Talk:Quantum mechanics/Archive 5

Prose
[http://en.wikipedia.org/w/index.php?title=Quantum_mechanics&diff=32568028&oldid=32410097 Can we please call a break here? I think I need to draw the line at Line 44] in the diff. This is a Featured Article. It took work to get it to this state. Line 44 is attempting to state in words what can be said in only a few math symbols. If we want some Simple English we might be able to work this out on the Talk Page. Another alternative is to create a link to some tutorial material on another page, much like a ' ' template to main article. It is also possible to lose Featured status on an article due to overworking the prose. --Ancheta Wis 08:10, 24 December 2005 (UTC)


 * One tactic we might use is something like the 'Ants and Martians' device that Feynman used with his son Carl. It ought to be possible to devise a transforming mechanism which allows the sympathetic editor some freedom in explanation for an absolute beginner. But again, the Line 44 diff highlights the need for a stylistic "zoom lens". Or perhaps a stylistic "instant replay" for explanations for the absolute beginner. One possibility that comes to mind is a Portal treatment for the prose of the Line 44 diff. --Ancheta Wis 08:22, 24 December 2005 (UTC)


 * Oops. I just re-read the Line 27 diff. We need to walk carefully here. One possible interpretation of the diff is to call into question the 'eigenstate' formulation, which is very basic to the QM picture and which does not contravene the Uncertainty principle. --Ancheta Wis 08:33, 24 December 2005 (UTC)
 * I have just re-read DenisDiderot's formulation above and believe it might be possible to use an optical cavity as a device for some explication of QM ideas. That is pretty concrete, is intimately tied to the ideas of QM and allows us to tie into some laser physics, which rests on QM ideas. --Ancheta Wis 08:56, 24 December 2005 (UTC)
 * OK - here is a possibility for some Line 44 rework: we use the [Black-body EM radiator], which is the experimental root of QM anyway, and then use optical cavity to explain some basics about waves. Just like the pictures of electron orbitals in the article. That frees us from the need to have a wave travel from position A to position B - a mechanical picture which we don't need (its Newtonian physics, well established and not the problem being addressed). What we do need is some explanation of the radiators -- that is what Planck was worrying about anyway. That is intimately tied to the laser physics I was referring to above -- which is matter-dependent, and where QM shines. --Ancheta Wis 09:19, 24 December 2005 (UTC)
 * Let's see if we can work through a revision which is sympathetic to beginners without destroying QM's Featured status. --Ancheta Wis 11:08, 24 December 2005 (UTC)

A contrary argument
 * Line 44 reverted back and as is now in the article: "The first type of quantum effect is the quantization of certain physical quantities. In the example we have given, of a free particle in empty space, both the position and the momentum are continuous observables. However, if we restrict the particle to a region of space (the so-called "particle in a box" problem), the momentum observable will become discrete; it will only take on the values $$n \frac{h}{2 L}$$, where $$L$$ is the length of the box, $$h$$ is Planck's constant, and $$n$$ is an arbitrary nonnegative integer number. Such observables are said to be quantized, and they play an important role in many physical systems. Examples of quantized observables include angular momentum, the total energy of a bound system, and the energy contained in an electromagnetic wave of a given frequency."


 * Ancheta Wis reverts from prose explanation because he says: "Line 44 is attempting to state in words what can be said in only a few math symbols."

Reasons this approach is flawed:


 * 1. There are many complaints in the "talk" regarding:  "plain English please".
 * 2. Mathematical formulae in the form of notation and symbols that can only be known and accessible to people who are already acquainted with the language of science is an inappropriate shortcut in encyclopedia article.
 * 3. It's true, perhaps the prose was too wordy, not concise enough, but necessary.  In other words, it should be re-worked but not eliminated.
 * 4. The QM featured status will not be destroyed by a more accessible explanation if it is worded expertly in the manner of the major encyclopedia. In other words, all the major encyclopedia do not simply state a formula to explain a concept.  It isn't done.  Especially without an accessible explanation of the scientific notation in the formula.
 * 5. Also, this article on QM underemphasizes the importance of Planck's constant by not explaining it.  In the Encyclopedia Brittanica, the second paragraph on QM says:  "In the equations of quantum mechanics, Max Planck's constant of action h = 6.626 ´ 10-34 joule-second plays a central role. This constant, one of the most important in all of physics, has the dimensions energy ´ time. The term “small-scale” used to delineate the domain of quantum mechanics should not be literally interpreted as necessarily relating to extent in space. A more precise criterion as to whether quantum modifications of Newtonian laws are important is whether or not the phenomenon in question is characterized by an “action” (i.e., time integral of kinetic energy) that is large compared to Planck's constant. Accordingly, if a great many quanta are involved, the notion that there is a discrete, indivisible quantum unit loses significance."
 * 6. It is important to delineate what Planck's constant is, its importance in QM, its pervasiveness throughout QM, BEFORE any mathematical formula are introduced that include the constant h and just say it is Planck's constant in some off-hand manner.
 * 7. Wikipedia is becoming an inaccessible encyclopedia due to the fact that many articles on physics are simply reduced to mathematical formulae.  This is very uncommon in the major encyclopedia like Encyclopedia Britannica and MSN Encarta where formulae are kept to a strict minimum and prose dominates.
 * 8. MSN Encarta before introducing any formulae states in its introductory article on QM: "Momentum is a quantity that can be defined for all particles. For light particles, or photons, momentum depends on the frequency, or color, of the photon, which in turn depends on the photon’s energy. The energy of a photon is equal to a constant number, called Planck’s constant, times the frequency of the photon. Planck’s constant is named for German physicist Max Planck, who first proposed the relationship between energy and frequency. The accepted value of Planck’s constant is 6.626 × 10-34 joule-second. This number is very small—written out, it is a decimal point followed by 33 zeroes, followed by the digits 6626. The energy of a single photon is therefore very small."  Again another example of properly explaining and emphasizing Planck's constant and its place in QM.
 * 9. So finally, the simple mathematical formula used alone as a definition for quantization is actually a faux pas in the world of major encyclopedia.


 * IN CONCLUSION, it is therefore a necessity, a priority, and a duty to find a way to include a prose explanation of Planck's constant before introducing mathematics. So instead of simply erasing my prose explanation, either edit it yourself or come up with a better prose explanation. I agree it needs honing down and was unnecessarily wordy. --Voyajer 13:32, 24 December 2005 (UTC)


 * I much appreciate your responsive edit. There is yet another custom in this community of encyclopedists, which is that we work together by consensus. But it actually harms us to see the prose of Encarta, which is a tertiary source (doubly filtered). Back when the encyclopedia started, we had nothing. So the prose you see is the result of the community. We agree not to publish original thoughts, but that does not prevent us from having thoughts which we think through on our own, and then find citations for the original sources, for which we can cite our (and their) thoughts. That means, on a practical level, that the other editors on this article have a say, and that we do not attempt to browbeat others. After all, they have something to contribute as well. (But we also do not slavishly copy Encarta, etc.) There is actually a template which exists to flag the existence of copied text.
 * Back to Planck's constant, h. Deutschland's greatest physicist, for which Max Planck Institute is aptly named, solved a problem which caused a crisis in the philosophical foundations of his science. We are privileged to have witnessed a century of development of a framework for that science which has not yet been integrated into our civilization, other than for a tiny fraction of the billions on our planet. In one of his popular lectures Richard Feynman wrote out a decimal point, followed by 33 zeros wandering around the board, with the significant figures for h. Perhaps that is why Encarta took the trouble to write out the value. And Planck actually came up with that number before (1899) the blackbody radiation calculations (1900).
 * So there is something going on here which we can write about, or simply link to. I personally hope that you and DenisDiderot can work out a conversation here, or perhaps on the article page itself, one thought leading to another, one editor talking to another, using the prose of the article, but doing justice to the subject, with a conscious effort to use the skills and viewpoints of each other as well as our own.
 * As an aside, I hope that someday Helgoland will somehow get a mention under the QM rubric, which is, after all, where QM was born. --Ancheta Wis 17:05, 24 December 2005 (UTC)


 * I thought the Encyclopedia Britannica quote was infinitely more important than the MSN Encarta quote because I agree whole-heartedly with you on that particular point.
 * I am not quite clear on what equation exactly that Heisenberg first formulated in Helgoland.
 * I am not sure exactly how to go about expanding the article on QM to include the information that DenisDiderot presented since it would look more like the length of a textbook than an article. You appear to be proposing a link to something else with more clarification on QM.  But I'm not quite sure what that would be either.  Do you have any ideas?  Personally, I think many readers would appreciate more clarification, more background on Planck, more history of development, and more illustrations relating to waves and acoustics, that is, if they were given enough background to understand the connection to QM. I just did a search and found an article that made me cringe when reading it called Quantum Mechanics - simplified.  I suppose this could used as a page for the type of further clarification we were discussing. --Voyajer 18:09, 24 December 2005 (UTC)


 * Well, what immediately comes to mind is the picture that DenisDiderot is filling in below this text: There is a medium (called the field -- but the name is immaterial, it is something that fills a manifold ) which is shaped/affected by the boundaries upon which the system (The Operator) works. For the picture below, it is air, but what the heck. For example, if we have a thing (and wave is not too far from my intuition) which is affected by the energy we/Nature are pouring into it, then the medium takes on shapes. Like the electron orbitals in the article. Now the geometrical part (the manifold) is the mathematical side, and the System (the Energy,etc Operators) is the physics side. Like in GR, where the math side (the pretty side)= the physics side (the messy part with Stress-Energy). So for an optical cavity we have light filling it, and that light is augmenting the energy states of the material in the cavity. But since we have artificially shaped the boundaries of the cavity, and since we are pumping in energy, we are getting back specific transitions of electron energy levels in the material, which is specific frequencies of light. The specific equations of QM, depending on the picture (Schrödinger picture or Heisenberg picture) describe the System under consideration. I have to admit more comfort with the wave (Schrödinger) description than the more rigorous state description (Heisenberg). But - and this is the non-intuitive part - the Heisenberg picture, which does not operate in space/time can be translated to a wave-like (more intuitive) view by the 'Ehrenfest procedure' (discussed by Guest in the talk page above) which allows the computations of time averages, occupying space. That's the crazy part which as a macroscopic being brought up on large time and space scales, I personally do not have a feeling for. The best description I have seen is Feynman's characterization of the Uncertainty principle as something like silly putty, where if you squeeze it, it (the medium/air/field) fights back. But that is where you have a nice description, given above. --Ancheta Wis 19:04, 24 December 2005 (UTC)
 * Some of the ramifications of "...which does not operate in space..." can be seen in Bell's theorem and quantum entanglement.


 * Although you gave a nice explanation, that isn't what I was asking. I meant where does Helgoland come into the picture?  What was Heisenberg working on in Helgoland? Heisenberg born 1901 Wuerzbrug, Bavaria was Sommerfeld's student and in 1922 he co-authored two papers on the atomic theory of X-ray spectra and the so-called anomalous Zeeman effect. In 1921 Heisenberg published his own paper on the anomalous Zeeman effect introducing half-interger quantum numbers. In 1923 Heisenberg collaborated with Born on perturbation methods to describe the helium atom after which Heisenberg went to Goettingen.  In 1924 Heisenberg went to Copenhagen to develop quantum theory with Bohr. In 1925 Heisenberg wrote "On a quantum theoretical re-interpretation of kinematical and mechanical relationships" based on anharmonic oscillators which became known as matrix mechanics.  In 1927 Heisenberg developed the uncertainty principle. QUESTION: Therefore, which of the above theories was developed in Helgoland?--Voyajer 21:21, 24 December 2005 (UTC)
 * Google tells me it was Matrix Mechanics. --Ancheta Wis 21:44, 24 December 2005 (UTC)


 * "In June 1925, while recuperating from an attack of hay fever on Helgoland, he solved the problem of the stationary (discrete) energy states of an anharmonic oscillator:
 * Heisenberg (1925), "Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen" ("About the Quantum-Theoretical Reinterpretation of Kinetic and Mechanical Relationships"), Zeitschrift für Physik.

Time-out.
OK folks, those of you who are watching the show, please go to the refrigerator or switch to another channel while we take a game-break. Otherwise, stay tuned while we work thru some details about this article's construction. What I am seeing is several sincere editors who have clearly stated programs for making this Featured article more accessible to beginners. Some of the differences I see are stylistic, others are philosophical.

I will be taking the phrases that pop-out at me and will attempt to place them in outline form, for starters. Please feel free to interrupt, re-arrange, etc. --Ancheta Wis 11:21, 24 December 2005 (UTC)
 * a standing wave,
 * A standing wave is considered a one-dimensional concept by many students, because of the examples (waves on a spring or on a string) usually provided. In reality, a standing wave is a synchronous oscillation of all parts of an extended object in which the oscillation profile (in particular the nodes and the points of maximal oscillation amplitude) doesn't change. This is also called a normal (= uncoupled) mode of oscillation. You can make the profile visible in Chladni's figures and in vibrational holography. In unconfined systems, i.e. systems without reflecting walls or attractive potentials, traveling waves may also be chosen as normal modes of oscillation (see boundary conditions).


 * why we view the electron as a standing wave,
 * An electron beam (accelerated in a cathode ray tube similar to TV) is diffracted in a crystal and diffraction patterns analogous to the diffraction of monochromatic light by a diffraction grating or of X-rays on crystals are observed on the screen. This observation proved de Broglie's idea that not only light, but also electrons propagate and get diffracted like waves. In the attracting potential of the nucleus, this wave is confined like the acoustic wave in a guitar corpus. That's why in both cases a standing wave (= a normal mode of oscillation) forms. An electron is an occupation of such a mode.


 * what an orbital actually is,
 * An orbital is a normal mode of oscillation of the electronic quantum field, very similar to a light mode in an optical cavity being a normal mode of oscillation of the electromagnetic field.


 * why an electron is said to be an occupation of an orbital, and
 * In my view, this is the main new idea in quantum mechanics, and it is forced upon us by observations of the states of electrons in multielectron atoms. Certain fields like the electronic quantum field are observed to allow its normal modes of oscillation to be excited only once at a given time, they are called fermionic. If you have more occupations to place in this quantum field, you must choose other modes (the spin degree of freedom is included in the modes), as is the case in a carbon atom, for example. Usually, the lower-energy (= lower-frequency) modes are favoured. If they are already occupied, higher-energy modes must be chosen. In the case of light the idea that a photon is an occupation of an electromagnetic mode was found much earlier by Planck and Einstein, see below.


 * the basics of wave-particle duality.
 * If you do a position measurement, the result is the occupation of a very sharp wavepacket being an eigenmode of the position operator. These sharp wavepackets look like pointlike objects, they are strongly coupled to each other, which means that they spread soon.

---
 * waves
 * superpositions of waves
 * Waves can go through each other without disturbing each other. It just looks like there were two superimposed realities each carrying only one wave and not knowing of each other. That's what is assumed if you use the superposition principle mathematically in the equations.


 * Chladni's figures.
 * On page Chladni's figures you find some very enlightening pictures in the links provided.


 * What about emission, absorption, particle processes?
 * All processes in nature can be reduced to the time evolution of modes and to (superpositions of) reshufflings of occupations, as described in the Feynman diagrams. For example in an emission of a photon by an electron changing its state, the occupation of one electronic mode is moved to another electronic mode of lower frequency and an occupation of an electromagnetic mode (whose frequency is the difference between the frequencies of the mentioned electronic modes) is created. --DenisDiderot 11:44, 25 December 2005 (UTC)


 * experiments with standing and propagating waves
 * fermionic and bosonic occupations
 * Electrons and photons become very similar in quantum theory, but one main difference remains: Electronic modes cannot be excited/occupied more than once (= Pauli exclusion principle) while photonic/electromagnetic modes can and even prefer to do so (= stimulated emission). --DenisDiderot 12:17, 24 December 2005 (UTC)
 * This property of electonic modes and photonic modes is called fermionic and bosonic, respectively. Two photons are indistinguishable and two electrons are also indistinguishable, because in both cases, they are only occupations of modes: all that matters is which modes are occupied. The order of the occupations is irrelevant except for the fact that in odd permutations of fermionic occupations, a negative sign is introduced in the amplitude.


 * Of course, there are other differences between electrons and photons:
 * The electron carries an electric charge and a rest mass while the photon doesn't.
 * In physical processes (see the Feynman diagrams), a single photon may be created while an electron may not be created without at the same time removing some other fermionic particle or creating some fermionic antiparticle. This is due to the conservation of charge. --DenisDiderot 11:44, 25 December 2005 (UTC)

---
 * Pauli exclusion principle and
 * stimulated emission
 * Now let's look at Planck's problem. From his article, we see that he was trying to solve a practical problem, which was to derive an expression for the energy radiating from a light bulb and the rest is history, (well-known, etc., etc.) ...
 * Planck was the first to suggest that the electromagnetic modes are not excited continuously but discretely by energy quanta $$h \nu$$ proportional to the frequency. By this assumption, he could explain why the high-frequency modes remain unexcited in a thermal light source: The thermal exchange energy $$k_B T$$ is just too small to provide an energy quantum $$h \nu$$ if $$\nu$$ is too large. Classical physics predicts that all modes of oscillation -- regardless of their frequency -- carry the average energy $$1/2 k_B T$$, which amounts to an infinite total energy (called ultraviolet catastrophe). This idea of energy quanta was the historical basis for the concept of occupations of modes, designated as photons by Einstein.  --DenisDiderot 12:59, 24 December 2005 (UTC)

--- What about operators, eigenstates, measurements and all that?
 * The system of modes to describe the waves can be chosen at will. Any arbitrary wave can be decomposed into contributions from each mode in the chosen system. For the mathematically inclined: The situation is analogous to a vector being decomposed into components in a chosen coordinate system. Decoupled modes or, as an approximation, weakly coupled modes are particlularly convenient if you want to describe the evolution of the system in time, because each mode evolves independently of the others and you can just add up the time evolutions. In many situations, it is sufficient to consider less complicated weakly coupled modes and describe the weak coupling as a perturbation.


 * In every system of modes, you must choose some (continuous or discrete) numbering (called "quantum numbers") for the modes in the system. In Chladni's figures, you can just count the number of nodes of the standing waves in the different space directions in order to get a numbering, as long as it is unique. For decoupled modes, the energy or, equivalently, the frequency might be a good idea, but usually you need further numbers to distinguish different modes having the same energy/frequency (this is the situation referred to as degenerate energy levels). Usually these additional numbers refer to the symmetry of the modes. Plane waves, for example -- they are decoupled in spatially homogeneous situations -- can be characterized by the fact that the only result of shifting (translating) them spatially is a phase shift in their oscillation. Obviously, the phase shifts corresponding to unit translations in the three space directions provide a good numbering for these modes. They are called the wavevector or, equivalently, the momentum of the mode. Spherical waves with an angular dependence according to the spherical harmonics functions (see the pictures) -- they are decoupled in spherically symmetric situations -- are similarly characterized by the fact that the only result of rotating them around the z-axis is a phase shift in their oscillation. Obviously, the phase shift corresponding to a rotation by a unit angle is part of a good numbering for these modes; it is called the magnetic quantum number m (it must be an integer, because a rotation by 360° mustn't have any effect) or, equivalently, the z-component of the orbital angular momentum. If you consider sharp wavepackets as a system of modes, the position of the wavepacket is a good numbering for the system. In crystallography, the modes are usually numbered by their transformation behaviour (called group representation) in symmetry operations of the crystal, see also symmetry group, crystal system.


 * The mode numbers thus often refer to physical quantities, called observables characterizing the modes. For each mode number, you can introduce a mathematical operation, called operator, that just multiplies a given mode by the mode number value of this mode. This is possible as long as you have chosen a mode system that actually uses and is characterized by the mode number of the operator. Such a system is called a system of eigenmodes, or eigenstates: Sharp wavepackets are no eigenmodes of the momentum operator, they are eigenmodes of the position operator. Spherical harmonics are eigenmodes of the magnetic quantum number, decoupled modes are eigenvalues of the energy operator etc. If you have a superposition of several modes, you just operate the operator on each contribution and add up the results. If you chose a different modes system that doesn't use the mode number corresponding to the operator, you just decompose the given modes into eigenmodes and again add up the results of the operator operating on the contributions. So if you have a superposition of several eigenmodes, say, a superposition of modes with different frequencies, then you have contributions of different values of the observable, in this case the energy. The superposition is then said to have an indefinite value for the observable, for example in the tone of a piano note, there is a superposition of the fundamental frequency and the higher harmonics being multiples of the fundamental frequency. The contributions in the superposition are usually not equally large, e.g. in the piano note the very high harmonics don't contribute much. Quantitatively, this is characterized by the amplitudes of the individual contributions. If there are only contributions of a single mode number value, the superposition is said to have a definite or sharp value.


 * In measurements of such a mode number in a given situation, the result is an eigenmode of the mode number, the eigenmode being chosen at random from the contributions in the given superposition. All the other contributions are supposedly eradicated in the measurement -- this is called the wave function collapse and some features of this process are questionable and disputed. The probability of a certain eigenmode to be chosen is equal to the absolute square of the amplitude, this is called Born's probability law. This is the reason why the amplitudes of modes in a superposition are called "probability amplitudes" in quantum mechanics. The mode number value of the resulting eigenmode is the result of the measurement of the observable. Of course, if you have a sharp value for the observable before the measurement, nothing is changed by the measurement and the result is certain. This picture is called the Copenhagen interpretation. A different explanation of the measurement process is given by Everett's many-worlds theory; it doesn't involve any wave function collapse. Instead, a superposition of combinations of a mode of the measured system and a mode of the measuring apparatus (an entangled state) is formed, and the further time evolutions of these superposition components are independent of each other (this is called "many worlds").
 * --DenisDiderot 21:12, 24 December 2005 (UTC)


 * As an example: a sharp wavepacket is an eigenmode of the position observable. Thus the result of measurements of the position of such a wavepacket is certain. On the other hand, if you decompose such a wavepacket into contributions of plane waves, i.e. eigenmodes of the wavevector or momentum observable, you get all kinds of contributions of modes with many different momenta, and the result of momentum measurements will be accordingly. Intuitively, this can be understood by taking a closer look at a sharp or very narrow wavepacket: Since there are only a few spatial oscillations in the wavepacket, only a very imprecise value for the wavevector can be read off (for the mathematically inclined reader: this is a common behaviour of Fourier transforms, the amplitudes of the superposition in the momentum mode system being the Fourier transform of the amplitudes of the superposition in the position mode system). So in such a state of definite position, the momentum is very indefinite. The same is true the other way round: The more definite the momentum is in your chosen superposition, the less sharp the position will be, and it is called Heisenberg's uncertainty relation.


 * Two different mode numbers (and the corresponding operators and observables) that both occur as characteristic features in the same mode system, e.g. the number of nodes of one of Chladni's figures in x direction and the number of nodes in y-direction or the different position components in a position eigenmode system, are said to commute or be compatible with each other (mathematically, this means that the order of the product of the two corresponding operators doesn't matter, they may be commuted). The position and the momentum are non-commuting mode numbers, because you cannot attribute a definite momentum to a position eigenmode, as stated above. So there is no mode system where both the position and the momentum (referring to the same space direction) are used as mode numbers.
 * --DenisDiderot 12:32, 25 December 2005 (UTC)


 * What about the Schrödinger equation, the Dirac equation etc?
 * As in the case of acoustics, where the direction of vibration, called polarization, the speed of sound and the wave impedance of the media, in which the sound propagates, are important for calculating the appearance and the frequency of modes as seen in Chladni's figures, the same is true for electronic or photonic/electromagnetic modes: In order to calculate the modes (and their frequencies or time evolution) exposed to potentials that attract or repulse the waves or, equivalently, exposed to a change in refractive index and wave impedance, or exposed to magnetic fields, there are several equations depending on the polarization features of the modes:
 * Electronic modes (their polarization features are described by Spin 1/2) are calculated by the Dirac equation, or, to a very good approximation in cases where the theory of relativity is irrelevant, by the Schrödinger equation and the Pauli equation.
 * Photonic/electromagnetic modes (polarization: Spin 1) are calculated by Maxwell's equations (You see, 19th century already found the first quantum-mechanical equation! That's why it's so much easier to step from electromagnetic theory to quantum mechanics than from point mechanics).
 * Modes of Spin 0 would be calculated by the Klein-Gordon equation.
 * --DenisDiderot 13:08, 25 December 2005 (UTC)

--- ---
 * But let's also look at Helmholtz. He ran an experiment which clearly showed the physical reality of resonances in a box. (He predicted and detected the eigenfrequencies.)


 * DenisDiederot and Ancheta Wis have put together some wonderful fundamental information on QM here. It seems a shame that it will eventually get lost in the "talk" discussion section.  Does anyone have any idea of how to include this information in Wikipedia? Will it bog down the current article?  If so, under what article title could this information be included, as a whole, as a further explanation of QM?  --Voyajer 18:26, 24 December 2005 (UTC)


 * Thank you very much. I'm sure we'll find a solution, maybe on a page like Quantum mechanics explained. Merry Christmas to everybody! --DenisDiderot 21:12, 24 December 2005 (UTC)
 * And Merry Christmas to you both! --Ancheta Wis 21:52, 24 December 2005 (UTC)


 * Merry Christmas ,From TW.--HydrogenSu 07:20, 25 December 2005 (UTC)

I moved the texts to the new page Quantum mechanics explained, because this talk page might otherwise get too large. Let's see if we can create a text providing an exact (non-simplified) explanation for beginners. --DenisDiderot 13:35, 25 December 2005 (UTC)