. Because the frequencies in the quantum motion are not multiples of a common frequency, the matrix elements cannot be interpreted as the Fourier coefficients of a sharp classical trajectory. Nevertheless, as matrices, X(t) and P(t) satisfy the classical equations of motion; also see Ehrenfest's theorem, below.

Matrix basics
When it was introduced by Werner Heisenberg, Max Born and Pascual Jordan in 1925, matrix mechanics was not immediately accepted and was a source of controversy, at first. Schrödinger's later introduction of wave mechanics was greatly favored.

Part of the reason was that Heisenberg's formulation was in an odd mathematical language, for the time, while Schrödinger's formulation was based on familiar wave equations. But there was also a deeper sociological reason. Quantum mechanics had been developing by two paths, one led by Einstein, who emphasized the wave–particle duality he proposed for photons, and the other led by Bohr, that emphasized the discrete energy states and quantum jumps that Bohr discovered. De Broglie had reproduced the discrete energy states within Einstein's framework—the quantum condition is the standing wave condition, and this gave hope to those in the Einstein school that all the discrete aspects of quantum mechanics would be subsumed into a continuous wave mechanics.

Matrix mechanics, on the other hand, came from the Bohr school, which was concerned with discrete energy states and quantum jumps. Bohr's followers did not appreciate physical models that pictured electrons as waves, or as anything at all. They preferred to focus on the quantities that were directly connected to experiments.

In atomic physics, spectroscopy gave observational data on atomic transitions arising from the interactions of atoms with light quanta. The Bohr school required that only those quantities that were in principle measurable by spectroscopy should appear in the theory. These quantities include the energy levels and their intensities but they do not include the exact location of a particle in its Bohr orbit. It is very hard to imagine an experiment that could determine whether an electron in the ground state of a hydrogen atom is to the right or to the left of the nucleus. It was a deep conviction that such questions did not have an answer.

The matrix formulation was built on the premise that all physical observables are represented by matrices, whose elements are indexed by two different energy levels. The set of eigenvalues of the matrix were eventually understood to be the set of all possible values that the observable can have. Since Heisenberg's matrices are Hermitian, the eigenvalues are real.

If an observable is measured and the result is a certain eigenvalue, the corresponding eigenvector is the state of the system immediately after the measurement. The act of measurement in matrix mechanics 'collapses' the state of the system. If one measures two observables simultaneously, the state of the system collapses to a common eigenvector of the two observables. Since most matrices don't have any eigenvectors in common, most observables can never be measured precisely at the same time. This is the uncertainty principle.

If two matrices share their eigenvectors, they can be simultaneously diagonalized. In the basis where they are both diagonal, it is clear that their product does not depend on their order because multiplication of diagonal matrices is just multiplication of numbers. The uncertainty principle, by contrast, is an expression of the fact that often two matrices A and B do not always commute, i.e., that AB − BA does not necessarily equal 0. The fundamental commutation relation of matrix mechanics, $$\sum_k ( X_{nk} P_{km} - P_{nk} X_{km}) = i\hbar \, \delta_{nm}$$ implies then that there are no states that simultaneously have a definite position and momentum.

This principle of uncertainty holds for many other pairs of observables as well. For example, the energy does not commute with the position either, so it is impossible to precisely determine the position and energy of an electron in an atom.

Nobel Prize
In 1928, Albert Einstein nominated Heisenberg, Born, and Jordan for the Nobel Prize in Physics. The announcement of the Nobel Prize in Physics for 1932 was delayed until November 1933. It was at that time that it was announced Heisenberg had won the Prize for 1932 "for the creation of quantum mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen" and Erwin Schrödinger and Paul Adrien Maurice Dirac shared the 1933 Prize "for the discovery of new productive forms of atomic theory".

It might well be asked why Born was not awarded the Prize in 1932, along with Heisenberg, and Bernstein proffers speculations on this matter. One of them relates to Jordan joining the Nazi Party on May 1, 1933, and becoming a stormtrooper. Jordan's Party affiliations and Jordan's links to Born may well have affected Born's chance at the Prize at that time. Bernstein further notes that when Born finally won the Prize in 1954, Jordan was still alive, while the Prize was awarded for the statistical interpretation of quantum mechanics, attributable to Born alone.

Heisenberg's reactions to Born for Heisenberg receiving the Prize for 1932 and for Born receiving the Prize in 1954 are also instructive in evaluating whether Born should have shared the Prize with Heisenberg. On November 25, 1933, Born received a letter from Heisenberg in which he said he had been delayed in writing due to a "bad conscience" that he alone had received the Prize "for work done in Göttingen in collaboration – you, Jordan and I." Heisenberg went on to say that Born and Jordan's contribution to quantum mechanics cannot be changed by "a wrong decision from the outside."

In 1954, Heisenberg wrote an article honoring Max Planck for his insight in 1900. In the article, Heisenberg credited Born and Jordan for the final mathematical formulation of matrix mechanics and Heisenberg went on to stress how great their contributions were to quantum mechanics, which were not "adequately acknowledged in the public eye."

Mathematical development
Once Heisenberg introduced the matrices for X and P, he could find their matrix elements in special cases by guesswork, guided by the correspondence principle. Since the matrix elements are the quantum mechanical analogs of Fourier coefficients of the classical orbits, the simplest case is the harmonic oscillator, where the classical position and momentum, X(t) and P(t), are sinusoidal.

Harmonic oscillator
In units where the mass and frequency of the oscillator are equal to one (see nondimensionalization), the energy of the oscillator is $$ H = {1 \over 2} \left(P^2 + X^2\right) .$$

The level sets of $H$ are the clockwise orbits, and they are nested circles in phase space. The classical orbit with energy $E$ is $$ X(t)= \sqrt{2E}\cos(t), \qquad P(t) = - \sqrt{2E}\sin(t) ~. $$

The old quantum condition dictates that the integral of $P dX$ over an orbit, which is the area of the circle in phase space, must be an integer multiple of Planck's constant. The area of the circle of radius $\sqrt{2E}$ is $2πE$. So $$ E = \frac{nh}{2\pi} = n \hbar \, ,$$ or, in natural units where $ħ = 1$, the energy is an integer.

The Fourier components of $X(t)$ and $P(t)$ are simple, and more so if they are combined into the quantities $$A(t) = X(t) + i P(t) = \sqrt{2E}\,e^{-it}, \quad A^\dagger(t) = X(t) - i P(t) = \sqrt{2E}\,e^{it}.$$ Both $A$ and $A^{†}$ have only a single frequency, and X and P can be recovered from their sum and difference.

Since $A(t)$ has a classical Fourier series with only the lowest frequency, and the matrix element $A_{mn}$ is the $(m − n)$-th Fourier coefficient of the classical orbit, the matrix for $A$ is nonzero only on the line just above the diagonal, where it is equal to $\sqrt{2E_{n}}|undefined$. The matrix for $A^{†}$ is likewise only nonzero on the line below the diagonal, with the same elements. Thus, from $A$ and $A^{†}$, reconstruction yields $$\sqrt{2} X(0)= \sqrt{\hbar} \; \begin{bmatrix} 0 & \sqrt{1} & 0 & 0 & 0 & \cdots \\ \sqrt{1} & 0 & \sqrt{2} & 0 & 0 & \cdots \\ 0 & \sqrt{2} & 0 & \sqrt{3} & 0 & \cdots \\ 0 & 0 & \sqrt{3} & 0 & \sqrt{4} & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{bmatrix},$$ and $$\sqrt{2} P(0) = \sqrt{\hbar} \; \begin{bmatrix} 0 & -i\sqrt{1} & 0 & 0 & 0 & \cdots \\ i\sqrt{1} & 0 & -i\sqrt{2} & 0 & 0 & \cdots \\ 0 & i\sqrt{2} & 0 & -i\sqrt{3} & 0 & \cdots \\ 0 & 0 & i\sqrt{3} & 0 & -i\sqrt{4} & \cdots\\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{bmatrix},$$ which, up to the choice of units, are the Heisenberg matrices for the harmonic oscillator. Both matrices are hermitian, since they are constructed from the Fourier coefficients of real quantities.

Finding $X(t)$ and $P(t)$ is direct, since they are quantum Fourier coefficients so they evolve simply with time, $$X_{mn}(t) = X_{mn}(0) e^{i(E_m - E_n)t},\quad P_{mn}(t) = P_{mn}(0) e^{i(E_m -E_n)t}~.$$

The matrix product of $X$ and $P$ is not hermitian, but has a real and imaginary part. The real part is one half the symmetric expression $XP + PX$, while the imaginary part is proportional to the commutator $$[X,P] = (XP - PX).$$ It is simple to verify explicitly that $XP − PX$ in the case of the harmonic oscillator, is $iħ$, multiplied by the identity.

It is likewise simple to verify that the matrix $$H ={1\over 2}(X^2 + P^2)$$ is a diagonal matrix, with eigenvalues $E_{i}$.

Conservation of energy
The harmonic oscillator is an important case. Finding the matrices is easier than determining the general conditions from these special forms. For this reason, Heisenberg investigated the anharmonic oscillator, with Hamiltonian $$H = {1\over 2} P^2 + {1\over 2} X^2 + \varepsilon X^3 ~.$$

In this case, the $X$ and $P$ matrices are no longer simple off-diagonal matrices, since the corresponding classical orbits are slightly squashed and displaced, so that they have Fourier coefficients at every classical frequency. To determine the matrix elements, Heisenberg required that the classical equations of motion be obeyed as matrix equations, $${dX \over dt} = P \quad {dP \over dt} = - X - 3 \varepsilon X^2 ~.$$

He noticed that if this could be done, then $H$, considered as a matrix function of $X$ and $P$, will have zero time derivative. $${dH\over dt} = P*{dP\over dt} + ( X + 3 \varepsilon X^2)*{dX\over dt} = 0 ~,$$ where $A∗B$ is the anticommutator, $$A*B = {1\over 2}(AB+BA) ~.$$

Given that all the off diagonal elements have a nonzero frequency; $H$ being constant implies that $H$ is diagonal. It was clear to Heisenberg that in this system, the energy could be exactly conserved in an arbitrary quantum system, a very encouraging sign.

The process of emission and absorption of photons seemed to demand that the conservation of energy will hold at best on average. If a wave containing exactly one photon passes over some atoms, and one of them absorbs it, that atom needs to tell the others that they can't absorb the photon anymore. But if the atoms are far apart, any signal cannot reach the other atoms in time, and they might end up absorbing the same photon anyway and dissipating the energy to the environment. When the signal reached them, the other atoms would have to somehow recall that energy. This paradox led Bohr, Kramers and Slater to abandon exact conservation of energy. Heisenberg's formalism, when extended to include the electromagnetic field, was obviously going to sidestep this problem, a hint that the interpretation of the theory will involve wavefunction collapse.

Differentiation trick — canonical commutation relations
Demanding that the classical equations of motion are preserved is not a strong enough condition to determine the matrix elements. Planck's constant does not appear in the classical equations, so that the matrices could be constructed for many different values of $ħ$ and still satisfy the equations of motion, but with different energy levels.

So, in order to implement his program, Heisenberg needed to use the old quantum condition to fix the energy levels, then fill in the matrices with Fourier coefficients of the classical equations, then alter the matrix coefficients and the energy levels slightly to make sure the classical equations are satisfied. This is clearly not satisfactory. The old quantum conditions refer to the area enclosed by the sharp classical orbits, which do not exist in the new formalism.

The most important thing that Heisenberg discovered is how to translate the old quantum condition into a simple statement in matrix mechanics.

To do this, he investigated the action integral as a matrix quantity, $$\int_0^T \sum_k P_{mk}(t) {dX_{kn} \over dt} dt \,\, \stackrel{\scriptstyle?}{\approx} \,\, J_{mn} ~.$$

There are several problems with this integral, all stemming from the incompatibility of the matrix formalism with the old picture of orbits. Which period T should be used? Semiclassically, it should be either m or n, but the difference is order $ħ$, and an answer to order $ħ$ is sought. The quantum condition tells us that Jmn is 2πn on the diagonal, so the fact that J is classically constant tells us that the off-diagonal elements are zero.

His crucial insight was to differentiate the quantum condition with respect to n. This idea only makes complete sense in the classical limit, where n is not an integer but the continuous action variable J, but Heisenberg performed analogous manipulations with matrices, where the intermediate expressions are sometimes discrete differences and sometimes derivatives.

In the following discussion, for the sake of clarity, the differentiation will be performed on the classical variables, and the transition to matrix mechanics will be done afterwards, guided by the correspondence principle.

In the classical setting, the derivative is the derivative with respect to J of the integral which defines J, so it is tautologically equal to 1. $$\begin{align} &{} {d \over dJ } \int_0^T P dX = 1 \\ &= \int_0^T dt \left( {dP\over dJ} {dX\over dt} + P{d\over dJ}{dX\over dt} \right) \\ &= \int_0^T dt \left( {dP\over dJ} {dX\over dt} - {dP\over dt}{dX\over dJ} \right) \end{align} $$ where the derivatives dP/dJ and dX/dJ should be interpreted as differences with respect to J at corresponding times on nearby orbits, exactly what would be obtained if the Fourier coefficients of the orbital motion were differentiated. (These derivatives are symplectically orthogonal in phase space to the time derivatives dP/dt and dX/dt).

The final expression is clarified by introducing the variable canonically conjugate to J, which is called the angle variable θ: The derivative with respect to time is a derivative with respect to θ, up to a factor of 2πT, $${2\pi\over T} \int_0^T dt \left( {dp \over dJ} {dX\over d\theta} - {dP \over d\theta} {dX\over dJ}\right) = 1 \, .$$ So the quantum condition integral is the average value over one cycle of the Poisson bracket of X and P.

An analogous differentiation of the Fourier series of P dX demonstrates that the off-diagonal elements of the Poisson bracket are all zero. The Poisson bracket of two canonically conjugate variables, such as X and P, is the constant value 1, so this integral really is the average value of 1; so it is 1, as we knew all along, because it is dJ/dJ after all. But Heisenberg, Born and Jordan, unlike Dirac, were not familiar with the theory of Poisson brackets, so, for them, the differentiation effectively evaluated {X, P} in J, θ coordinates.

The Poisson Bracket, unlike the action integral, does have a simple translation to matrix mechanics−−it normally corresponds to the imaginary part of the product of two variables, the commutator.

To see this, examine the (antisymmetrized) product of two matrices A and B in the correspondence limit, where the matrix elements are slowly varying functions of the index, keeping in mind that the answer is zero classically.

In the correspondence limit, when indices m, n are large and nearby, while k,r are small, the rate of change of the matrix elements in the diagonal direction is the matrix element of the J derivative of the corresponding classical quantity. So its possible to shift any matrix element diagonally through the correspondence, $$A_{(m+r) (n+r)} - A_{mn} \approx r\; \left({dA\over dJ}\right)_{m n} $$ where the right hand side is really only the (m − n)'th Fourier component of dA/dJ at the orbit near m to this semiclassical order, not a full well-defined matrix.

The semiclassical time derivative of a matrix element is obtained up to a factor of i by multiplying by the distance from the diagonal, $$ik A_{m (m+k)} \approx \left({T\over 2\pi} {dA\over dt}\right)_{m (m+k)} =\left({dA\over d\theta}\right)_{m (m+k)}\, .$$ since the coefficient Am(m+k) is semiclassically the k'th Fourier coefficient of the m-th classical orbit.

The imaginary part of the product of A and B can be evaluated by shifting the matrix elements around so as to reproduce the classical answer, which is zero.

The leading nonzero residual is then given entirely by the shifting. Since all the matrix elements are at indices which have a small distance from the large index position (m,m), it helps to introduce two temporary notations: $A[r,k] = A_{(m+r)(m+k)}$ for the matrices, and $(dA/dJ)[r]$ for the r'th Fourier components of classical quantities, $$\begin{align} (AB - BA)[0,k] &= \sum_{r=-\infty}^{\infty} \left( A[0,r] B[r,k] - A[r,k] B[0,r] \right) \\ &= \sum_r \left(\; A[-r+k,k] + (r-k){dA \over dJ}[r]\; \right) \left(\; B[0,k-r] + r {dB\over dJ}[r-k] \; \right) - \sum_r A[r,k]B[0,r]\,. \end{align}$$

Flipping the summation variable in the first sum from $r$ to r' = k − r, the matrix element becomes, $$ \sum_{r'} (\;A[r',k] - r' {dA \over dJ}[k-r']\;)\left(\; B[0,r'] +(k-r'){dB\over dJ}[r']\;\right)- \sum_r A[r,k] B[0,r] $$ and it is clear that the principal (classical) part cancels.

The leading quantum part, neglecting the higher order product of derivatives in the residual expression, is then equal to $$\sum_{r'} \left(\; {dB\over dJ}[r'](k-r')A[r',k] - {dA\over dJ}[k-r'] r' B[0,r']\right)$$ so that, finally, $$ (AB - BA)[0,k] =\sum_{r'} \left(\; {dB\over dJ}[r']i{dA\over d\theta}[k-r'] - {dA\over dJ}[k-r'] i {dB\over d\theta}[r']\right) $$ which can be identified with $i$ times the $k$-th classical Fourier component of the Poisson bracket.

Heisenberg's original differentiation trick was eventually extended to a full semiclassical derivation of the quantum condition, in collaboration with Born and Jordan. Once they were able to establish that $$i\hbar \{X,P\}_\mathrm{PB} \qquad \qquad \longmapsto \qquad \qquad [ X, P ] \equiv XP - PX = i\hbar \, ,$$ this condition replaced and extended the old quantization rule, allowing the matrix elements of P and X for an arbitrary system to be determined simply from the form of the Hamiltonian.

The new quantization rule was assumed to be universally true, even though the derivation from the old quantum theory required semiclassical reasoning. (A full quantum treatment, however, for more elaborate arguments of the brackets, was appreciated in the 1940s to amount to extending Poisson brackets to Moyal brackets.)

State vectors and the Heisenberg equation
To make the transition to standard quantum mechanics, the most important further addition was the quantum state vector, now written |ψ⟩, which is the vector that the matrices act on. Without the state vector, it is not clear which particular motion the Heisenberg matrices are describing, since they include all the motions somewhere.

The interpretation of the state vector, whose components are written $ψ_{m}$, was furnished by Born. This interpretation is statistical: the result of a measurement of the physical quantity corresponding to the matrix $A$ is random, with an average value equal to $$\sum_{mn} \psi_m^* A_{mn} \psi_n \,.$$ Alternatively, and equivalently, the state vector gives the probability amplitude $ψ_{n}$ for the quantum system to be in the energy state $n$.

Once the state vector was introduced, matrix mechanics could be rotated to any basis, where the $H$ matrix need no longer be diagonal. The Heisenberg equation of motion in its original form states that $A_{mn}$ evolves in time like a Fourier component, $$A_{mn}(t) = e^{i(E_m - E_n)t} A_{mn} (0) ~,$$ which can be recast in differential form $${dA_{mn}\over dt} = i(E_m - E_n ) A_{mn} ~,$$ and it can be restated so that it is true in an arbitrary basis, by noting that the $H$ matrix is diagonal with diagonal values $E_{m}$, $${dA\over dt} = i( H A - A H ) ~ .$$ This is now a matrix equation, so it holds in any basis. This is the modern form of the Heisenberg equation of motion.

Its formal solution is: $$A(t) = e^{iHt} A(0) e^{-iHt} ~.$$

All these forms of the equation of motion above say the same thing, that $A(t)$ is equivalent to $A(0)$, through a basis rotation by the unitary matrix $e^{iHt}$, a systematic picture elucidated by Dirac in his bra–ket notation.

Conversely, by rotating the basis for the state vector at each time by $e^{iHt}$, the time dependence in the matrices can be undone. The matrices are now time independent, but the state vector rotates, $$| \psi(t) \rangle = e^{-iHt} | \psi(0) \rangle, \;\;\;\; {d |\psi \rangle \over dt} = - i H | \psi \rangle \,.$$ This is the Schrödinger equation for the state vector, and this time-dependent change of basis amounts to transformation to the Schrödinger picture, with ⟨x|ψ⟩ = ψ(x).

In quantum mechanics in the Heisenberg picture the state vector, |ψ⟩ does not change with time, while an observable A satisfies the Heisenberg equation of motion,

The extra term is for operators such as $$ A = (X + t^2 P) $$which have an explicit time dependence, in addition to the time dependence from the unitary evolution discussed.

The Heisenberg picture does not distinguish time from space, so it is better suited to relativistic theories than the Schrödinger equation. Moreover, the similarity to classical physics is more manifest: the Hamiltonian equations of motion for classical mechanics are recovered by replacing the commutator above by the Poisson bracket (see also below). By the Stone–von Neumann theorem, the Heisenberg picture and the Schrödinger picture must be unitarily equivalent, as detailed below.

Further results
Matrix mechanics rapidly developed into modern quantum mechanics, and gave interesting physical results on the spectra of atoms.

Wave mechanics
Jordan noted that the commutation relations ensure that P acts as a differential operator.

The operator identity $$[a,bc] = abc - bca = abc - bac + bac - bca = [a,b]c + b[a,c]$$ allows the evaluation of the commutator of P with any power of X, and it implies that $$[P,X^n] = - i n~ X^{n-1} $$ which, together with linearity, implies that a P-commutator effectively differentiates any analytic matrix function of X.

Assuming limits are defined sensibly, this extends to arbitrary functions−−but the extension need not be made explicit until a certain degree of mathematical rigor is required,

Since X is a Hermitian matrix, it should be diagonalizable, and it will be clear from the eventual form of P that every real number can be an eigenvalue. This makes some of the mathematics subtle, since there is a separate eigenvector for every point in space.

In the basis where X is diagonal, an arbitrary state can be written as a superposition of states with eigenvalues x, $$|\psi\rangle = \int_x \psi(x)|x\rangle \,,$$ so that ψ(x) = ⟨x|ψ⟩, and the operator X multiplies each eigenvector by x, $$X |\psi\rangle = \int_x x \psi(x) |x\rangle ~ .$$

Define a linear operator D which differentiates $ψ$, $$D \int_x \psi(x) | x\rangle = \int_x \psi'(x) |x\rangle\,,$$ and note that $$(D X - X D) |\psi\rangle = \int_x \left[ \left(x \psi(x)\right)' - x \psi'(x) \right] |x\rangle = \int_x \psi(x) |x\rangle = |\psi\rangle\,,$$ so that the operator −iD obeys the same commutation relation as P. Thus, the difference between P and −iD must commute with X, $$[P+iD,X]=0\,,$$ so it may be simultaneously diagonalized with X: its value acting on any eigenstate of X is some function f of the eigenvalue x.

This function must be real, because both P and −iD are Hermitian, $$(P+iD ) |x\rangle = f(x) |x\rangle\,,$$ rotating each state $$|x\rangle$$ by a phase $f(x)$, that is, redefining the phase of the wavefunction: $$\psi(x) \rightarrow e^{-if(x)} \psi(x)\,.$$ The operator iD is redefined by an amount: $$iD \rightarrow iD + f(X)\,,$$ which means that, in the rotated basis, P is equal to −iD.

Hence, there is always a basis for the eigenvalues of X where the action of P on any wavefunction is known: $$P \int_x \psi(x) |x\rangle = \int_x - i \psi'(x) |x\rangle\,,$$ and the Hamiltonian in this basis is a linear differential operator on the state-vector components, $$\left[{P^2\over 2m} + V(X) \right] \int_x \psi_x |x\rangle = \int_x \left[-{1\over 2m}{\partial^2 \over \partial x^2} + V(x)\right] \psi_x |x\rangle$$

Thus, the equation of motion for the state vector is but a celebrated differential equation,

Since D is a differential operator, in order for it to be sensibly defined, there must be eigenvalues of X which neighbors every given value. This suggests that the only possibility is that the space of all eigenvalues of X is all real numbers, and that P is iD, up to a phase rotation.

To make this rigorous requires a sensible discussion of the limiting space of functions, and in this space this is the Stone–von Neumann theorem: any operators X and P which obey the commutation relations can be made to act on a space of wavefunctions, with P a derivative operator. This implies that a Schrödinger picture is always available.

Matrix mechanics easily extends to many degrees of freedom in a natural way. Each degree of freedom has a separate X operator and a separate effective differential operator P, and the wavefunction is a function of all the possible eigenvalues of the independent commuting X variables. $$\begin{align} \left[X_i, X_j\right] &= 0 \\[1ex] \left[P_i, P_j\right] &= 0 \\[1ex] \left[X_i, P_j\right] &= i\delta_{ij} \,. \end{align}$$

In particular, this means that a system of N interacting particles in 3 dimensions is described by one vector whose components in a basis where all the X are diagonal is a mathematical function of 3N-dimensional space describing all their possible positions, effectively a much bigger collection of values than the mere collection of N three-dimensional wavefunctions in one physical space. Schrödinger came to the same conclusion independently, and eventually proved the equivalence of his own formalism to Heisenberg's.

Since the wavefunction is a property of the whole system, not of any one part, the description in quantum mechanics is not entirely local. The description of several quantum particles has them correlated, or entangled. This entanglement leads to strange correlations between distant particles which violate the classical Bell's inequality.

Even if the particles can only be in just two positions, the wavefunction for N particles requires 2N complex numbers, one for each total configuration of positions. This is exponentially many numbers in N, so simulating quantum mechanics on a computer requires exponential resources. Conversely, this suggests that it might be possible to find quantum systems of size N which physically compute the answers to problems which classically require 2N bits to solve. This is the aspiration behind quantum computing.

Ehrenfest theorem
For the time-independent operators X and P, $∂A/∂t = 0$ so the Heisenberg equation above reduces to: $$i\hbar{dA \over dt} = [A,H]= AH - HA,$$ where the square brackets denote the commutator. For a Hamiltonian which is $$p^2/2m + V(x)$$, the X and P operators satisfy: $${d X\over dt} = {P\over m},\quad {d P \over dt} = - \nabla V ,$$ where the first is classically the velocity, and second is classically the force, or potential gradient. These reproduce Hamilton's form of Newton's laws of motion. In the Heisenberg picture, the X and P operators satisfy the classical equations of motion. You can take the expectation value of both sides of the equation to see that, in any state |ψ⟩: $$\begin{align} \frac{d}{dt} \langle X\rangle &= \frac{d}{dt} \langle \psi|X|\psi \rangle = \frac{1}{m} \langle \psi|P|\psi \rangle = \frac{1}{m} \langle P \rangle \\[1.5ex] \frac{d}{dt} \langle P\rangle &= \frac{d}{dt} \langle \psi|P|\psi \rangle = \langle \psi | (-\nabla V) |\psi\rangle = -\langle\nabla V\rangle \,. \end{align} $$

So Newton's laws are exactly obeyed by the expected values of the operators in any given state. This is Ehrenfest's theorem, which is an obvious corollary of the Heisenberg equations of motion, but is less trivial in the Schrödinger picture, where Ehrenfest discovered it.

Transformation theory
In classical mechanics, a canonical transformation of phase space coordinates is one which preserves the structure of the Poisson brackets. The new variables $x',p'$ have the same Poisson brackets with each other as the original variables $x,p$. Time evolution is a canonical transformation, since the phase space at any time is just as good a choice of variables as the phase space at any other time.

The Hamiltonian flow is the canonical transformation: $$\begin{align} x &\rightarrow x+dx = x + \frac{\partial H}{\partial p} dt \\[1ex] p &\rightarrow p+dp = p - \frac{\partial H}{\partial x} dt ~. \end{align}$$

Since the Hamiltonian can be an arbitrary function of x and p, there are such infinitesimal canonical transformations corresponding to every classical quantity $G$, where $G$ serves as the Hamiltonian to generate a flow of points in phase space for an increment of time s, $$\begin{align} dx &= \frac{\partial G}{\partial p} ds = \left\{ G, X \right\} ds \\[1ex] dp &= -\frac{\partial G}{\partial x} ds = \left\{ G, P \right\} ds \,. \end{align}$$

For a general function $A(x, p)$ on phase space, its infinitesimal change at every step ds under this map is $$dA = {\partial A \over \partial x} dx + {\partial A\over \partial p} dp = \{ A,G\} ds \, .$$ The quantity $G$ is called the infinitesimal generator of the canonical transformation.

In quantum mechanics, the quantum analog $G$ is now a Hermitian matrix, and the equations of motion are given by commutators, $$dA = i [G,A] ds \, .$$

The infinitesimal canonical motions can be formally integrated, just as the Heisenberg equation of motion were integrated, $$A' = U^{\dagger} A U $$where $U = '' e^{iGs}$ and $s$ is an arbitrary parameter.

The definition of a quantum canonical transformation is thus an arbitrary unitary change of basis on the space of all state vectors. $U$ is an arbitrary unitary matrix, a complex rotation in phase space, $$U^{\dagger} = U^{-1} \, .$$ These transformations leave the sum of the absolute square of the wavefunction components invariant, while they take states which are multiples of each other (including states which are imaginary multiples of each other) to states which are the same multiple of each other.

The interpretation of the matrices is that they act as generators of motions on the space of states.

For example, the motion generated by P can be found by solving the Heisenberg equation of motion using P as a Hamiltonian, $$\begin{align} dX &= i[X,P] ds = ds \\[1ex] dP &= i[P,P] ds = 0 \,. \end{align}$$ These are translations of the matrix X by a multiple of the identity matrix, $$X\rightarrow X+s I ~.$$ This is the interpretation of the derivative operator D: $e^{iPs} = e^{D}$, the exponential of a derivative operator is a translation (so Lagrange's shift operator).

The X operator likewise generates translations in P. The Hamiltonian generates translations in time, the angular momentum generates rotations in physical space, and the operator $X + P$ generates rotations in phase space.

When a transformation, like a rotation in physical space, commutes with the Hamiltonian, the transformation is called a symmetry (behind a degeneracy) of the Hamiltonian−−the Hamiltonian expressed in terms of rotated coordinates is the same as the original Hamiltonian. This means that the change in the Hamiltonian under the infinitesimal symmetry generator L vanishes, $${dH\over ds} = i[L,H] = 0\, .$$

It then follows that the change in the generator under time translation also vanishes, $${dL\over dt} = i[H,L] = 0 $$ so that the matrix L is constant in time: it is conserved.

The one-to-one association of infinitesimal symmetry generators and conservation laws was discovered by Emmy Noether for classical mechanics, where the commutators are Poisson brackets, but the quantum-mechanical reasoning is identical. In quantum mechanics, any unitary symmetry transformation yields a conservation law, since if the matrix U has the property that $$U^{-1} H U = H $$ so it follows that $$UH = HU$$ and that the time derivative of U is zero—it is conserved.

The eigenvalues of unitary matrices are pure phases, so that the value of a unitary conserved quantity is a complex number of unit magnitude, not a real number. Another way of saying this is that a unitary matrix is the exponential of i times a Hermitian matrix, so that the additive conserved real quantity, the phase, is only well-defined up to an integer multiple of 2π. Only when the unitary symmetry matrix is part of a family that comes arbitrarily close to the identity are the conserved real quantities single-valued, and then the demand that they are conserved become a much more exacting constraint.

Symmetries which can be continuously connected to the identity are called continuous, and translations, rotations, and boosts are examples. Symmetries which cannot be continuously connected to the identity are discrete, and the operation of space-inversion, or parity, and charge conjugation are examples.

The interpretation of the matrices as generators of canonical transformations is due to Paul Dirac. The correspondence between symmetries and matrices was shown by Eugene Wigner to be complete, if antiunitary matrices which describe symmetries which include time-reversal are included.

Selection rules
It was physically clear to Heisenberg that the absolute squares of the matrix elements of $X$, which are the Fourier coefficients of the oscillation, would yield the rate of emission of electromagnetic radiation.

In the classical limit of large orbits, if a charge with position $X(t)$ and charge $q$ is oscillating next to an equal and opposite charge at position 0, the instantaneous dipole moment is $q X(t)$, and the time variation of this moment translates directly into the space-time variation of the vector potential, which yields nested outgoing spherical waves.

For atoms, the wavelength of the emitted light is about 10,000 times the atomic radius, and the dipole moment is the only contribution to the radiative field, while all other details of the atomic charge distribution can be ignored.

Ignoring back-reaction, the power radiated in each outgoing mode is a sum of separate contributions from the square of each independent time Fourier mode of $d$, $$P(\omega) = {2\over 3} {\omega^4} |d_i|^2 ~.$$

Now, in Heisenberg's representation, the Fourier coefficients of the dipole moment are the matrix elements of $X$. This correspondence allowed Heisenberg to provide the rule for the transition intensities, the fraction of the time that, starting from an initial state $i$, a photon is emitted and the atom jumps to a final state $j$, $$P_{ij} = {2 \over 3} (E_i -E_j)^4 |X_{ij}|^2\, .$$

This then allowed the magnitude of the matrix elements to be interpreted statistically: they give the intensity of the spectral lines, the probability for quantum jumps from the emission of dipole radiation.

Since the transition rates are given by the matrix elements of $X$, wherever $X_{ij}$ is zero, the corresponding transition should be absent. These were called the selection rules, which were a puzzle until the advent of matrix mechanics.

An arbitrary state of the Hydrogen atom, ignoring spin, is labelled by |n;ℓ,m ⟩, where the value of ℓ is a measure of the total orbital angular momentum and $m$ is its $z$-component, which defines the orbit orientation. The components of the angular momentum pseudovector are $$L_i = \varepsilon_{ijk} X^j P^k $$ where the products in this expression are independent of order and real, because different components of X and P commute.

The commutation relations of L with all three coordinate matrices X, Y, Z (or with any vector) are easy to find, $$[L_i, X_j] = i\varepsilon_{ijk} X_k\,,$$ which confirms that the operator L generates rotations between the three components of the vector of coordinate matrices X.

From this, the commutator of Lz and the coordinate matrices X, Y, Z can be read off, $$\begin{align} \left[L_z, X\right] &= iY\,, \\[1ex] \left[L_z, Y\right] &= -iX\,. \end{align}$$

This means that the quantities $X + iY, X − iY$ have a simple commutation rule, $$\begin{align} \left[L_z, X + iY\right] &= (X + iY)\,, \\[1ex] \left[L_z, X - iY\right] &= -(X - iY)\,. \end{align}$$

Just like the matrix elements of X + iP and X − iP for the harmonic oscillator Hamiltonian, this commutation law implies that these operators only have certain off diagonal matrix elements in states of definite m, $$L_z ( (X+iY)|m\rangle )= (X+iY)L_z|m\rangle + (X+iY) |m\rangle = (m+1) (X+iY)|m\rangle $$ meaning that the matrix $(X + iY)$ takes an eigenvector of $L_{z}$ with eigenvalue $m$ to an eigenvector with eigenvalue $m$ + 1. Similarly, $(X− iY)$ decrease $m$ by one unit, while $Z$ does not change the value of $m$.

So, in a basis of |ℓ,m⟩ states where $L^{2}$ and $L_{z}$ have definite values, the matrix elements of any of the three components of the position are zero, except when $m$ is the same or changes by one unit.

This places a constraint on the change in total angular momentum. Any state can be rotated so that its angular momentum is in the $z$-direction as much as possible, where m = ℓ. The matrix element of the position acting on |ℓ,m⟩ can only produce values of m which are bigger by one unit, so that if the coordinates are rotated so that the final state is |ℓ',ℓ' ⟩, the value of ℓ’ can be at most one bigger than the biggest value of ℓ that occurs in the initial state. So ℓ’ is at most ℓ + 1.

The matrix elements vanish for ℓ’ > ℓ + 1, and the reverse matrix element is determined by Hermiticity, so these vanish also when ℓ’ < ℓ - 1: Dipole transitions are forbidden with a change in angular momentum of more than one unit.

Sum rules
The Heisenberg equation of motion determines the matrix elements of P in the Heisenberg basis from the matrix elements of X.

$$P_{ij} = m{d\over dt} X_{ij} = im (E_i - E_j) X_{ij} \,,$$ which turns the diagonal part of the commutation relation into a sum rule for the magnitude of the matrix elements: $$\sum_j P_{ij}x_{ji} - X_{ij}p_{ji} = i \sum_j 2m(E_i - E_j) |X_{ij}|^2 = i \,.$$

This yields a relation for the sum of the spectroscopic intensities to and from any given state, although to be absolutely correct, contributions from the radiative capture probability for unbound scattering states must be included in the sum: $$\sum_j 2m(E_i - E_j) |X_{ij}|^2 = 1\,.$$

title: Matrix mechanics


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Template talk:Disambiguation/Archive 1

Wording of the msg
Will an admin remove the word "just" from this message? Seems to unnecessarily devalue pages which are actually necessary and useful Jamesday 18:23, 10 Dec 2003 (UTC)

 * Done. You might want to fix the boilerplate (ie boilerplate etc) in other places if you feel strongly about it. I was going to suggest fixing the pages that have it on, but I realised there may 2000 of them according to Google. :) Angela. 06:45, 11 Dec 2003 (UTC)

Thanks. I've done the boilerplate for this and a couple of others. There's also a change request at MediaWiki talk:Legal. I suppose if I really wanted 2,000 edits to the circa 1,500 I have so far I could do the rest, but I think not...:) Jamesday 18:26, 12 Dec 2003 (UTC)

Shouldn't the tense in the message be the present perfect? This is, "If you HAVE followed a link here" – since the time is not defined, nor is it important, and the action has been completed already, a perfective tense ought to be used. Could someone amend this? Sinuhe 20:53, 16 Mar 2004 (UTC)
 * No, it shouldn't. This is a single, simple past action, and perfect tenses are for refering to events either

 * completed relative to other events or times:
 * "Tommorow i will have seen a battle, and be considered a veteran.";
 * "By December, i had experienced all five of Vermont's seasons.";
 * or completed, but without a specific time:
 * "I've been here before.";
 * "I've told you repeatedly."
 * We are in fact interested in the possibility that the reader just did get here via a link in the last few minutes, and is likely therefore to remember, or be able to back-up to, the page with that disambiguable link. (We're not interested in any of the times when the reader has followed a link here in the past, because they are unlikely to remember what link they came on, and thus probably not in a position to fix that link.) --Jerzy(t) 00:15, 2004 Mar 22 (UTC)

 * Oh, OK, another British and American English difference I suppose. In British English, one would most certainly say (and other variants are considered wrong) "the reader has just got here" (yes, the past participle of 'to get' is got, not gotten), exactly as in all other sentences when just is used in its time sense. Sinuhe 12:55, 22 Mar 2004 (UTC)

 * No, Jerzy, the word was has, not had. It was still the present, a span rather than a spot. However, using the simple past makes the suggestion apply after any time. No, Sinuhe, the past participle is not "got" outside corrupted English. This is an objective, not dialectoregional, assessment notwithstanding that I say that English got messed up (corrupted; mea culpa <- my bad; quasi <- as if; circa <- around <- agird...) after Latin crept in with its fastenish mindlessness (religious nonsense). lysdexia 20:44, 9 Nov 2004 (UTC)

However, the wording
 * other pages that might otherwise have the same name

is ambiguous, inviting the reader to ask how those pages could have the same name [as each other]; what kind of otherwise would make that feasible (a non-deterministic architecture?....)

The real otherwise is "if only one of them had an article in WP", but we can
 * keep the "otherwise what?" from being asked very often, and
 * point the reader in the right direction for answering it themself if it is asked:
 * other pages that might otherwise have this article's name

or
 * other pages that might otherwise have this article's title

directs attention to where the title can be seen, and if the reader is interested in the "otherwise", they'll infer "if it weren't in use by this dab", and it's a short leap from there to the formulation (BTW, too long to include in the boilerplate) "if a dab weren't needed, which would only be the case if there were only one article for which the ideal name would be this one, or if the other articles' claims to this title could somehow be ignored." --Jerzy(t) 00:15, 2004 Mar 22 (UTC)

Format of the Boilerplate
Revert. Older version was better. This multi-line message is harder to format nicely than the short stubone. — Sverdrup 18:23, 18 Mar 2004 (UTC)

 * I'd prefer if the multi-line text isn't centered. Maybe a  is sufficient to separate the note (in the initial format) from the page. -- User:Docu

 * I think I prefer the old one as well. Perhaps we should have a poll? Sarge Baldy 03:24, Mar 19, 2004 (UTC)

 * I think this is rather unprofessional-looking. Also, I don't really see a reason for drawing attention to the message; what a disambiguation page is is readily apparent just by looking at one. As for separating the message from the page, I think that the italics do that just fine. -- Djinn112 07:16, Mar 20, 2004 (UTC)

Pretty much the same discussion is in MediaWiki talk:Stub; the format should probably be consistent. --Shallot 12:27, 20 Mar 2004 (UTC)

 * More or less, except that these message is bit long (maybe even too long) .. Anyways, in the meantime, I reverted it to the previous version and added the  . -- User:Docu

I liked the "boldly edited" box, but i'll be happy as long as something distinguishes the dab b-plt from dab'g text. I'd prefer more than italics; how about centered & italic, with a borderless, transparent box?

But just don't drop formatting completely; i just went thru manually checking 29 dabs yesterday, that i kept a record of having worked on, to be sure their B-plt was a msg and my ""-coded hr had been removed; i feel entitled to ask for clear separation.--Jerzy(t) 00:15, 2004 Mar 22 (UTC)

A Special Case
Oh, i don't suggest doing anything about it until this settles down, but Corn has long had a variant dab b-plt; someone deleted the b-plt completely, i think in response to the push to standardize; i restored it and converted it to the pale-blue-box format. Someone should decide what to do about it at the end of this transitional period. --Jerzy(t) 00:15, 2004 Mar 22 (UTC)

title: Template talk:Disambiguation/Archive 1


text:
Punk zine

A punk zine (or punkzine) is a zine related to the punk subculture and hardcore punk music genre. Often primitively or casually produced, they feature punk literature, such as social commentary, punk poetry, news, gossip, music reviews and articles about punk rock bands or regional punk scenes.

1970s: origins
Starting in the 1970s, the DIY aesthetic of the punk subculture created a thriving underground press. Amateur magazines related to punk were inspired by the rock fanzines of the early 1970s, which were inspired by zines from the science fiction fan community. Perhaps the most influential of the fanzines to cross over from science fiction fandom to rock and, later, punk rock and new wave music was Greg Shaw's Who Put the Bomp, founded in 1970.

One of the earliest punk zines was Punk, founded in New York City by John Holmstrom, Ged Dunn and Legs McNeil. Debuting in January 1976, the zine championed the early New York underground music scene and helped associate the word "punk" with these bands, most notably the Ramones. Other early punkzines from the United States included Search & Destroy (later REsearch), Flipside and Slash.

An early United Kingdom punk zine was Sniffin' Glue, produced by Mark Perry, who also founded the band Alternative TV, in 1976. Perry produced the first photocopied issue of Sniffin' Glue in London after attending the Ramones concert on 4 July 1976 at the Roundhouse. Punk zines were produced in many European countries in subsequent years. The first Irish one was published in March 1977.

In Australia in 1977, inspired by the Saints and Radio Birdman, Bruce Milne and Clinton Walker fused their respective first zines Plastered Press and Suicide Alley to launch Pulp; Milne later went on to invent the cassette zine with Fast Forward, in 1980. Another early publication was Self Abuse first published in Sydney in December 1977.

1980s
The politically charged Maximum RocknRoll and the anarchist Profane Existence were notable punkzines that were founded in the 1980s. By that time, most local punk scenes had at least one punkzine. The magazine Factsheet Five chronicled thousands of underground publications and "zines" in the 1980s and 1990s.

In the 1980s, the punk self-publication scene was quickly expanding to include numerous different subcultures within the genre. For example, the birth of the Queercore movement: inspired by the desire for social change, the subculture was represented by zines that sought to accept those within the LGBT community who were also involved in punk and also had overwhelming themes of promoting individual rights. The topics discussed in the issues often ignited forums and chatrooms where readers could share their opinions. This genre of zine was self-sustaining and produced in a DIY manner. Queercore is often accredited to a Toronto-based zine entitled J.D.s, an abbreviation for "juvenile delinquents", created by H. Quinn and co-published with Bruce Wayne. H.Q.s was a cut-and-paste-style zine that featured manifestos and dialogue about identifying as queer within the realms of the punk community. Other zines that instigated this movement are Chainsaw (punk zine), Outpunk, and Homocore. The queercore zines influenced the Riot Grrrl zines of the late 1980s and 1990s, as well.

Riot Grrrl zines
The "Riot Grrrl" movement emerged from the punk scene in the United States when women began to produce zines with feminist themes. The "riot grrrl" wave was influential for pinkzines as it called for women to publish and produce content in the male dominated culture. Featuring political issues from a personal standpoint, the zines arose in popularity amongst the underground world of punk. The format of the "riot grrrl" zines was similar to that of queercore zines, in that they were cut and paste and xeroxed with many featuring collages. Self-published punkzines from this era such as Bikini Kill, Girl Germs, Le Tigre, and Jigsaw were put out by members of riot grrrl bands who supported the notion of women learning to play music and feeling self-empowered. Other apparent themes in this category of zine include activism, social change, sexuality, body image, and the discussion of controversial topics such as racism and abuse.

List of punk zines

 * Absolutely Zippo
 * Artcore
 * Black Market Magazine
 * Capitol Crisis
 * Chainsaw
 * Cometbus
 * Damage
 * Flipside
 * Forced Exposure
 * Gadgie
 * Girl Germs
 * HeartattaCk
 * Homocore
 * Jamming
 * J.D.s
 * Kill Your Pet Puppy
 * Lights Go Out
 * Lobotomy
 * Love & Molotov Cocktails
 * Matter
 * Maximum RocknRoll
 * New York Rocker
 * No Cure
 * Outpunk
 * Ox-Fanzine
 * Pork
 * Profane Existence
 * Punk Magazine
 * Rancid News
 * Rave On
 * Razorcake
 * RE/Search
 * Ripped and Torn
 * Slash
 * Slug and Lettuce
 * Sluggo!
 * Sniffin' Glue
 * Spuno
 * Substitute
 * Suburban Voice
 * Suburban Rebels
 * Take It
 * TNSrecords Fanzine
 * Touch and Go
 * T.V.O.R.

title: Punk zine


text:
Bowen

Bowen may refer to:

Places

 * Bowen Mountain (disambiguation), including Mount Bowen
 * Bowen Park (disambiguation)
 * Bowen River (disambiguation)

Australia

 * Bowen Bridge, crossing the Derwent River in Tasmania

Queensland

 * Bowen, Queensland, a town
 * Bowen Orbital Spaceport (BOS), Abbot Point, Bowen
 * Bowen Hills, Queensland, a suburb
 * Bowen Hills railway station, a railway station in Bowen Hills
 * Bowen Park, Brisbane, a park in Bowen Hills
 * Mount Bowen (Queensland)

Canada

 * Bowen Island, British Columbia
 * Bowen Lake, a lake in Alberta

United States

 * Plant Bowen, a major coal-fired power plant in Georgia, U.S.
 * Bowen, Illinois
 * Bowen, Missouri
 * Bowen, Nebraska
 * Lake Bowen, a lake in South Carolina, U.S.
 * Bowen, West Virginia

Colorado

 * Bowen, Colorado (disambiguation)
 * Bowen, Colorado (Las Animas County)
 * Bowen, Colorado (Rio Grande County)
 * Bowen Mountain (Colorado), a summit

Other places

 * Bowen (crater), a lunar crater
 * Mount Bowen, a peak on Mount Joyce, Victoria Land, Antarctica
 * Bowen, Mendoza, a district in the General Alvear Department, Argentina
 * Bowen Road, Hong Kong
 * Bowen's Court, County Cork, Ireland
 * Bowen University, Iwo, Nigeria
 * Bowen Secondary School, a secondary school in Hougang, Singapore

People

 * Bowen (surname)
 * Judge Bowen (disambiguation)
 * Justice Bowen (disambiguation)

People with the family name

 * Anthony Bowens (born 1990), American professional wrestler
 * George Bowen (1821–1899), British viceregal
 * Isabelle Bowen (1899–1969), American painter and floriculturist
 * Jarrod Bowen (born 1996), English footballer
 * John Bowen (pirate) (16??–1704), pirate of Créole origin active during the Golden Age of Piracy
 * Natasha Bowen, Nigerian Welsh writer
 * York Bowen (1884-1961), composer

People with the given name

 * Bowen Ebenezer Aylsworth (1835–1914), Ontario political figure
 * Bowen Byram (born 2001), Canadian ice hockey player
 * Bowen Huang (born 1987), Chinese footballer
 * Bowen Leung (born 1949), former Hong Kong civil servant
 * Bowen Lockwood (born 1978), former Australian rules footballer
 * Bowen Ouyang (born 1992), Chinese male tennis player
 * Bowen Southwell (1713–1796), Irish politician
 * Bowen Stassforth (1926–2019), American former competition swimmer
 * Bowen Yang (born 1990), American comedian, SNL cast member

Other

 * Bowen knot, an emblem
 * Bowen ratio, used to describe energy flux
 * Bowen technique, an alternative massage therapy
 * Bowen Theory, developed by Murray Bowen, M.D.
 * Bowen's Disease, a sunlight-induced skin disease
 * Bowen's Kale, a calibration substance
 * Bowen's reaction series, in geology
 * Bowens International, a manufacturer of professional photographic lighting equipment
 * HMAS Bowen, an Australian warship named after Bowen, Queensland
 * Bowen script (僰文) traditionally used for writing the Bai language

title: Bowen