Talk:Wave function/Archive 7

§§

Please keep this talk page organized
Chjoaygame, I see that you keep adding text to messy old huge threads. Please start new threads, one for each new topic that you have. (A single post of yours can be about $10^{250}$ different things. Walls of text doesn't automatically give you right because nobody is able to respond.) YohanN7 (talk) 11:10, 14 January 2016 (UTC)


 * Copied from above:


 * Am I reading your meaning aright if I am prompted to ask please would you very kindly give a link here to where a non-probabilistic interpretation of the wave function is discussed?Chjoaygame (talk) 20:19, 30 January 2015 (UTC)


 * Sure. Walter Greiner, Relativistic Quantum Mechanics, page 1, item 3. One more: Landau & Lifshitz, Quantum Electrodynamics pp 1-4. (L&L kills it off pretty good.)
 * The probability interpretation simply doesn't work well with relativity. You can have it only when applying relativistic equations (e.g. Dirac) to essentially non-relativistic problems (e,g hydrogen atom). I do not intend to explain to you why. YohanN7 (talk) 21:15, 30 January 2015 (UTC)


 * With much respect for your erudition in the mathematical direction, I have given some thought to your proposal that "Landau & Lifshitz, Quantum Electrodynamics pp 1-4. (L&L kills it off pretty good.)" It is the Born rule that is said to be killed off. I think the scattering matrix is a formulation that keeps account of data found in experimental set-ups, but I think the fundamental physics of observation is still given by the Born rule. That rule tells how to relate an actual physical particle detection to some mathematical wave functions. The scattering matrix keeps account of such detections. The scattering matrix is in the phyicist's notebook or computer. The particle detections are in the laboratory apparatus. Also I have looked at your cited place in Greiner's Relativistic Quantum Mechanics. I don't think this kills the Born rule. It means one must reconsider, but not discard the Born rule. More precisely, that item is about probability density, not the Born rule as such, though closely related.
 * Chjoaygame (talk) 17:35, 9 January 2016 (UTC)Chjoaygame (talk) 12:18, 14 January 2016 (UTC)

Same mistake again Chjoaygame. Goddamm it, start a thread with fresh text. Please. YohanN7 (talk) 12:24, 14 January 2016 (UTC)

Probability density and Born rule in relativity
Following your above statements, my question is 'what alternative is there to the Born rule?' I think neither of those cited items offers an alternative.Chjoaygame (talk) 14:53, 14 January 2016 (UTC)


 * Okay, for the reader unaware of the context of this: I have stated that the standard interpretation in non-relativistic theory of the wave function squared (or more properly $Ψ^{&dagger;}Ψ$) in configuration space as a probability density is impossible in a relativistic theory.


 * The physical reason for this is pair production and other phenomena that relativity allows for (and even forces I believe). Consider for definiteness an electron in an external potential (hydrogen atom). The solution initially offered by Dirac through the Dirac equation yields a wave function with much of the non-relativistic machinery intact, including that of a probability density. One has a standing wave (sort of). But what really happens is that there is a constant exchange of photons between the electron and the proton. While these virtual photons propagate, they may temporarily create an electron-positron pair through vacuum polarization. Reiterate the reasoning (draw more complicated Feynman diagrams) and the result is that at any time the number of particles (and their species) is known. Thus it should be clear that any attempt at interpreting the wave function (which is essentially constant in time apart from a sinusoidal oscillation) as the probability amplitude of finding the particle at $x$ cannot stand up for rigorous inspection.


 * One might object that in the hydrogen example there is one real electron while all other particles are virtual. A measurement will disturb this situation. Either a strong enough field (see Hawking radiation for an extreme example) or a passing by real particle may annihilate a virtual antiparticle in (for example) a (would–be) vacuum bubble while the virtual particle acquires reality and escapes. (This latter scenario occurs in a perturbative expansion (when iterating the Lippmann–Schwinger equation) when a scattering center is modeled by an external potential, see Greiner's Quantum Electrodynamics, chapter two. Not too sure it is a valid process in full QFT.)


 * For a reasonably down to earth example that needs neither measurements of virtual particles do demonstrate the impossibility of the probability density interpretation, see Klein paradox, where one starts out with a single real particle and end up with more real particles - all following directly from the Dirac equation or the Klein–Gordon equation. It should be noted that a potential barrier as given in the linked article need not have the shape of a Heaviside step function. It need just be sufficiently steep and high, but can otherwise be smooth. While these equations both do have a preserved norm (not positive definite in the KG case), the number of particles they describe varies with time - even with no virtual particles involved.


 * Another example is provided by a wave packet of Gaussian type, initially consisting of (superpositions of) states describing electrons with, say, spin up. This is initially interpreted as a localized electron. If the state is allowed to evolve according to the Dirac equation, it will describe a mixture of "spin up electron", but also "positron with spin both up and down" goes into the mixture. This eample can be found in Greiner's RQM.


 * A mathematical reason (which needs physical interpretation) is that the Heisenberg uncertainty principle quite literally acquires a new dimension in relativity; the product of the uncertainty in energy and the uncertainty of time is greater than zero. This somehow (don't ask me how) allows for nature to cheat with the rules that it is supposed to follow, at least a little bit during a short time. A particle that is localized within its Compton wavelength automatically has companions.


 * The Born rule is as far as I can see left intact by this, but its application needs extreme care if one wants to be rigorous. For one thing, position eigenstates are rarely used in relativistic quantum mechanics, an inherently inconsistent collection of theories by the way, full QFT is needed. As far as the scattering matrix goes, it is a complete subject of its own to formulate a scattering process rigorously. The key point is that one turns (mathematically) interactions off in the distant past and distant future. This corresponds well with how scattering experiments are actually performed. There are of course interactions that cannot be turned off because they are present in the vacuum. I believe (but do not know the details) that these effects are collected into the renormalization parameters of the theory. Another key point is that no attempt whatsoever is made to follow what actually happens during a scattering process. One does not solve for any time dependent wave function and try to interpret it at some instant during the collision. One merely calculates the overlap
 * $$\langle \Psi_{out}|\Psi_{in}\rangle$$
 * between the "in" states and the "out" states in the Heisenberg picture, or less rigorously but much simpler, in the interaction picture. These states are in the distant past and distant future states with a fixed number of non-interacting particles. This quantity squared yields the scattering matrix element – interpreted as the transition probability. This is exactly the Born rule.


 * I stated also way up that I don't think this article should discuss this subject. I have four reasons for this that taken together might make sense:
 * It is beyond the scope of the article
 * People get upset (Whut - Dirac wrong? Dude ya kidding!)
 * Textbooks most often present things in the historical order of events
 * Much wave function = much likelihood of finding particle still holds true
 * The first bullet should be uncontroversial. The second bullet is taken from experience from Wikipedia and also elsewhere. There are people that are sentimental about Dirac and his equation (perhaps they taught classes in the fifties or whatever on the Dirac equation). They will not under any circumstances be told that Dirac's original rationale, finding a positive definite probability density, was wrong (he couldn't possibly have known at that time). The same applies to freshly baked undergraduate students (the most likely readers and editors of this article), but for different reasons. (They just nailed QM101 and are close to knowing everything.) Note: This is not Dirac bashing. Dirac later turned the one-particle theory into a multi-particle theory through Hole theory. Amazingly, this theory works mathematically (the Greiner series of books uses it throughout), though most believe today that it is physical nonsense. (Dirac himself (according to Weinberg) in the seventies finally abandoned hole theory.)


 * Avoiding endless debates and even conflicts is not good enough rationale, but taken together with the last bullet it makes some sense. Take for instance Greiner. He warns on the first page that the probability density will have to go. Then he proceeds in the footsteps of Dirac. It is probability density (and the lack of it with the Klein–Gordon equation) for a several hundreds of pages. Only in chapter twelve the probability density and the one-particle interpretation is finally kicked out. Other books do the same, but some of them simply refrain form using the term probability density. The only exception I know of that have written complete introductions to modern QM is Weinberg. He simply ignores original Dirac one-particle theory (and all RQM) because it at best leads nowhere and at worst introduces misconceptions. This is perhaps logical, but not very pedagogical. The Dirac equation, the Klein–Gordon equation and all equations of RQM then reappears in a different guise in QFT.


 * To wrap up the last bullet: People can and will find "norm of wave function" and "probability density" on the same page in the same book by well renowned authors. Thus it is wiki-correct to call the thing in question "probability density". Besides, it is in the non-relativistic limit (e.g hydrogen to good approximation!) correct to call it probability density. This is what the last bullet means. YohanN7 (talk) 10:16, 18 January 2016 (UTC)

position and momentum wave functions represent the "same" object?
The article claims, "As elements of abstract physical Hilbert space, whose elements are the possible states of the system under consideration, they represent the same object." What does this even mean? Certainly all infinite-dimensional separable Hilbert spaces are isometrically isomorphic to ℓ2, and hence to each other. However there are uncountably many such isomorphisms between any two such spaces. What distinguishes the isomorphism that brings these two wave functions, drawn from distinct-but-isomorphic Hilbert spaces, into coincidence, and is it a natural isomorphism? Vaughan Pratt (talk) 04:58, 30 January 2016 (UTC)


 * It means that they represent identical physical states. States are more precisely rays in Hilbert space, so "state space" is really "ray space". Maps between ray spaces and the requirements on such maps (most importantly, preservation of the Born rule) under symmetry transformations are discussed in Wigner's theorem. For such maps, there are corresponding maps between the underlying Hilbert spaces (this is the content of Wigner's theorem), but these are decidedly not unique. For momentum and position wave functions, the map is the Fourier transform. You can also here view the Fourier transform as a unitary transformation on the same Hilbert space of square integrable functions. Technical details and rigor (and what else?) are fading from my memory, so please forgive inconsistencies in what I say here. I don't know how precise this article should be. As you can see above, there are complaints about too much math and mathematical (attempts at) rigor already. YohanN7 (talk) 10:36, 30 January 2016 (UTC)


 * I am guilty of writing "walls of text" here. I think Editor Vaughan Pratt is probably right here. I think that the article suffers from a serious muddle. For more or less obvious reasons, I have not tried to edit the muddle away. But I will say something here that I think supports Editor Vaughan Pratt.


 * A state vector is a symbol that belongs to a certain set of abstract objects called the set of state vectors. The set has valuable mathematical properties. The Dirac notation is well adapted for dealing with this.


 * The symbols of the set indicate or designate or signify (but are not themselves) some other quite different mathematical objects called wave functions, that were invented by Schrödinger. A wave function is a solution of the Schrödinger equation. Each wave function describes an instance of a pure state of a quantum mechanical system. There are two kinds of state vector and thus two kinds of pure states: symbolized by bras and kets. The theory is symmetrical between bras and kets, but let us, for sake of definiteness, follow Dirac and others in saying that a ket denotes a preparative device and a bra denotes an observing device. A pure state refers to a macroscopic physical thing, either a preparative device or an observing device. A beam, composed of independent instances of inscrutable quantum systems, passes from preparative to observing device.


 * The beam is essential to the reasoning. The beam is necessary as the carrier of the ensemble that is the sample space for the Born transition probabilities. The word beam occurs about 39 times in Dirac's Chapter 1 (fourth edition), which is devoted to physical meaning. We usually don't get to directly see the contents of the beam; they are usually too small to see. We only know how the preparative device was built and how we feed and drive it, and how the observing device was built and how we operate and read it, and how we place the output port of the preparative device up against the input port of the observing device; or how we instruct others to do so; it's not us as individual persons. That is the core and correct part of the Copenhagen interpretation, and I think it is obvious common sense.


 * There are three basic device elements: analyzers, ovens, and detectors (= anti-ovens).


 * An analyzer is for example a calcite crystal with two output channels for one input channel. The crystal analyzes one degree of freedom into its eigenstates. Each output channel lets out instances of a system in a labeling eigenstate characteristic of the crystal's degree of freedom. A lifetime of investigation can go into finding a suitable crystal, Stern-Gerlach magnet, prism, or whatever.


 * An oven is full of some hot vapor of a chemically pure species, with a hole in the side to let out a beam of independent instances of quantum systems, atoms, or molecules (not cats, you can't vaporize a cat and then expect it still to be a cat).


 * An anti-oven (= detector) is a macroscopic chamber full of special stuff in a metastable state that can be triggered to boil, form a droplet, cascade, click, whatever, by the arrival of a suitable system, a particle or somesuch.


 * A filter is an analyzer with all output channels blocked except one. It lets out a beam in one eigenstate of its degree of freedom.


 * The filters/analyzers are passive and reversible, and the oven and anti-oven are irreversible.


 * To prepare a pure state, the preparative device is typically an oven with a small hole in the wall, from which issues a desperately and utterly mixed beam; that is then filtered by a sufficient (maximal) succession of mutually compatible (commutative) analyzers configured as filters (all except one output channel blocked) that eventually supply a beam of independent systems, each an instance of the system in a pure state, denoted by the ket and described by the wave function. This is in a simultaneous eigenstate of all the analyzer/filters. It is defined by a list of degrees of freedom (crystals etc.) and the corresponding list of respective unblocked eigenvalues, one eigenvalue for each degree of freedom. Such a preparative device is fine in theory, but not so easy in practice.


 * The observing device is more or less a mirror image of this. It takes in the beam and passes it through a sufficient (maximal) succession of mutually compatible (commutative) analyzers (just like the preparative analyzer/filters, but with all output channels open and able to pass instances of the system, not blocked as they are for filtering) that eventually supplies sub-beams of independent instances of systems that go to respective anti-ovens = detectors. Any one final detector defines the investigator's chosen degree of freedom and the eigenstate of the instance (specimen) of the system that is detected, denoted by the bra and described by the dual wave function.


 * The wave function (bra or ket) is identified by (1) the generalized configuration space (= list of degrees of freedom) determined by the preparative or observing filter/analyzers, (2) the list of eigenvalues of the respective degrees of freedom, and (3) by the pure substance that is put into the preparative oven.


 * If the preparative filter and observing analyzer are copies of one another, then the pure beam goes into precisely one output channel of the analyzer. That's how we check the apparatus and know we have a pure state.


 * Of more exciting interest is when the observing analyzer is incompatible with the preparative filters. Then the beam is resolved into many sub-beams. In principle they could be re-assembled by being put back through a further copy of the analyzer. The proportions of the beam that go into the various sub-beams are the coefficients in the expansion of the pure state into a superposition according to the analyzer. The choice of analyzer is called the choice of representation. The proportions jointly are are called the representative.


 * The choice of preparative filter determines once and for all which pure state we will deal with, assuming the same pure substance is fed into the preparative oven. The preparative filter determines, for example, that we will deal with system instances with an up spin in the y direction and with a certain z-momentum. The y-spin and z-momentum determine the unique configuration space for that pure state. The up value and the certain value of the z-momentum tell exactly which pure state within that configuration space. The uniqueness is what defines the state as pure.


 * The investigator can combine identically prepared beams and still have the same pure state with the same definite unique wave function. But if the preparations are not identical, the combined beam is mixed, and is not described by a single wave function. The slightest departure from identical preparation creates a mixture. A combination of beams from preparative devices with different filters or different substances in the ovens produces a mixture, not a superposition, while a combination from preparative devices with the same filters and substances produces a pure beam.


 * The choice of representation assumes in advance that the pure state has been already defined and fixed by choice of preparative device. A different choice of preparative devices produces a different pure state with its respectively different wave function. There is no tampering with the original beam. If it is tampered with, it becomes a different beam with a different wave function, or a mixture.


 * But any choice of representation is permitted, within reason. The process of representation is a matter of splitting = analyzing the original pure state into an array of sub-beams. If the sub-beams are left intact (no detector) so they can be re-assembled, they remain in the original pure state. If they are treated by external agents, such as detectors, the splitting is reconsidered and re-described, as a new preparative filter, and the configuration space is new, and they are no longer in the original pure state. They are now in new pure states, with respect to the new filtration. With due perversity, this is transmogrified into a crazy story called "collapse". It means that the instance of the system is changed by passage through the filters, heaven forbid us to say so, by a randomly determined quantum transfer of translational momentum according to Duane's principle. That momentum transfer can be reversed by a copy of the filter that produced it, to re-assemble the original beam if there is no external tampering, such as by a detector; the re-assembly reminds us that we are thinking about superposition when we forbid tampering. It is strictly forbidden for hard-line Niels Bohr Copenhagenists to mention or think about Duane's principle, because it is demystificatory, and makes physical sense, and does away with teasing puzzles about "collapse" and is therefore counter-revolutionary, revanchist, revisionist, backsliding, and generally frowned upon. But Heisenberg accepted it and put it in his 1930 textbook. (For some odd reason I haven't worked out, Born was very hard on Landé about this, and of course Shimony jumped on that bandwagon.)


 * The foregoing is, I think, childishly simple and obvious. But I think there are those who will think it wrong or beneath the dignity of Wikipedia.


 * A given original state vector, denoting a corresponding wave function, describing a corresponding physical state, defined by a preparation by a sufficient (maximal) set of compatible filters and pure chemical substance, is taken and analyzed into a superposition of a complete set of wave functions, described by the list of weights of the constituents. Any one of that set is found in a specific beam, and if separately tampered with, is thereby in a different state. Any sufficient (maximal) set of compatible analyzers is permitted to be chosen and nominated as representation, at the investigator's whim. It is the whole list of weights that is the representative of the original object, not the component wave functions as referred to separately and considered one by one.


 * I think this supports Editor Vaughan Pratt.Chjoaygame (talk) 16:18, 30 January 2016 (UTC)


 * As for which Hilbert spaces? Schrödinger's wave functions are native born citizens of L2. But they have also the special property that they are solutions of the Schrödinger equation. Consequently, discussion of them doesn't need the vastness of L2. A more specialized Hilbert space (that fits well and truly inside L2) is enough. The space of state vectors is isomorphic to (but of course not the same thing as) the latter more specialized Hilbert space, and forgets all about the birth of the wave functions in L2.Chjoaygame (talk) 06:55, 31 January 2016 (UTC)


 * Replying specifically to Editor YohanN7. I guess I am the subject of your comment "there are complaints about too much math and mathematical (attempts at) rigor already". I am sorry if that has seemed to be my concern. No, I am not complaining about too much math. I see nothing wrong with math. My concern, that applies particularly to the article Quantum state, but also to this one, is that there is a lack of a physical account. Dirac spends the bulk of his Chapter 1 on a physical account. But hardly a word on it in Wikipedia in these articles, in which it is needed. Quantum mechanics is a branch of physics. Without a physical account, the math is opaque.Chjoaygame (talk) 23:20, 31 January 2016 (UTC)


 * Quoting the part of the article to which Editor Vaughan Pratt refers:
 * The position-space and momentum-space wave functions are Fourier transforms of each other, therefore both contain the same information, and either one alone is sufficient to calculate any property of the particle. As elements of abstract physical Hilbert space, whose elements are the possible states of the system under consideration, they represent the same object, but they are not equal when viewed as square-integrable functions.
 * Looking closely at these two sentences, one can see a possible miscommunication: in the words "object", and "state". I would say that in these sentences, the word 'object' makes sense as referring to the chemical species that is chemically pure in the oven. A beam of the chemical species, the same 'object' in that sense, can be prepared in a pure state of momentum and it can be prepared in a pure state of position. The two states are physically different. The state that is pure with respect to momentum is not pure with respect to position. It is a superposition of countless states each pure with respect to position. And vice versa. Thus the two states refer to the same 'object', but to different states of that object. Physically speaking, instances of a species in different states are different objects, in that they have different physical properties, though they belong to the same chemical substance. A difference of states is physically significant, though it does not signify a chemical difference. And we are here doing physics. Thus there is room for muddle in the word 'object'. It is like room for muddle with the word 'it'. Used without due precaution, the word 'it' can confuse: to which of several preceding items does it refer? Hintikka calls this problem 'unresolved anaphora'. Aristotle would distinguish between οὐσία and ἕξις. I am not clued up enough on his theory of material substances to say if he distinguished physical from chemical properties. But that is the relevant distinction here.


 * I think Editor Vaughan Pratt is concerned about this. Personally, I disagree with the above sentence "It means that they represent identical physical states."Chjoaygame (talk) 00:17, 1 February 2016 (UTC)

Thanks, Chjoaygame. Let's see what the editor responsible for "they represent identical physical states" has to say. If he or she can't cite a source to that effect or explain it consistently it should probably be deleted as meaningless. Vaughan Pratt (talk) 06:12, 1 February 2016 (UTC)


 * Ignoring Chjoaygame's new wall of text, complete with anti-ovens: this is quite simple.


 * The state of a quantum system is described by a vector in Hilbert space. This is postulate 1 of quantum mechanics in and in every modern text on QM unless they call it postulate 0, or they don't use postulates, in which case it is still the main point to make. A wave function is the projection of a state vector onto a specific set of coordinate axes. I. e. it is a coordinate vector. See, and Shankar again for a good intuitive introductory level explanation in Chapter 1, for this. This is no different from a triple of numbers representing an arrow in three-dimensional mathematical space modeling a physical arrow in physical space, with reference to three designated basic arrows. The collection of all wave functions with respect to a particular coordinate axes behaves like another Hilbert space, usually a closed subspace of $L^{2}$.


 * The wave function contains all information about a quantum state because it is the coordinates of the state vector that, according to postulate 1, describes the quantum system. See the first few pages in for this. Naturally, any sets of coordinates will do. Different coordinate axes - different wave functions, but they are derived from the same state vector. Analogously, choosing a different set of basic arrows in three-dimensional space, you generally get a different triple of numbers describing the same arrow.


 * The relation between (in particular, this is not the only possibility) wave functions in the coordinate representation (i.e. in terms of eigenfunctions of the position operator) and the momentum representation is simple to derive because it is known what the eigenfunctions of the momentum operator is in the coordinate representation. What the momentum and position operators are, by the way, the content of postulate 2 in Shankar. Using the three-dimensional analogy again, if the expression of three new basic arrows in terms of the three old basic arrows, then the new triple of numbers can be computed from the old triple of numbers.


 * Thus the two wave functions represent the same state vector. Going back to postulate 1, the physical state is identical, either of which wave function you chose to represent it. This is to say that the physical arrow in three-dimensional space remains the same. Sure, the state can refer to the ground state of any of e.g. all hydrogen atoms in existence, but when viewed as physical states of the system under consideration (here a hydrogen atom), these are identical. Same thing with arrows. Duplicate the arrow and parallel transport the copy to Copenhagen. They are identical as far as their vectorial properties go when mathematically described. (In the quantum case, this is as complete description as you will get.)


 * Now, is your thesis that the wave function in momentum space represents a quantum state vector different from the state vector the coordinate wave function does? If so, do you also claim that, in the analogous thee-dimensional case, the two triples of numbers do not represent the same arrow? The burden is on you to provide references supporting your point of view. Why? This is all basic and contained in all of the general references given in the article. YohanN7 (talk) 10:33, 1 February 2016 (UTC)
 * Online copy


 * An eigenfunction of position cannot be, and cannot describe, the same state as, an eigenfunction of momentum. And vice versa.


 * An eigenfunction of position can, however, be expanded as a weighted sum or integral of several eigenfunctions of momentum. Such an expansion is called a momentum wave function, or a wave function in the momentum representation. It is not an eigenfunction of momentum. A weighted sum or integral of several eigenfunctions of position is a position wave function or wave function in the position representation. It is not an eigenfunction of position.


 * Physically, how can such states be produced? One produces, verifies, and analyses pure states as I have sketched in the above wall of text. But above I have said only how to produce a beam of systems in a state that is a simultaneous eigenfunction of chosen available quantum analyzers.


 * To prepare a more general pure state, one can, just as above, prepare a beam pure in a simultaneous eigenstate, and pass it to a further device, before passing it to one's chosen observing device. The further device splits the original pure beam, by passing it through some suitably chosen diffracting item. One puts various attenuators and phase delayers into the paths of the several resulting sub-beams, and passes the resulting re-weighted but otherwise undisturbed, and hence still coherent, sub-beams through copies of the diffracting item so as to re-assemble them. This forms a new superposition, not a mixture, because one has carefully not interposed incompatible non-commuting objects that would break the coherence of the sub-beams. The new superposition is no longer in a simultaneous eigenstate of the original preparative device. But it is still in a pure state. This can be verified by passing it through copies of analyzers as indicated above. It has a wave function that belongs to the suitably chosen diffracting item. For example, that diffracting item might be a momentum analyzer; then the resulting pure state is naturally described by a momentum wave function. Moreover, it is in an eigenstate of the operator that indicates the combination of the original preparation device and the suitably chosen diffraction item, the weighting, and the re-assembly device.Chjoaygame (talk) 23:46, 1 February 2016 (UTC)Chjoaygame (talk) 05:54, 2 February 2016 (UTC)


 * Chjoaygame, this topic is about whether a position representation wave function, and its Fourier transform represent the same physical state (they do). It is not whether eigenstates of position and momentum are equal (they aren't). It is not whether there exists simultaneous eigenstates of position and momentum (there doesn't). There has been no such claim, and you should know that . There is no need to hijack a discussion on the present topic to demonstrate that you are cleverer. It is extremely frustrating.


 * Then create a new topic if you want to discuss other totally different matters, such as how to prepare particular pure states. The universe of discourse in this article is an isolated quantum system, hence a system described by a Hilbert space state and accompanying wave functions – a pure state! This is established in the first few sentences in the lead. The particular state is either given to us or up to us to chose for the purpose of discussing wave functions.


 * That said, I am happy that you seem to (finally) distinguish the concepts of pure state and simultaneous eigenstate of a set of commuting observables, but it remains to see if you have gotten there in full. Our history warns me that you me that you might not be there yet, especially considering the below paragraph.


 * I see in the same passage above that you have problems distinguishing momentum representation wave functions and eigenstates of momentum. These are indeed very different concepts. I find it unbelievable that you can be so extremely engaged here over so long time and not be knowing this. Are you trolling? I must ask, since this has been going on for so long long time. Your message: The article is overall very bad. Every section, every paragraph is bad. In fact, every sentence is bad and the choice of individual words is disastrous. This has been you message. All of the time. Time and time again. Is there a single letter, say, perhaps, a, that is acceptable? No? Yet, not at any point where an actual matter (such as the disputed sentence above) have been discussed have you had any actual point to make. This leads me to suspect trolling. YohanN7 (talk) 09:45, 2 February 2016 (UTC)


 * Editor YohanN7 has put various epithets on me. I think that relieves me of a duty to explain myself to him. It may be of value to him to know that I have not found his epithets helpful to my understanding.


 * He may be surprised and perhaps glad to know that I have now recognized, and I think understood, my mistake. Rather, no longer surprised, since he has just above acknowledged my progress, at least to some extent. In particular, I no longer think that, as I wrote above, "the article suffers from a serious muddle".


 * I had partly reached this conclusion a little before Editor Vaughan Pratt posted his question, and I was considering ways to break the good news. That question seems to have been helpful to me in further clarifying my thinking. I now think that the answer to his question is 'yes', not 'no' as I have posted above. Thank you, Editor Vaughan Pratt. My just previous comment is specifically intended to address your question. I guess not exactly as you posed it, but still so intendedChjoaygame (talk) 11:05, 2 February 2016 (UTC)


 * You have mysterious ways of "breaking your good news". Your first reply to Vaughan was another rant on how poor the article is, not nearly a εὕρηκα moment on your part at all. Your further posts do not address Vaughan's posts. You lecture me on trivial matters in quantum mechanics. This behavior is fine one time. Ten times. A hundred times? Maybe. But not forever. I tend to get rude on occasion. But I am not permanently rude. Childish anger goes away quickly. But your style of debating on talk pages is perpetually rude in a sense that you may not understand. Constant off-topic walls of text. Hundreds of misinterpreted quotes taken out if context, willingly or unwillingly, doesn't matter, meant to support that all scientists from Bohr to Weinberg share your (mis)conceptions. An excerpt illustrating what you have written: As I read it, this part of the present article flatly contradicts the consensus of Born, Bohr, Heisenberg, Rosenfeld, Kramers, Messiah, Weinberg, and Dirac. Of course, it was my contributions you attacked. Always my contributions.


 * It is a horrible climate for discussion of improving the article. You invade every topic and fill it with quotes, allegations, Stern–Gerlach magnets (repeated ad infinitum), and walls of sometimes impenetrable text. The same occurred over at the Quantum mechanics talk page. A crackpot entered with the message that that article was badly mistaken (on all accounts). You quickly jumped on that train too. Have you noticed that almost everyone else stays away from here? YohanN7 (talk) 11:43, 2 February 2016 (UTC)

corrected version of the above faulty one

 * Considering that I have just very lately now recognized a big conceptual mistake that has been the cause of some puzzlement to me, and that made the original version of the above wall of text wrong in some important respects, I am now amending and correcting the text as follows.


 * A state vector is a symbol that belongs to a certain set of abstract objects called the set of state vectors. The set has valuable mathematical properties.


 * Each symbol of the set indicates or designates or signifies a large set of some other quite different mathematical objects called wave functions, that were invented by Schrödinger. A wave function is a solution of the Schrödinger equation or its equivalent. Each state vector refers uniquely to a possible instance of a pure state of a quantum mechanical system. In the sense that the state vector does not specify a particular wave function to define it, the state vector is an abstraction from the set of wave functions that it designates. This abstraction was invented by Dirac. He also invented a notation well adapted to express it.


 * There are two kinds of state vector and thus two kinds of pure states: symbolized by bras and kets. The theory is symmetrical between bras and kets, but let us, for sake of definiteness, follow Dirac and others in saying that a ket denotes a preparative device and a bra denotes an observing device. A pure state refers to a macroscopic physical thing, either a preparative device or an observing device. A beam, composed of independent instances of inscrutable quantum systems, passes from preparative to observing device.


 * The wave functions are solutions of the Schrödinger equation or its equivalent for the chemical species of the system. A given quantum state, as prepared, can be observed in many different ways. The many ways of observation, called representations, give rise to many different corresponding wave functions, all for the one given quantum state. It is a remarkable and important fact of quantum mechanics that from any one such wave function, there can be derived by calculation every other one that belongs to the given quantum state. The many several wave functions for the state provide different descriptions of the given state, that are, however, equivalent in this sense. This is partly analogous to the situation in classical mechanics, in which a given kinematical description, in terms of given dynamical coordinates, can be transformed by canonical transformations, into many other kinematical descriptions in terms of other dynamical coordinates. There is, however, an important difference between classical and quantum kinematics. In general, a classical description is exhaustive and exact. In quantum mechanics, a description by a wave function is much less informative. In general, it allows prediction of observational results only loosely, in terms of probability. The degree of precision allowed for the probabilistic prediction is exactly restricted and determined by the Heisenberg uncertainty principle. The quantum mechanical formalism, in terms of wave functions, is by design and in principle, precisely constructed so as to exactly preserve and express this restriction. Consequently, importantly, and remarkably, the dynamical evolution of wave functions described by the Schrödinger equation or its equivalent is deterministic, not probabilistic. This contrast, between probabilistic and deterministic accounts, is the cause of much discussion. Because of the deterministic evolution prescribed by the Schrödinger equation, there is no room to add further variables to the quantum mechanical description by wave functions. In particular, such an addition could not make the probabilistic prediction of observational results any less probabilistic than, as just remarked, the Heisenberg uncertainty principle determines. These facts are the excellent and specific virtue of the kinematical description provided by wave functions.


 * The beam is essential to the reasoning. The beam is necessary as the carrier of the ensemble that is the sample space for the Born transition probabilities. The word beam occurs about 39 times in Dirac's Chapter 1 (fourth edition), which is devoted to physical meaning. We usually don't get to directly see the contents of the beam; they are usually too small to see. We only know how the preparative device was built and how we feed and drive it, and how the observing device was built and how we operate and read it, and how we place the output port of the preparative device up against the input port of the observing device; or how we instruct others to do so; it's not us as individual persons. That is the core and correct part of the Copenhagen interpretation, and I think it is obvious common sense.


 * There are three basic device elements: analyzers, ovens, and detectors (= anti-ovens).


 * An analyzer is for example a calcite crystal with two output channels for one input channel. The crystal analyzes one degree of freedom into its eigenstates. Each output channel lets out instances of a system in a labeling eigenstate characteristic of the crystal's degree of freedom. A lifetime of investigation can go into finding a suitable crystal, Stern-Gerlach magnet, prism, or whatever.


 * An oven is full of some hot vapor of a chemically pure species, with a hole in the side to let out a beam of independent instances of quantum systems, atoms, or molecules (not cats, you can't vaporize a cat and then expect it still to be a cat).


 * An anti-oven (= detector) is a macroscopic chamber full of special stuff in a metastable state that can be triggered to boil, form a droplet, cascade, click, whatever, by the arrival of a suitable system, a particle or somesuch.


 * A filter is an analyzer with all output channels blocked except one. It lets out a beam in one eigenstate of its degree of freedom.


 * The filters/analyzers are passive and reversible, and the oven and anti-oven are irreversible.


 * To prepare a pure state, the preparative device is typically an oven with a small hole in the wall, from which issues a desperately and utterly mixed beam; that is then filtered by a sufficient (maximal) succession of mutually compatible (commutative) analyzers configured as filters (all except one output channel blocked) that eventually supply a beam of independent systems, each an instance of the system in a pure state, denoted by the ket and described by the wave function. This is in a simultaneous eigenstate of all the analyzer/filters. It is defined by a list of degrees of freedom (crystals etc.) and the corresponding list of respective unblocked eigenvalues, one eigenvalue for each degree of freedom. Such a preparative device is fine in theory, but not so easy in practice.


 * The observing device is more or less a mirror image of this. It takes in the beam and passes it through a sufficient (maximal) succession of mutually compatible (commutative) analyzers (just like the preparative analyzer/filters, but with all output channels open and able to pass instances of the system, not blocked as they are for filtering) that eventually supplies sub-beams of independent instances of systems that go to respective anti-ovens = detectors. Any one final detector defines the investigator's chosen degree of freedom and the eigenstate of the instance (specimen) of the system that is detected, denoted by the bra and described by the dual wave function.


 * The state vector (bra or ket) is identified by (1) the generalized configuration space (= list of degrees of freedom) determined by the preparative or observing filter/analyzers, (2) the list of eigenvalues of the respective degrees of freedom, and (3) by the pure substance that is put into the preparative oven.


 * If the preparative filter and observing analyzer are copies of one another, then the pure beam goes into precisely one output channel of the analyzer. That's how we check the apparatus and know we have a pure state.


 * Of more exciting interest is when the observing analyzer is incompatible with the preparative filters. Then the beam is resolved into many sub-beams. In principle they could be re-assembled by being put back through a further copy of the analyzer. The proportions of the beam that go into the various sub-beams are the coefficients in the expansion of the pure state into a superposition according to the analyzer. The choice of analyzer is called the choice of representation. The proportions jointly are are called the representative.


 * The choice of preparative filter determines once and for all which pure state we will deal with, assuming the same pure substance is fed into the preparative oven. The preparative filter determines, for example, that we will deal with system instances with an up spin in the y direction and with a certain z-momentum. The y-spin and z-momentum determine the unique configuration space for that pure state. The up value and the certain value of the z-momentum tell exactly which pure state within that configuration space. The uniqueness is what defines the state as pure.


 * If one desires to produce a pure state more general than the foregoing simultaneous eigenstate, one can, just as above, prepare a beam pure in a simultaneous eigenstate, and pass it to a further device, before passing it to one's chosen observing device. The further device splits the original pure beam, by passing it through some suitably chosen diffracting item. One puts various attenuators and phase delayers into the paths of the several resulting sub-beams, and passes the resulting re-weighted but otherwise undisturbed, and hence still coherent, sub-beams through copies of the diffracting item so as to re-assemble them. This forms a new superposition, not a mixture, because one has carefully not interposed incompatible non-commuting objects that would break the coherence of the sub-beams. The new superposition is no longer in a simultaneous eigenstate of the original preparative device. But it is still in a pure state. This can be verified by passing it through copies of analyzers as indicated above. It has a wave function that belongs to the suitably chosen diffracting item. For example, that diffracting item might be a momentum analyzer; then the resulting pure state is naturally described by a momentum wave function. Moreover, it is in an eigenstate of the operator that indicates the combination of the original preparation device and the suitably chosen diffraction item, the weighting, and the re-assembly device.


 * The investigator can combine identically prepared beams and still have the same pure state with the same definite unique wave function. But if the preparations are not identical, the combined beam is mixed, and is not described by a single wave function. The slightest departure from identical preparation creates a mixture. A combination of beams from preparative devices with different filters or different substances in the ovens produces a mixture, not a superposition, while a combination from preparative devices with the same filters and substances produces a pure beam.


 * The choice of representation assumes in advance that the pure state has been already defined and fixed by choice of preparative device. A different choice of preparative devices produces a different pure state with its respectively different wave function. There is no tampering with the original beam. If it is tampered with, it becomes a different beam with a different wave function, or a mixture.


 * A wave function is determined by a choice of a way of observation, which is called a representation. Any choice of representation is permitted, within reason. This is why there are many wave functions for each quantum state. The process of representation is a matter of splitting = analyzing the original pure state into an array of sub-beams. If the sub-beams are left intact (no detector) so they can be re-assembled, they remain in the original pure state. If they are treated by external agents, such as detectors, the splitting is reconsidered and re-described, as a new preparative filter, and the configuration space is new, and they are no longer in the original pure state. They are now in new pure states, with respect to the new filtration. With due perversity, this is transmogrified into a crazy story called "collapse". It means that the instance of the system is changed by passage through the filters, heaven forbid us to say so, by a randomly determined quantum transfer of translational momentum according to Duane's principle. That momentum transfer can be reversed by a copy of the filter that produced it, to re-assemble the original beam if there is no external tampering, such as by a detector; the re-assembly reminds us that we are thinking about superposition when we forbid tampering. It is strictly forbidden for hard-line Niels Bohr Copenhagenists to mention or think about Duane's principle, because it is demystificatory, and makes physical sense, and does away with teasing puzzles about "collapse" and is therefore counter-revolutionary, revanchist, revisionist, backsliding, and generally frowned upon. But Heisenberg accepted it and put it in his 1930 textbook. (For some odd reason I haven't worked out, Born was very hard on Landé about this, and of course Shimony jumped on that bandwagon.)


 * The foregoing is, I think, childishly simple and obvious. But I think there are those who will think it wrong or beneath the dignity of Wikipedia.Chjoaygame (talk) 12:38, 3 February 2016 (UTC)