Talk:Mathematical formulation of quantum mechanics/Archive 1

Untitled
The first sentence in this article as just revised is very abrupt! How 'bout a bit of context-setting first? Michael Hardy 00:16, 16 November 2003 (UTC)

I've put back in some basic information (postulates) cut out in the November 2003 rewrite. It makes somewhat better sense now - still quite ruough.

Charles Matthews 11:55, 14 May 2004 (UTC)

By tradition observables are associated to self-adjoint operators? I think it is more than tradition. CSTAR 03:09, 8 July 2004 (UTC)

Also I think it is misleading to include the composite system -- tensor product correspondence in the Schrodinger picture. It doesn't belong there and particularly, the spaces of multi-particle systems is more complex than is suggested in the article (Boso statistics, Fermion statistics, etc). I would suggest remove that item. CSTAR 03:33, 8 July 2004 (UTC)

Impenetrable phrasing
states are points in phase space, observables are real-valued functions on phase space and the dynamics 'is given by a one-parameter group of transformations of the phase space. '  This seems a bit obtuse. What is the parameter? What are the transformations? Shouldn't there be a link to group? Mr. Jones

A sentence seems out-of-place
The sentence "We cannot assume that the operator is defined on the whole of H: the Hellinger-Toeplitz theorem) that an operator is continuous, then it is a bounded linear map from H to H." seems to be a bit out-of-place. I think it should read "We cannot assume that the operator is defined on the whole of H: the Hellinger-Toeplitz theorem states that a self-adjoint operator defined everywhere on H must be continuous, or equivalently it should be a bounded linear map."

Other formulations
Thr backward reference to 'postulate 4.2' has been lost, after some edit.

Charles Matthews 15:10, 12 July 2004 (UTC)

New intro: Schrödinger's wave mechanics
The following paragraph taken from the current intro makes a radical assertion about the History of QM:


 * Schrödinger's wave mechanics originally did not represent a radical departure from the mathematical framework of classical mechanics. His wave function can be seen to be strongly related to the classical Hamilton-Jacobi equation, and Schrödinger himself initially did not understand the fundamental probabilistic nature of quantum mechanics, as he thought the fact that the (squared) wavefunction of an electron must be interpreted as the charge density of an extended object. It was Max Born who introduced the correct probabilistic interpretation of the (squared) wave function as the probebility that a pointlike object.

Geesh, is the historical claim of the above graf true? Agreed, the relation between the asymptotics of the Schrödinger equation and the Hamilton-Jacobi eqn has been known since the 1930's and Maslov-Hormander's theory of propagation of singularities of the 1970' formalizes the relation between the two via Fourier integral operators. Therefore, I don't object to the validity of the sentence


 * His wave function can be seen to be strongly related to the classical Hamilton-Jacobi equation


 * I was not referring to this stuff at all. I was referring to the fact that if you write the wavefunction as $$A\exp(iS/\hbar)$$ then S satisfies the Hamilton-Jacobi equation up to a term proportional to $$\hbar$$.
 * Schrödinger must have had a reason to represent momentum as a derivative, though, I wonder what it originally was (modern ways to argue that that's the right thing to do are a dime a dozen)
 * Darn you, CSTAR, now you've piqued my interest and I won't be able to rest until I read Schrödinger's original paper, or die trying.
 * &mdash; Miguel 06:14, 19 November 2004 (UTC)

However, as a matter of historical development I think it would be desirable to give a specific reference to Schrödinger's work to justify the historical claim.

Also doesn't the paragraph terminate prematurely


 * interpretation of the (squared) wave function as the probebility that a pointlike object.

CSTAR 05:44, 19 November 2004 (UTC)


 * Ok, ok, the beginning of the article claims that the old quantum mechanics did not depart from the toolds and concepts of classical mechanics. then there is 1) Schrödinger's equation is a PDE; and 2) Schrödinger interpreted the wavefunction as a classical charge density (conservative idea) and did not understand the probabilistic interpretation (radical idea) until Born pointed it out. This is how i would justify the claim that Schrödinger was not radically departing from what was then known. As for going to the original paper, I have heard it repeatedly that it is basically unreadable, but it might be worth a try.


 * That part I believe; I'm just not sure about whether Schrödinger derived the Schrödinger eqn by finding a PDE with the right high-frequency asymptotics. Maybe he did.  I just don't know.  But somebody here should try to determine this for sure.CSTAR 06:02, 19 November 2004 (UTC)


 * it is definitely worth investigating what exactly Schrödinger was smoking when he wrote his paper ;-) I have also introduced the relationship to the Hamilton-Jacobi equation "in hindsight"
 * anyway, these are good calls, good thing you're paying attention. &mdash; Miguel 06:07, 19 November 2004 (UTC)


 * I have toned down the historical interpretation, though, by calling De Broglie radical and stating that Schrödinger successfully formulated wave-particle duality mathematically.


 * &mdash;Miguel 05:59, 19 November 2004 (UTC)

Electron as extended object
The intro also refers to Schrödinger's picture of an electron as an extended object. This I think is historically right, however more explanation is needed on what that means, for instance a phrase such as the electron as an object smeared out over an extended, possibly infinite, volume of space CSTAR 15:14, 19 November 2004 (UTC)

Article introduction
I'm still not satisfied with the introduction of the article, before the table-of-contents. Some additional explanation: The nature of that departure should be better explained in the introduction.CSTAR 00:36, 23 November 2004 (UTC)
 * Mature physical theories are formulated in mathematical or quantitative language.
 * Quantum mechanics represents a departure from the languages used in previous physical theories.

I have added some more. Charles Matthews 07:01, 23 November 2004 (UTC)

The introduction, before the ToC (not to be confused with the WoT) should be able to stand on its own. Why split off a section on old-quantum theory, when there is already an article on it? 17:04, 23 November 2004 (UTC)


 * Oops and I forgot to sign the comment above ... or something.  CSTAR 17:50, 23 November 2004 (UTC)


 * I certainly don't agree with discussion of the photoelectric effect here; it's not quantum mechanics, in the sense that the mechanics only came around 1925. So I think I agree with the comment above (?). By all means write something about the 'old' theory in another article. But the continuities there seem to me (not an expert) to be in the physics, while in the mathematical formulation there was more like a break. Charles Matthews 17:44, 23 November 2004 (UTC)

The problem is that CSTAR wants to be very specific about the state of physics prior to the break, which forces you to discuss a bunch of stuff irrelevant to the mathematics of the new quantum theory.

There is no article on the old quantum theory in the form that it needs to be in, including Plank's quanta, Einstein's photons, Bohr's atom, and the Bohr-Sommerfeld and Sommerfeld-Wilson-Ishiwara quantizations. The article old quantum theory currently redirects to Bohr model which is taking the part for the whole. The Bohr model was the first phenomenological model of the atom, but later Bohr himself (and others) developed a more precise mathematical formulation based on allowing only closed orbits in phase space enclosing an area equal to an integer multiple of planck's constant. That is as far as classical mechanics could take you, and is the right point of comparison with Schroedinger's wave mechanics. So we are not at the point where you can write the introduction in the form you want, CSTAR.

I can do one of two things: either expand the section here and then refactor it into a new article leaving a summary, or go off and write the article on the old quantum mechanics first, and then come back and summarize it here.

What I want is not a full discussion of the photoelectric effect, but maybe a few words about where Planck's constant came from and how Einstein used it to invent photons. You can then point out that neither Planck nor Einstein took wave-particle duality seriously, and it took a physicist of the younger generation (de Broglie) to do that.

&mdash; Miguel 18:46, 23 November 2004 (UTC)


 * The introduction now is certainly a lot better. Your changes address my concerns.  Thanks. CSTAR 19:05, 23 November 2004 (UTC)

It was I who made old quantum theory a redirect. It obviously can be an article in its own right, perhaps under a more 'professional' name. Charles Matthews 20:29, 23 November 2004 (UTC)


 * the name old quantum theory is what you find in physics textbooks. There can be more 'professionally-named' articles about the subtopics, such as Bohr model and Bohr-Sommerfeld quantization. &mdash; Miguel 20:41, 23 November 2004 (UTC)

Epistemology and separability
The assertion about the relation between separability of H and sufficiency of countably many observations is interesting, but may require more elucidiation. However, how is it related to epistemology? I assume one could argue that there are sequences of observations Ai which ultimately distinguish a pair of states. The problem with such a claim is that it will be inevitably subject to much critical examination.CSTAR 20:38, 24 November 2004 (UTC)

The relationship to spistemology is that we can only have access to a finite amount of information, hence a finite amount of experiments. A countable number is necessary to reason mathematically about finitely many experiments. That is exactly the point of separability in the theory of stochastic processes, except that nobody introduces it in that way so it sounds like a completely unmotivated assumption made for mathematical convenience.

You can remove the mention of epistemology if you think it will lead to too much discussion (on second thought, I agree it will), but leave there the statement about countability. I don't know in how much more detail we can get in this article before having to spin off other pages. &mdash; Miguel 21:00, 24 November 2004 (UTC)

Ab-initio
Whazzat? Do you mean derivation from first principles? This is an expression so often used by physicists (and other scientists) so maybe it should have a separate article? How is this different from derivation from axioms? Note that I put in a wikilink to phenomenology (science) to explain its (correct) use in this article. A similar link is needed here, I think.CSTAR 16:12, 25 November 2004 (UTC)


 * While we're at it, I want a separate phenomenology (physics) article. There is much to be said about the scope, goals and level of success of phenomenology in different branches of physics. High-energy physics phenomenology is broad, deep and successful enough to be an independent branch of physics in its own right. &mdash; Miguel 18:38, 25 November 2004 (UTC)


 * Whoops. I reistated the old link.  Sorry, I though you made a mistake. Fui yo que me equivoqué. I will revert.18:51, 25 November 2004 (UTC)

There are about 200,000 google hits for ab-initio calculation. It does mean "from first principles", but it is, in fact, customary in atomic and molecular and solid-state physics. &mdash; Miguel 16:34, 25 November 2004 (UTC)


 * I don't dispute that it's use is customary in certain areas of physics. But for example the difference between a derivation from first principles and a derivation within an axiomatic system requires some explanation &mdash; it's certainly not obvious to me. These are important cultural nuances (for example, differences between the culture of physicists and mathematicians) which an encyclopedia has to deal with and explain to be useful as a mirror of not only knowledge, but culture.CSTAR 16:55, 25 November 2004 (UTC)


 * Can you explain what you have in mind about the difference between from first principles and within an axiomatic system? I grok not.


 * Physics is not axiomatic, so people such as von Neumann talk about postulates to weasel out of that fact.
 * A calculation from first principles or ab initio seems to mean with "no" experimental input.


 * I anxiously await your elucidacion of this philosophical tangle.
 * &mdash; Miguel 17:13, 25 November 2004 (UTC)


 * This is a very tentative response


 * Physics is not axiomatic. Agreed, but there is such a thing as axiomatic physics, which consists of experimentally testable assertions as axioms in a formal theory and deriving everything else purely by rigorous mathematics (in principle formalizable within a formal logical system).


 * A derivation from first principles makes reasonable guesses about what should be true in nature. For example the relative sizes of things (where a mathematically rigorous calculation would be purely asymptotic), or excluding certain cases as being physically unreasonable.


 * Now this distinction does require more elaboration, but my purpose is to provide a plausible argument as to why these two concepts are different.CSTAR 17:33, 25 November 2004 (UTC)

On a different matter:


 * I put in a link to ab initio which states


 * in sciences (especially physics and chemistry): from first principles. A calculation is said to be "ab initio" (or "from first principles"") if it only assumes basic and established laws and does not assume the validity of further assumptions such as models.


 * That's OK except further assumptions such as models should be changed to further assumptions includeing additional models.CSTAR 17:33, 25 November 2004 (UTC)


 * I added the only experimental input is the values of fundamental constants. &mdash; Miguel 18:02, 25 November 2004 (UTC)

Measurement
What happened to the caveat that the operator was assumed to have pure point spectrum? What about measurement for operators (such as position or momentum for a free particle) with continuous spectrum? CSTAR 04:14, 26 November 2004 (UTC)

Measurements of position or momentum result not in point values but in ranges. The collapse is represented by the orthogonal projector associated to the observed range by the spectral theorem. (This is the interpretation of the projector-valued measure in the spectral theorem)

However, if we can't write down the measurement postulate without assuming a discrete spectrum maybe we should think of a different way to discuss measurement. Something like this: in the Heisenberg picture, if the system is in state $$\left|\psi\right\rangle$$ and observables $$A_i$$ are measured at times $$t_i<t_{i+1}$$, then we consider the quantity


 * $$\left\langle\psi\mid A_n(t_n)\cdots A_1(t_1)\mid\psi\right\rangle$$

where the $$A_n(t_n)$$ are Heisenberg operators.

I'm not sure, though. Von Neumann's postulate has always seemed suspect to me. Moreover, I think a discussion of quantum measurement belongs in interpretation of quantum mechanics rather than here. We already have a complete description of the mathematical formalism of quantum mechanics. I was wondering what ever happened to the canonical commutation relations and the usual Schrödinger equation, but IMHO those topics belong in quantization or relation between classical and quantum mechanics, or classical limit of quantum mechanics.

I would personally remove the measurement discussion from this article, but I hesitate to do it because that's a very radical step to take. But it would save us a lot of mathematical and philosophical heasaches. In any case, quantum measurement is an active research area, not a postulate of quantum theory (even if von Neumann thought he could wrap it all up neatly in his book). &mdash; Miguel 05:23, 26 November 2004 (UTC)

Actually, there is already an article on measurement in quantum mechanics. I think we should remove all detailed discussion of measurement from this article and just link to that. Sooner or later I'll have to go and pitch in on quantum measurement. The current article is very limited and does not really go beyond von Neumann. That is neglecting 70 years of development of the theory. &mdash;Miguel 05:34, 26 November 2004 (UTC)

By the way, after collpse one would have to rescale the "collapsed" state vector to unit length, which makes the mathematical description of collapse all the more awkward, as it is now orthogonal projection followed by rescaling to unit norm. Can we not allow non-normalized states and density matrices and divide by $$\left\langle\psi\mid\psi\right\rangle$$ or $$\operatorname{tr}\rho$$ in the expected value of an observable? Also, we should maybe be more explicit and say that states are really equivalence classes of vectors in H, even though physicists cringe at the mention of equivalence classes. &mdash; Miguel 05:53, 26 November 2004 (UTC) -

Well the article on quantum operation is a more comprehensive discussion of general measurement (not just projective measurements). There is an intermediate discsuuion in quantum logic and quantum statistical mechanics. The quantum operation operation is completely equivalent to the relative state approach.CSTAR 06:02, 26 November 2004 (UTC)


 * Actually, there is already an article on measurement in quantum mechanics


 * It's not very good.CSTAR 06:05, 26 November 2004 (UTC)


 * I totally agree. Let's finish this one and then move to that one ;-) &mdash; Miguel 06:12, 26 November 2004 (UTC)

-


 * I would personally remove the measurement discussion from this article, but I hesitate to do it because that's a very radical step to take.


 * Yes, it would be very radical and no, I don't think it should be removed. It could remain as is, with enough pointers to other articles.CSTAR 06:17, 26 November 2004 (UTC)

Where the article is going
I should really just let you guys write what you're going to write.


 * This sounds like there is some criticism (besides the issue of formalism) that you decided to keep to yourself on second thought. I'd still like to know what you had in mind. &mdash; Miguel 18:46, 26 November 2004 (UTC)


 * Not really. If this is going to be a long article, that's OK. It seems a little strange to have the major discussion of the physics of 1925-1930 in an article about its mathematics. I'm not familiar with the WP articles on Dirac and so on; so I was just going to wait a little. Charles Matthews 19:38, 26 November 2004 (UTC)


 * I have begun to worry about the length of the article, too, and I wouldn't mind moving most of the references to physics and mathematics before 1925 to a different article. I wouldn't call what is in this article a major discussion of the physics, though. In any case, to me the physics and the mathematics of quantum mechanics are inseparable: you cannot really understand one without the other (well, you can understand the mathematics on its own, but then it's meaningless). I also suspect that most readers of this page would expect a discussione of the physical interpretation.
 * I personally like to discuss the history of a subject first, not last, but would you prefer to see the mathematical structure section be the first after the TOC?
 * If I had to strip this article to its bare bones, I would leave it at the first paragraph of the introduction and the mathematical structure section, moving eveyrthing else to other article and leaving behind very short pointers. In that case I would rename this article "Hilbert-space formulation of quantum mechanics", because the C^*-algebra formulation needs an article of its own.
 * &mdash;Miguel 20:06, 26 November 2004 (UTC)
 * My personal approach is quite historicist (within mathematics). So it doesn't worry me, really. In the end the site should contain these 'stories'. Charles Matthews


 * The length of the article doesn't concern me at all. And I don't like the idea of changing the pre ToC material by addition or deletion.  That section looks good to me.


 * The C*-algebra approach really does require a more extensive article (explaining some of the concepts, particularly how it relates to Jordan's view of QM.) Certianly if Quantum logic has its own article then the C*-algebra approach to QM should also have its own article.


 * I also like the historicist perspective. Unfortunately, that's very hard to get right.CSTAR 20:35, 26 November 2004 (UTC)

I have expanded 1.3 Later developments, and would like to use that as a replacement for 3 Other formulations and 4 The problem of measurement. That is, can we agree to remove sections 3 and 4 or merge their content with appropriate other WP articles? &mdash; Miguel 01:20, 27 November 2004 (UTC)

"Formalism"
I did want to make one point: I object to formalism in relation to mathematics. It obscures a distinction: it can mean a notation, or I suppose it can mean the result of a formal development of a topic. There is a big problem in relation to mathematics/physics (I think) in that 'formal meaning' means opposite things depending on who you are. 'Mathematical formulation' implies to me that bra-ket notation is not just a formalism, but something with a very definite semantic content. Dirac's delta function was a formally-introduced symbol; then we had what you could call 'mathematical formulation of generalised functions' and δ(x) is a Schwartz distribution with a definite semantics. I realise it depends which side of the fence you stand, whether this is an issue or not. Given the article's title I feel that 'formalism' could be banished.

Charles Matthews 09:32, 26 November 2004 (UTC)


 * The article formalism does not address this, but it would help me if I could read an article about it. &mdash;Miguel 16:00, 26 November 2004 (UTC)


 * I put in a bullet in formalism. This probably should be expanded in an article such as scientific formalism.CSTAR 18:03, 26 November 2004 (UTC)


 * Point taken. I think we should address this concern.  I tend to melange the two, but I am aware of the distinction and am also picky about it when I notice.CSTAR 14:20, 26 November 2004 (UTC)

Bargmann Segal
I thought one of the points of the Bargmann's paper was to show that the (symmetrized) Fock Representation was unitarily equivalent to the Bargmann Segal representation. Am I confused about something? CSTAR 04:13, 27 November 2004 (UTC)


 * OK I guess one shouldn't interpret your statement as asserting that these representations are inequivalent. I misread it.CSTAR 04:38, 27 November 2004 (UTC)


 * Precisely. &mdash; Miguel 04:56, 27 November 2004 (UTC)


 * The article now looks pretty good.


 * I really want to yank the measurement and other formulations sections. I think the first is actually a physical, not mathematical, problem, and that the second is superseded by the later developments section. &mdash;Miguel 06:50, 28 November 2004 (UTC)


 * I might start an article on Bargmann Segal. It fits naturally with the article Stone-von Neumann theorem; unfortunately, the notation I used in my notes on Bargmann Segal aren't quite consistent with the notation used in the Stone-von Neumann article (even though I wrote most of that one). I may have to do a little work to make them consistent.  Geometric quantization (Weyl-Maslov-Hormander-Voros-Guilleman-Sternberg Quantization) might be another useful article although I have nothing written on those.CSTAR 06:22, 28 November 2004 (UTC)

C*-algebraic formulation

 * I think I'm going to concentrate on the C^*-algebraic quantization now. &mdash;Miguel 06:50, 28 November 2004 (UTC)


 * Ah. That sounds more interesting, although I'm not sure where one should begin. Jordan Algebras? Or part 6 of Boguliubov, Axiomatic Quantum Fied Theory?CSTAR 06:58, 28 November 2004 (UTC)


 * The weyl relations and the GNS construction?
 * Anyway, you seem to know more than I do about this stuff. &mdash; Miguel 15:21, 28 November 2004 (UTC)
 * I do NOT particularly wish to see that developed in this article, though, but in a separate one. Given the large number of formulations listed under later developments, singling out quantum logic and C^* algebras seems wrong. Moreover, people would expect to find the Schroedinger/Heisenber/Dirac/von Neumann formulation under the current title. &mdash; Miguel 20:31, 2004 November 28 (UTC)
 * I have no objection. Go ahead and delete those subsections. This is legacy from previous versions.CSTAR 20:44, 28 November 2004 (UTC)


 * Well I was bold and deleted them.CSTAR 22:04, 28 November 2004 (UTC)

Of course there is the wikipedia article on GNS construction.


 * Yes, but that is not C-star algebraic quantization. &mdash; Miguel 20:31, 2004 November 28 (UTC)


 * Whatever you do, call it C*-algebraic quantization. C*-algebra looks weird and I have been consistently using  C*-algebra.CSTAR 20:44, 28 November 2004 (UTC)

It would be nice to start a C*-algebraic quantization article with a comphrehensible account of Heisenberg's matrix mechanics or at least refer to such an account. In the past, when I tried reading these accounts in various places, my eyes glaze over. These accounts also are supposed to reflect the Zeitgeist of operationalism of which Heisenberg was apparently a follower.

There is also a brief account of matrix mechanics in Alain Connes' book, Noncommutative Geometry (pp 33-39) which I have looked at but there too, my eyes glaze. Anyway Connes tries to bring in groupoids, but of course these weren't around for Heisenberg to play around with. CSTAR 17:31, 28 November 2004 (UTC)

Old quantum mechanics
It would be nice to find out how in the world Heisenberg stumbled upon matrix mechanics. I just don't see how that would be likely. I can understand how Schrödinger came up with wave mechanics. That's a pretty natural consequence of de Broglie's ideas, but how did Heisenberg come up with matrix mechanics? Phys 05:37, 29 November 2004 (UTC)

Another picture?
I noticed this in the article


 * closely related to the Schrödinger picture where time is not a parameter but an operator. Here, time, t and energy (which is not the Hamiltonian, which is a function of position and momenta) are canonically conjugate operators which commute with position and momenta. Here, we look at the rigged Hilbert space and insist that the valid states are those which satisfy (H-E)|?>=0. Physical observables are operators which commute with (H-E).

O.K. I'm stumped. Notice that in this "new" picture, the operator E is by the uniqueness of the CCRs unitarily equivalent to a multiple of 1/i d/dt. In particilar it has spectrum R.


 * You could make E be a bounded operator, or bounded below, and make T ~ id/dE. That would solve your problem. &mdash; Miguel 20:13, 29 November 2004 (UTC)


 * My confusion comes from the remark


 * Here, time, t and energy (which is not the Hamiltonian, which is a function of position and momenta) are canonically conjugate operator


 * If time T and energy E are canonically conjugate operators &mdash; which I take to mean [T,E] = r I, that forces E to have a spectrum R.


 * I suppose in operator-theoretic terms, one could take a compression of E, T to some appropriate subspace H and get a physically realizable (e.g. non-negative) energy operator projH E projH on H. CSTAR 20:34, 29 November 2004 (UTC)


 * How about x and p for a particle in a box? Isn't x bounded, p unbounded and [x,p]=i hbar? &mdash; Miguel 20:47, 29 November 2004 (UTC)


 * It depends on what you mean by [x,p]=i hbar. The eqn


 * $$ [x,p] \phi =i \hbar \phi $$


 * holds for functions &phi; which are zero near the boundary of the box. However, the operator x is not essentially self-adjoint on that domain (it has many self-adjoint extensions depending on the bdary conditions). So the commutation relations cannot be integrated to get them in Weyl form for which uniqueness holds.


 * I am aware of the essential self-adjointness issue, but you can fix that with (for instance) periodic boundary conditions. I do not know what a compression is in a technical sense. &mdash; Miguel 21:04, 29 November 2004 (UTC)


 * No, you can't! Then you have functions on a circle and x is not really well-defined even as a classical variable (it is defined modulo 2pi) &mdash; Miguel 21:06, 2004 November 29 (UTC)


 * Well yes it does have many self-adjoint extensions (periodic, Dirichlet,Neumann, Robin etc). But for any of these self-adjoint extensions the eqn


 * Correction. Yes it does have many self-adjoint extensions, corresponding to the fact that the space of square-intregrable distributional solutions of


 * $$ \frac{d}{dx} u = i \pm u $$


 * form a 1-dimensional space. To talk about Dirichlet,Neumann bdary conditions for a first order operator doesn't make too much sense.


 * $$ [x,p] \phi =i \hbar \phi $$


 * no longer holds in any reasonable way, since for all them the extensions have pure point spectrum. Compression of an operator T to a subspace H just \ means projH T projH .CSTAR 21:13, 29 November 2004 (UTC)


 * I guess in that sense, the compression to the kernel of H-E leads to E being bounded below. The quantization of systems with constraints is a thorny issue, though. &mdash; Miguel 21:19, 29 November 2004 (UTC)


 * In that case, what I said in the previous paragraph is true: one could take a compression (in some sense) of x and p on L2(R) to L2(TheBox).


 * Ooops, I meant the operator p. x is essentially self adjoint there.CSTAR 21:03, 29 November 2004 (UTC)


 * Hmmm, you can formulate classical nonrelativistic mechanics by adding t and E as variables satisfying CCRs and another unphysical parameter s. Classically, s parameterizes the trajectory of the system in x-t space, and the equation H(p,q)=E becomes a constraint (in the sense of Dirac) whose Poisson brackets generates changes in the parameter s.
 * The Schroedinger representation of that is no different from the ordinary schroedinger representation of the CCR, it is just applied to a larger phase space with coordinates (q,p,t,E) instead of (q,p).
 * That is the problem: the Schroedinger representation of the CCR is part of the problem of quantization and has nothing to do with pictures of dynamics or the overarching mathematical framework of quantum mechanics.
 * &mdash; Miguel 20:13, 29 November 2004 (UTC)

Also the set of ψ satisfying (H-E)|ψ>=0 is not a vector space, or a direct sum of vector spaces so this is an odd kind of condition. Of course anything is possible. CSTAR 15:01, 29 November 2004 (UTC)


 * SOrry for the triplicate. My crippleware version of Opera really sucks.CSTAR 15:03, 29 November 2004 (UTC)

Whatever the merits
I don't think this particular graf on E, t belongs here in thiis article, maybe in a separate article about issues in canonical quantization.CSTAR 21:36, 29 November 2004 (UTC)

I agree. It is not wrong, but I have argued above what that is not really part of the overarching mathematical framework. Similarly, I do think that measurement is also not part of the mathematical framework. Von Neumann thought he could dispatch it with a postulate, but 70 years of research into quantum optics, hidden variables, decoherent histories, quantum cosmology and quantum information theory have shown him wrong. &mdash; Miguel 06:14, 30 November 2004 (UTC)

ready to move on
I am pretty happy with the article at this point. I think I am going to move on to quantization or something like that. &mdash; Miguel 07:05, 30 November 2004 (UTC)

Relative state interpretation
It's really equivalent to the mathematical machinery of quantum operations (via the Stinespring-Choi-Kraus represntation theorm).CSTAR 15:47, 30 November 2004 (UTC)

Symmetries
This is still an introductory article. Given that, why does it include the following sentence in the symmetries section


 * Gauge symmetries are not exactly and are dealt with in a more complicated manner using BRST.


 * This is actually about field theory - yank it from the article, or put it in the "later developments" section. &mdash; Miguel 21:07, 11 December 2004 (UTC)

Are gauge symmetries explained anywhere in the article? CSTAR 05:16, 11 December 2004 (UTC)

Decomposition of reps of symmetries
What's all this stuff about irreducibles? One is interestd in the

primary decomposition = factor decomposition = central decomposition = decomposition into multiples of irreducible representations (in type I case)

not so much of the group of symmetries as of the group of symmetries + algebra of local observables which it normalizes.

CSTAR 03:19, 27 December 2004 (UTC)

Schrodinger equation
Shouldn't the equation be a partial differential equation?

Masud 01:56, 1 December 2005 (UTC)


 * I suppose, but by the time one works at this level, the reader is assumed to know that everything is a partial differential equation, and so informally, little distinction is made. linas 17:23, 24 December 2005 (UTC)


 * The distinction is still important. Masud 04:13, 19 February 2006 (UTC)


 * Sure, but no one makes it. linas 07:04, 19 February 2006 (UTC)


 * Sigh, I just looked at this article, and your edits, and I believe they are incorrect. In this context, the derivative w.r.t. time is *not* a partial derivative, as there is only one variable (time); there are no other variables; in particular, kets do not have a space dependence, so it is an ordinary diffeq not a partial diffeq, since there are no other variables at work here. There are half-a-dozen places in the article where this confusion is made. Sigh. :-( The college-level articles in WP tend to be filled with mistakes because the $#%^& college students keep making them. Oh well. WP needs a page protection and review system to lock out this kind of stuff. linas 07:14, 19 February 2006 (UTC)


 * Linas is right. There is only one variable. The notation that was used in the article was completely correct and standard; Please fix this. --CSTAR 07:37, 19 February 2006 (UTC)


 * But surely the ket must have some other dependence than time -- otherwise the momentum operator would have only zero eigenvalue(s). Masud 15:10, 25 February 2006 (UTC)


 * The ket is generally some element of an abstract Hilbert space of finite or infinite dimensions. It doesn't have to be a Hilbert space of functions on R3. It makes in general no sense to say the ket depends on anything. In this section we are considering how kets vary in time. The derivative used is that of a function with values in a Hilbert space. --CSTAR 16:11, 25 February 2006 (UTC)

Maybe I should soften my stance; although not for the reason that Masud gives (CSTAR is right, the kets to not depend on a position coordinate). Sometimes, in problems of physics, one wants to consider a family of Hamiltonians, with some "free parameter" (lets call it &theta;): e.g. the strength of the magnetic field, which is slowly, adiabatically varied. In this case, the kets in Hilbert space may be labelled with &theta;, and derivatives of things may be taken with respect to &theta;. If one imagines that the parameter &theta; is also time-varying, then there is potential for confusion. To avoid this confusion, some textbooks use the partial derivative for time. The point is that the explicitly time varying part of the wave-function is given by Schroedinger's equation, and nothing more, and any (adiabatic) time variation of any parameters must be treated independently, as its unrelated. This is a somewhat subtle point, but I believe it explains why partial derivatives are sometimes seen in this context. Perhaps a paragraph explaining this needs to be added to the article. (BTW, the article to link is then to Berry phase, which is how integrals of adiabatically varying parameters (in Hilbert space or other spaces) are treated.)linas 17:01, 26 February 2006 (UTC)


 * Thanks CSTAR, point taken. I'm fully aware that the dimensionality of the Hilbert space doesn't have to be related to the dimensionality of space. So the Hamiltonian definitely generates infinitesimal translations in time (of the ket), but we can't say the same for momentum operator since the ket may not depend on space. Is this correct? Masud 23:00, 27 February 2006 (UTC)


 * Yes, but in non-relativitistic quanutm mechanics, time is treated very very differently than space. In quantum mechanics, a ket $$|\psi>$$ is an element of an infinite dimensional vector space, the Hilbert space. It does not depend on position. Do not confuse the ket with the wave function, which does have a "position dependence". Here, the wave function $$=\psi(x)$$ is just the component of the vector $$|\psi>$$ in the $$|x>$$ direction. The ket $$|\psi>$$ does not have a dependence on x.


 * The momentum operator P is an operator in this infinite dimensional vector space. The expectation value of the momentum of a given state is $$<\psi|P|\psi>$$. Note that usually one may write


 * $$<\psi|P|\psi>=\int dx \int dy <\psi |y>$$


 * In this vector basis, the momentum operator P has the following matrix elements:


 * $$=-i \hbar \delta(x-y)\frac{\partial}{\partial x}$$


 * where &delta; is the Dirac delta function. So although the momentum operator can be thought of as the generator of infinitessimal displacements in space, this does not imply that the ket "depends on a spatial coordinate". The infinite-dimensional Hilbert space "encodes" the properties of 3D space in a curious way. The Schrodinger equation is about the time evolution of vectors in this inf. dimensional space (which just happens to have, "by accident", as it were, a 3D representation in which the momentum operator is diagonal). Perhaps this clarifies things. linas 00:58, 28 February 2006 (UTC)

Perhaps someone can add a short note in the article about commutators, since commutativity(or representations thereof) is not straightforward for unbounded operators. —This unsigned comment was added by mct_mht (talk • contribs).

Mixing history with theory
Please try not to mix history with theory, when someone is reading theory, he doesn't need to read names of scientists and dates interspersed in between, it would be much better to have a seperate section for all historical events, who did what and when etc... Thanks —Preceding unsigned comment added by 41.235.85.72 (talk) 23:38, 17 May 2008 (UTC)

bra-ket notation
bra-ket notation (a convention for brackets, denoted "", later to be re-used in http's html)
 * I refer specifically to the HTML element "A" form
 *  Text displayed for the HTTP link
 * which clearly shows the provenance respected by Tim Berners-Lee, who was at CERN where the bra-ket notation is well-known and very complicated QM expressions involving multiple items follow this form. --Ancheta Wis 16:51, 29 December 2005 (UTC)


 * I am sorry, but I find this justification extremely weak. Anyway, the proper place to mention this connection in my opinion is bra-ket notation, but if you do not have more direct proof, I'd rather that it isn't mentioned at all. -- Jitse Niesen (talk) 15:04, 3 January 2006 (UTC)

Measurement
The section on measurement needs to be modified, in terms of projection valued measures(PVM's) and/or positive operator valued measures(POVM's). No point in the whole particle being mathematically correct except one section. —This unsigned comment was added by mct_mht (talk • contribs).

Sign unsigned
How does one go back and signed unsigned comments? Thanks. Mct mht 08:23, 3 April 2006 (UTC)
 * Use the substituted Template:unsigned. That is, where you want to put a signature, insert username . -lethe talk [ +] 13:25, 3 April 2006 (UTC)

English?
Can anyone translate this for me? I think it is a little over the head of the most of the world.

71.71.124.75 01:12, 18 September 2006 (UTC)


 * Why are you trying to read "Mathematical formulation of quantum mechanics" rather than "Introduction to quantum mechanics"? —Keenan Pepper 19:08, 20 September 2006 (UTC)

tone
This article, at least as far as I got into it, had been written with a tone wholly inappropriate for wikipedia. A lead is supposed to summarize the information of the article, not expound on how remarkable the stuff it's about to talk about is. It badly needs a rewrite to bring it into a style free of overwrought prose that hides the facts. Night Gyr (talk/Oy) 08:38, 17 October 2006 (UTC)


 * seems pretty good to me. the intro gives a overview of the history and does give a summary. how exactly does it "expound on how remarkable the stuff it's about to talk about is" and "hides the facts"? Mct mht 09:28, 17 October 2006 (UTC)

The facts in this case would be "what is the mathematical formulation of quantum mechanics?,"


 * the first coupla sentence alludes to it. the subject matter requires laying out some background. how would you write it? Mct mht 09:56, 17 October 2006 (UTC)
 * Any necessary background would be technical; I don't think the history is necessary to say what comes in the final sentence of the first paragraph. Isn't there a second half to the sentence "The mathematical formulation of quantum mechanics is..." or if there's more than one a slight variation? Night Gyr (talk/Oy) 10:02, 17 October 2006 (UTC)


 * one can fill a volume completing that sentence (see references). a short summary would be along the lines of "Mathematical structure of quantum mechanics" section in article. seems to me that starting the article saying "The mathematical formulation of quantum mechanics is that the state space of a quantum system is a subspace of a Hilbert space, and observables are..." is not advisable. Mct mht 17:54, 17 October 2006 (UTC)

and the article seems kind of slow in getting to a clear answer on that. I took out the really flowery language ("one of the most remarkable things") when I did my edit, but it still as a ton of unnecessary or inappropriate wordiness that could be dropped to provide a more concise and just as readable piece, like "in brief," "clearly," "so-called" etc.

Skimming the rest of the article turns up awkward, disjointed sentences that could be combined to improve the flow, like "A related topic is the relationship to classical mechanics. Any new physical theory is supposed to reduce to successful old theories in some approximation. For quantum mechanics, this translates into the need to study the so-called classical limit of quantum mechanics."

The article just feels like it was written as an essay, not as an encyclopedic reference, but I couldn't find the "This article reads like an essay" tag. Night Gyr (talk/Oy) 09:46, 17 October 2006 (UTC)


 * I totally disagree with Oy's assesmnent. History is important, even for an encyclopedia article. Frankly, this is one the oddest criticisms I have ever heard. --CSTAR 14:54, 17 October 2006 (UTC)


 * second above. history is important for this particular article. i am gonna remove that tone tag (article can always use knowledgable copyedit). Mct mht 17:54, 17 October 2006 (UTC)

Canonical quantization
The 2nd graf currently starts off with
 * This formulation of quantum mechanics, called canonical quantization, continues to be used today, and still forms the basis of ab-initio calculations in atomic, molecular and solid-state physics.

Canonical quantization I that referred to an attempt at "functorializing" the relation between classical and quantum mechanics. It is one area of MFQM, but the sentence seems to suggest this is equivalent to it. Am I misunderstanding something here?--CSTAR 17:26, 19 October 2006 (UTC)


 * maybe remove that phrase from the sentence. a brief note explaining how one goes from CM to QM via quantization, with link, can be added somewhere. Mct mht 05:02, 20 October 2006 (UTC)

many worlds resolved paradoxes?
"the Copenhagen interpretation of quantum mechanics which held until the Many Worlds Interpretation which resolved its many paradoxes."

Is it true that the many worlds interpretaiton of Quantum Mechanics is the favored interpretation? As far as I know, the Copenhagen interpetation is still the most favored one and according to the page of the many worlds interpretation there are presently no way found to distinguish one from the other. --Sikory 15:44, 15 July 2008 (UTC)


 * I only discovered your question today. I think your reservations with respect to the many worlds interpretation are justified. I have attended many conferences on the Foundations of Quantum Mechanics, but I cannot remember to have met any physicist who took the many worlds interpretation seriously. Contrary to the many worlds interpretation most physicists take von Neumann projection seriously, either as something "really" happening to an individual microscopic object, or as a selection of a subensemble (within an ensemble interpretation). In my observation the general feeling is that the many worlds interpretation is not backed by any observational evidence, and is just explaining problems away.WMdeMuynck (talk) 13:39, 13 September 2008 (UTC)

Functional analysis
The article says: "Many of these structures were drawn from functional analysis, a research area within pure mathematics that developed in parallel with, and was influenced by the needs of, quantum mechanics."

Is this true? Functional analysis was founded by Stefan Banach, a pure mathematician who was not interested in physics. So considering that it was not founded with any intention of being used for quantum mechanics but rather it was applied to quantum mechanics by physicists after it had been developed, how can it be claimed that it was "developed in parallel with, and was influenced by the needs of, quantum mechanics." selfworm Talk ) 06:37, 13 September 2008 (UTC)


 * Yes by most accounts it is true. Stefan Banach developed one area of functional analysis, but Weyl's and von Neumann's contributions to functional analysis (for instance Rings of Operators) were most certainly influenced by applications to QM.--CSTAR (talk) 18:26, 23 October 2008 (UTC)

Probability flux
Isn't this an axiom implicitly assumed for the probability amplitude function? --82.112.218.84 (talk) 12:44, 22 October 2008 (UTC)