User:ItsTheEquations/sandbox

The Entire Discussion under "Uncertainty Principle...[edit]

...is nonideal and perhaps incomplete or erroneous in some respects. (Revised: I deleted these explanations on refactoring.)
I propose to replace this section text and examples with the following here (I am 99.999% done at this point.):

Heisenberg's Uncertainty Principle is another core concept of Quantum Mechanics where Planck's constant plays a key role. Perhaps Fourier Analysis^[1] best demonstrates the idea of an indeterminacy, i.e., something inherently indeterminable for a reason other than apparatus error or the Observer effect. In Fourier Analysis, as in Quantum Mechanics, these indeterminacies are associated with observing wave phenomena, although Heisenberg himself may not have initially understood this.^[2]

Fourier Analysis involves transform / inverse transform pairs between the time and frequency domains, between time and energy, between position and momentum, and between other examples of what physicists call canonical conjugates. One may calculate a set of transformed frequency data from a set of measured time data and view either on a 2D plot – but not both at once; it takes all the time data to calculate the transform for each frequency. However, plotting a 3D surface of frequency data (a spectrum) vs time (in frames $\scriptstyle \Delta t_{F}$ long) seems to reveal all of it: signal power vs frequency and at each frame time. This 3D plot, sometimes called a waterfall or a spectrogram, is commonly found in Speech, Music, Communications, and Geology; and it is how voiceprints^[3] are made. The Spectrogram is a rich example of uncertainty and waves.

The Fourier Transform is defined with integrals from $\scriptstyle -\infty$ to $\scriptstyle +\infty$ ; however, in real measurements, the apparatus goes on at $\scriptstyle t=0$ and measures samples at intervals of $\scriptstyle \Delta t_{S}$ until $\scriptstyle N$ samples are collected. Data are analyzed by the Discrete Fourier Transform^[4] (DFT), typically with an FFT algorithm. Since the $\scriptstyle t$ domain was sampled, the $\scriptstyle f$ domain also becomes discrete, having increments of $\scriptstyle \Delta f\,=\,1/\Delta t_{F}\,=\,1/(N\Delta t_{S})$ ; and values of $\scriptstyle f(k)=k\Delta f$ for $\scriptstyle k=0\,to\,(N-1)$ . However, these measurement parameters result in aliasing in the frequency domain that compels one to drop the terms for $\scriptstyle k=0$ because time-framing under-samples frequencies $\scriptstyle f\,<\,1/\Delta t_{F}$ , and to drop the $\scriptstyle k>(N/2)-1$ terms because the time data are real and $\scriptstyle f_{H}\,\,\thickapprox \,1/(2\Delta t_{S})\,=\,f_{S}/2$ is the Nyquist frequency. Thus, there are uncertainties limiting both ends of the calculated spectrum as well as the resolution: $\scriptstyle f_{L}\,\thickapprox \,\Delta f\,=\,1/\Delta t_{F}$ and $\scriptstyle f_{H}\,\,\thickapprox \,1/(2\Delta t_{S})$ .

A voiceprint with 10ms frames and a 20kSa/s sampling rate will produce a series of spectra each of which goes from 100Hz to 10kHz in 100Hz increments. Put another way, regardless of how accurately you measure; you cannot detect 100Hz (or a difference of 100Hz) in less than ~10ms $because it takes that long$ for 100Hz to happen. Thus, the indeterminacy of frequency in time means two things:

Within a given time frame $\scriptstyle \Delta t_{i}$ , if an event were detected, there can be no more precise knowledge of when it happened than to observe it was during that particular time frame $\scriptstyle \Delta t_{i}$ , and
Within any time frame $\scriptstyle \Delta t$ , the actual frequency can be no more precisely known than to observe it is within a range $\scriptstyle \Delta f\,\,\gtrsim \,\,1/\Delta t$ , while frequencies $\scriptstyle f\,\,<\,\,1/\Delta t$ are indeterminable.

In actual lab practice, scientists often use window functions, overlapping data, and other advanced processing to overcome some indeterminacy limitations. Technical choices of windows and strategies affect the $\scriptstyle \Delta t$ and the $\scriptstyle \Delta f$ in various ways to balance requirements for signal levels, noise levels, spectral leakage, etc. For more details and examples see the Short-time Fourier transform (STFT), the Discrete Fourier transform (DFT), the Discrete-time Fourier transform (DTFT), and the Fast Fourier Transform (FFT).

By introducing Planck's constant, one may obtain Heisenberg's Energy vs Time and Momentum vs Position Uncertainty Relations in one step:

The Fourier Analysis indeterminacies for temporal frequency

\scriptstyle f

and spatial frequency

\scriptstyle \xi

:

\scriptstyle f_{o}\,\,=\,\,1/T_{o}

\scriptstyle \to

\scriptstyle \Delta f\Delta t\,\,\gtrsim \,\,1

and

\scriptstyle \xi _{o}\,\,=\,\,1/\lambda _{o}

\scriptstyle \to

\scriptstyle \Delta \xi \Delta x\,\,\gtrsim \,\,1

The de Broglie Relations^[5] introduce Plancks constant:

\scriptstyle f\,\,=\,\,E/h

and

\scriptstyle \lambda \,\,=\,\,h/p

(or

\scriptstyle \xi \,\,=\,\,p/h

where

\scriptstyle \lambda \,\,=\,\,1/\xi

)

Substitution reveals the familiar inequalities:

\scriptstyle \Delta E\Delta t\,\,\gtrsim \,\,h

and

\scriptstyle \Delta p\Delta x\,\,\gtrsim \,\,h

Comparing these results with $\scriptstyle \Delta f\Delta t\,\,\gtrsim \,\,1$ , one would expect the following also to hold in Quantum Mechanics due to indeterminacies associated with observing wave phenomena characterized by Planck's constant:

$\Delta E\Delta t\,\,\gtrsim \,\,h$ (the indeterminacy of energy in time)	$\Delta p\Delta x\,\,\gtrsim \,\,h$ (the indeterminacy of momentum in space)
1. If energy were detected during time frame $\scriptstyle \Delta t_{i}$ , there can be no more precise knowledge of when it happened than to observe it was during that particular time frame $\scriptstyle \Delta t_{i}$ .	1. If momentum were detected within the segment of space $\scriptstyle \Delta x_{i}$ , there can be no more precise knowledge of where it happened than to observe it was inside that particular spatial segment $\scriptstyle \Delta x_{i}$ .
2. Within any time frame $\scriptstyle \Delta t$ , the actual energy can be no more precisely known than to observe it is within a range $\scriptstyle \Delta E\,\,\gtrsim \,\,h/\Delta t$ , while energies $\scriptstyle E\,\,<\,\,h/\Delta t$ are indeterminable.	2. Within any spatial segment $\scriptstyle \Delta x$ , the actual momentum can be no more precisely known than to observe it is within a range $\scriptstyle \Delta p\,\,\gtrsim \,\,h/\Delta x$ , while momenta $\scriptstyle p\,\,<\,\,h/\Delta x$ are indeterminable.

There are other ways like this one that show how Plancks constant relates the Heisenberg Uncertainty Principal inequalities to the Fourier transforms familiar to a broader technical audience. Nevertheless, Heisenberg's first derivation of $\scriptstyle \Delta p\Delta x\,\,\gtrsim \,\,h$ in 1927^[6] was complex, utilizing the Matrix Mechanics published in 1925-26 by him, Max Born, and Pascual Jordan.^[7] Matrix Mechanics is regarded as the first conceptually autonomous and logically consistent formulation of quantum mechanics, comparable to the Wave Mechanics formulation based on the Schrödinger wave equation, published in 1927.^[8] The form $σ p σ x \geq ħ /2$ , where the $σ p$ and $σ x$ are standard deviations and $ħ$ is the reduced Planck constant, is often called the exact solution of the Uncertainty Principle, and it has been shown valid for all wave functions – not just Gaussian waves as Heisenberg later had shown. In the modern mathematical formulation of quantum mechanics, any pair of non-commuting self-adjoint operators representing observables are subject to similar uncertainty limits; for example, $\scriptstyle [{\hat {p}}_{i},{\hat {x}}_{j}]=-i\hbar \delta _{ij}$ where $\scriptstyle {\hat {p}}$ is the momentum operator, $\scriptstyle {\hat {x}}$ is the position operator, and $δ ij$ is the Kronecker delta.

The Uncertainty Principle enables an alternate way of looking at certain classical wave problems, for example, diffraction. Assume the width of the central bright spot is determined by uncertainties in the momentum wave in space introduced by a single slit. The slit, with $width = a$ , causes position uncertainty $Δ y = a$ in both the positive and negative $y$ -directions, which in turn causes momentum uncertainty $Δ p y = h / Δ y$ in each $y$ -direction. The momentum in the $x$ -direction towards the screen is given by $p x = h / λ$ (de Broglie). If $θ$ is the angle from the center of the central bright spot to the first minimum on one side, then $θ = arctan(Δ p y / p x) = arctan( λ / a )$ . When the screen is very far from the slit and $θ$ is small (Fraunhofer diffraction), then $θ \approx sin θ \approx tan θ$ , and this reduces to $θ = λ / a$ . This is the same result obtained from classical methods for diffraction (e.g., Huygens' principle) for the minima of single slit diffraction, $a sin θ = nλ$ , when $n=1$ and $θ$ is small, and for the envelope of the interference pattern in the multi-slit case where each $width = a$ . The uncertainty approach, however, is applicable to matter waves and not only to light. Interestingly, diffraction and interference patterns are the Fourier transforms of the slit patterns that produce them.

References[edit]

^ Fourier, Joseph (1822). Theorie Analytique de la Chaleur. Paris. ISBN [[Special:BookSources/ISBN 978-1-108-00180-9|'"`UNIQ--templatestyles-00000072-QINU`"'[[ISBN (identifier)|ISBN]] [[Special:BookSources/978-1-108-00180-9 |978-1-108-00180-9]]]]. {{cite book}}: Check |isbn= value: invalid character (help); templatestyles stripmarker in |isbn= at position 1 (help)
^ Heisenberg, Werner (1930). The Physical Principles of Quantum Theory. Chicago: University of Chicago Press..
^ Schafer, Lawrence R. Rabiner, Ronald W. (1978). Digital processing of speech signals. Englewood Cliffs, N.J.: Prentice-Hall. ISBN 978-0132136037.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Buck, Alan V. Oppenheim, Ronald W. Schafer with John R. (1999). Discrete-time signal processing (2nd ed. ed.). Upper Saddle River, N.J.: Prentice-Hall. ISBN 0-13-754920-2. {{cite book}}: |edition= has extra text (help)CS1 maint: multiple names: authors list (link)
^ L. de Broglie, Recherches sur la théorie des quanta (Researches on the quantum theory), Thesis (Paris), 1924; L. de Broglie, Ann. Phys. (Paris) 3, 22 (1925).
^ Heisenberg, W (1927-03-01). "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik". Zeitschrift für Physik A Hadrons and Nuclei. 43 (3–4). Berlin / Heidelberg: Springer: 192–198. doi:10.1007/BF01397280. ISSN 0939-7922. {{cite journal}}: Unknown parameter |month= ignored (help)CS1 maint: date and year (link)
^ M. Born, W. Heisenberg, and P. Jordan, Zur Quantenmechanik II, Zeitschrift für Physik, 35, 557–615 (1925). The paper was received on 16 November 1925. [English translation in: B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1]
^ Schrödinger, E. (1926-12-01). "An Undulatory Theory of the Mechanics of Atoms and Molecules". Physical Review. 28 (6): 1049–1070. doi:10.1103/PhysRev.28.1049.

ItsTheEquations (talk) 08:43, 6 May 2012 (UTC)
Revised — ItsTheEquations (talk) 01:58, 15 May 2012 (UTC), Revised — ItsTheEquations (talk) 04:29, 15 May 2012 (UTC), Revised — ItsTheEquations (talk) 01:51, 17 May 2012 (UTC), Revised — ItsTheEquations (talk) 20:45, 17 May 2012 (UTC), Revised — ItsTheEquations (talk) 21:59, 17 May 2012 (UTC), Revised — ItsTheEquations (talk) 08:39, 18 May 2012 (UTC), Revised — ItsTheEquations (talk) 11:01, 19 May 2012 (UTC), Revised — ItsTheEquations (talk) 15:31, 19 May 2012 (UTC), Revised — ItsTheEquations (talk) 18:33, 22 May 2012 (UTC), Revised — ItsTheEquations (talk) 19:53, 24 May 2012 (UTC), Revised — ItsTheEquations (talk) 06:55, 25 May 2012 (UTC), Revised — ItsTheEquations (talk) 23:02, 26 May 2012 (UTC), Revised — ItsTheEquations (talk) 22:08, 27 May 2012 (UTC), Revised — ItsTheEquations (talk) 03:23, 29 May 2012 (UTC)
— user

I have not read all your post WP:TLDR, but to answer your final point, the article already gives a link to where this is more fully discussed at Uncertainty principle. You are welcome to improve this, or any other, article. Please take a look at our verifiability policy before you do so. Spinning Spark 12:58, 6 May 2012 (UTC)

Verifiability per Wiki is a great way to do a pedia. Looking toward providing something really useful and clear. Will address Verifiability as I correct the errors in my talk/discussion above. Many folks infer many additional meanings to the Heisenberg Uncertainty statement. Thus, a concise approach may seem unbelievable, so very good citations are needed here. But if the task is to make especially clear why Planck's constant is in there, then it is a worthwhile task. Having this insight goes to the heart of "waves vs particles." — ItsTheEquations (talk) 16:24, 10 May 2012 (UTC)

I am developing a proposal to replace this section. As I have refactored, I replaced my original Talk material with the proposal above for a new section. I will not alter Spinningspark or other editors remarks. Thx for your patience. Feedback is desirable at this point. — ItsTheEquations (talk) 01:58, 15 May 2012 (UTC)

Revised: ItsTheEquations (talk) 18:33, 22 May 2012 (UTC)

Revised ItsTheEquations (talk) 01:51, 17 May 2012 (UTC)

It does look well written, and a good effort, though you should:

change the statement of "seeing 100Hz" to "measuring 100Hz",
where you have linked "conjugate variables (explained)" that should be a proper wiki-link, not an external link, like this (and preferably with a simpler and more direct title):

"These are examples of what physicists call conjugate variables

that is;

These are examples of what physicists call [[conjugate variables]].

You can link articles using two square brackets, in the following general way:

[[article#section|paragraph text]]

Furthermore don't use so much bold, the article will not have a uniform appearance if there is a lot of bold in one section - its only for real emphasis. In the above case you can just use italics for the key words/phrases but not for full sentences, see Wikipedia:Manual of Style.

(Presumably you were going to fix these anyway? Just making sure)... =) F = q(E+v×B) ⇄ ∑_ic_i 16:34, 17 May 2012 (UTC)

The use of quotation marks should indicate quotations exclusively. Perhaps some of your words in quotes should be set in italics instead, per MOS:WORDSASWORDS. --TSchwenn (talk) 18:30, 17 May 2012 (UTC)

Newcomer is deeply grateful for the gentle advice. Please look for points needing my attention based on new draft that overwrote the old one above. Revised: ItsTheEquations (talk) 18:33, 22 May 2012 (UTC)
ItsTheEquations (talk) 20:45, 17 May 2012 (UTC)

I should have mentioned that you should create a sandbox: User:ItsTheEquations/sandbox (just start typing in it and the page will be created). You can edit this as many times as you like, rather than here on the talk page. About feedback -

don't get bogged down into calculations,
just keep it all simple and use summary style: search out the relevant articles and link to them in the text,
if there is no room for what you have to say, just write see also...[[page]] (or words to effect), never begin and continue from there to emphasize inessential detials when it is covered elsewhere (for example you mentioned slit calculations - there is the article Diffraction formalism you could link to)
Its good to have some analagous applications/occurances, but use your judgement to decide the scope: usually a sentence or few are eneogh (any more easily becomes paragraphs). I think what you have written is generally fine, but again get rid of the extra bolding, for the above example:

inherently indeterminable

indeterminacy of frequency in time

indeterminacies are associated with measuring wave phenomena

do not need to be bolded or italixed. F = q(E+v×B) ⇄ ∑_ic_i 16:15, 18 May 2012 (UTC)

You may also have some use for Help:Displaying a formula Spinning Spark 18:35, 18 May 2012 (UTC)

Thanks. I defer to the ones who comment: I will remove bold in paragraphs. (It seems to be allowed in actual practice, and the results look okay to me.) MOS is a lot of disorganized reading, but I will consume it. I write for readers, but I publish for editors; I understand that: somebody will just change it. I converted some of the bold to italics, the rest to plain, and it looks okay.

Objectives – The purpose of the article section is two-fold: (1) to provide a quick low-math way to see Heisenberg using Planck's constant (which requires mentioning canonical conjugate variables and Fourier Analysis for technical legitimacy), and (2) to provide a candidate for others' link-to or perhaps even template-from (since the Planck emphasis is not too obvious). Revised: Only at the end is there mention of the advanced methods used by H and others, and of the supposedly exact form of the HUP: σ_pσ_x ≥ ħ/2. This is by design; it is an historical note. Readers who come to Planck|Uncertainty need not to be pushed aside by mumbo jumbo (like Matrix Mechanics) if a simple explanation (like this comparison to Fourier) also is appropriate. Because we are in the Planck article, we have an obligation to do the nutshell section treatment on Heisenberg – and to show it really doesn't happen without Planck (via deBroglie) to get the frequency and wavelength for phenomena in Quantum Mechanics. The whole Uncertainty issue is explained through the more familiar

\scriptstyle \Delta f\Delta t\,\,\gtrsim \,\,1

uncertainty, which is clearly about waves. Then Planck's constant converts this one into the two HUP's in one step.

ItsTheEquations (talk) 11:01, 19 May 2012 (UTC)

Revised: — ItsTheEquations (talk) 18:33, 22 May 2012 (UTC)

Revised: – ItsTheEquations (talk) 06:42, 24 May 2012 (UTC)

It seems to be allowed in actual practice... The quality of Wikipedia articles varies enormously, almost anything will survive, at least for a while. If you want to see what is considered good, look at the articles listed at WP:FA. These have all been very thouroughly reviewed (or an earlier version of them was) and are considered the very best Wikipedia has to offer. Bolding in formulae is reserved for sets, matrices, vectors etc. It is almost certain someone will remove the bolding eventually if you put it in the article like that. MOS:MATH has some good advice on formatting formulae. A small font that approximately matches the inline text size can be forced with the TeX command \scriptstyle but it is not very popular because it can be a nightmare to align properly and is also not genuinely text and will not be searchable. You may prefer to use the {{math}} template. Spinning Spark 11:32, 19 May 2012 (UTC)

I get it. No bold ever, anywhere, period. Someday they may elucidate good rules for this; they even use it on their own pages about best articles. But how to explain it to all editors... I appreciate your warning; I removed it; and results looks fine to me. I appreciate your encouragement and advice, Sparks. ItsTheEquations (talk) 15:31, 19 May 2012 (UTC)

Revised: ItsTheEquations (talk) 06:42, 24 May 2012 (UTC)

I have converted every equation and every variable mentioned in the proposed section to TEX with \scriptstyle as mentioned by SpinningSpark. Looks pretty good. Thanks for the tip. The in-line alignment issue is visible and serious, but it is a tracked bug. All these things notwithstanding, I believe the exclusive availability of the " $\scriptstyle \gtrsim$ " symbol is an overriding consideration in this case. Hopefully, as it is conforming, it also will not be bait for font pedants or rules robots. I realize these formatting conventions are well-known basics, but I am a Newbie; so, thanks for your help.

This edit includes other minor changes and corrections. I removed the popular and intuitive word slice and replaced it with the also popular but less intuitive word chunk, which is used in Wiki the links I included. I am using S as the subscript for sampling, which is rather universal. I use the subscript C for chunk because it seems reasonable. I added a thin summary history of HUP calculations to explain the other HUP with sigmas rather than deltas. Finally, I refactored my remarks here on the talk page to help the reduce the sprawl I cause.

Re: Single Slit – Revised: A new paragraph for this was added at the end.

This draft is the Release Candidate: Thank you for your comments.

— ItsTheEquations (talk) 18:33, 22 May 2012 (UTC)

— Revised: ItsTheEquations (talk) 06:42, 24 May 2012 (UTC)
— Revised: ItsTheEquations (talk) 19:53, 24 May 2012 (UTC)
— Revised: ItsTheEquations (talk) 06:55, 25 May 2012 (UTC)

Fixed the slit example, finally. And I made various minor edits.

— ItsTheEquations (talk) 23:02, 26 May 2012 (UTC)

This should have been entirely hashed out on the uncertainty principle page before being brought here. While I agree that the HUP uncertainty principle is just the Fourier uncertainty with h serving to translate frequency into energy, at the same time it's not at all clear why if f*t < 1, why E*t < h, not h/4pi. In any case, when you do get that little matter worked out, it's only going to be a small part of this article (in the little subsection on HUP) and need not be discussed on this article's talk page. Do it on the HUP talk page. S B H arris 23:18, 26 May 2012 (UTC)

Thank you for your feedback. I feel this subarticle may be pushing the size limit for subarticles, but keep in mind the encyclopedia user is starting at Planck constant and THEN stepping into HUP. In a somewhat tutorial style, this article explains HUP from the Planck perspective. I wrote this simply as a response to the frustration I felt as I was trying to better understand the origins and meaning of the HUP. THE ONLY PLACE THIS WAS CLEAR WAS THE MAIN ARTICLE. But it was not very readable, and it did not meet my needs. The Planck/HUP subarticle has ineffective examples (which explain nothing); it has an incorrect equation at the top; and the correct equation at the bottom that only grad students of physics or Math would recognize. Most other HUP subarticles are extremely specific to the article... or they are wrong. THERE IS A NEED TO POINT OUT AUTHORITATIVELY that HUP does NOT mean anything but what it means: it is about observing waves. Period. Many technical users "get" Fourier, and they might be surprised that HUP has more to do with Fourier (1822) than Heisenberg (~1924). For the many people with skills and knowledge of Fourier Analysis, an understanding of HUP is just a Planck's constant away — which is why this subarticle is in this article. This was my thinking.

Keep in mind, that for a long time, even Heisenberg himself thought HUP was about apparatus error (1930). This is why there is an historical paragraph at the end. While it was only vandalism, last night somebody tried to add something about Depak Chopra saying HUP means "anything can happen." It is THIS misconception about HUP I hoped to extinguish within this "easier" context.

Finally, I believe the ΔE*Δt>h form you referred to is correct, because ΔE is not the same as σ_E. When one uses σ_E and σ_t, the right side becomes

h /4π = ħ /2

where

ħ = h /2π

. If I had found a simple reference for how the 1/4π comes out of the variance calculation, I would have included it, too. However, since this subarticle pushes the size envelope, I am glad I did not find it.

ItsTheEquations (talk) 18:13, 27 May 2012 (UTC)

Minor edits for clarity, sentence integrity, etc.
ItsTheEquations (talk) 22:08, 27 May 2012 (UTC)

Changed chunk to frame and the variable $Δ t C$ to $Δ t F$ . Significantly improved misc words, sentences, WikiLinks, and equation formats.
— ItsTheEquations (talk) 03:23, 29 May 2012 (UTC)

changed f_L formula to approx, corrected "compell", — user

[1] Fourier, Joseph (1822). Theorie Analytique de la Chaleur. Paris. ISBN [[Special:BookSources/ISBN 978-1-108-00180-9|'"`UNIQ--templatestyles-00000072-QINU`"'[[ISBN (identifier)|ISBN]] [[Special:BookSources/978-1-108-00180-9 |978-1-108-00180-9]]]]. {{cite book}}: Check |isbn= value: invalid character (help); templatestyles stripmarker in |isbn= at position 1 (help)

[2] Heisenberg, Werner (1930). The Physical Principles of Quantum Theory. Chicago: University of Chicago Press..

[3] Schafer, Lawrence R. Rabiner, Ronald W. (1978). Digital processing of speech signals. Englewood Cliffs, N.J.: Prentice-Hall. ISBN 978-0132136037.{{cite book}}: CS1 maint: multiple names: authors list (link)

[4] Buck, Alan V. Oppenheim, Ronald W. Schafer with John R. (1999). Discrete-time signal processing (2nd ed. ed.). Upper Saddle River, N.J.: Prentice-Hall. ISBN 0-13-754920-2. {{cite book}}: |edition= has extra text (help)CS1 maint: multiple names: authors list (link)

[5] L. de Broglie, Recherches sur la théorie des quanta (Researches on the quantum theory), Thesis (Paris), 1924; L. de Broglie, Ann. Phys. (Paris) 3, 22 (1925).

[6] Heisenberg, W (1927-03-01). "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik". Zeitschrift für Physik A Hadrons and Nuclei. 43 (3–4). Berlin / Heidelberg: Springer: 192–198. doi:10.1007/BF01397280. ISSN 0939-7922. {{cite journal}}: Unknown parameter |month= ignored (help)CS1 maint: date and year (link)

[7] M. Born, W. Heisenberg, and P. Jordan, Zur Quantenmechanik II, Zeitschrift für Physik, 35, 557–615 (1925). The paper was received on 16 November 1925. [English translation in: B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1]

[8] Schrödinger, E. (1926-12-01). "An Undulatory Theory of the Mechanics of Atoms and Molecules". Physical Review. 28 (6): 1049–1070. doi:10.1103/PhysRev.28.1049.

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