Talk:Mathematical formulation of quantum mechanics

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Talk:Mathematical formulation of quantum mechanics/archive 1 up to 2004 November 19. Talk:Mathematical formulation of quantum mechanics/archive 2 Nov. & Dec. 2004.

Contents 1 Schrodinger equation 2 bra-ket notation 3 Measurement 4 Sign unsigned 5 English? 6 tone 7 Canonical quantization

[edit] 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 θ): 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 θ, and derivatives of things may be taken with respect to θ. If one imagines that the parameter θ 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 | ψ > 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 < x | ψ > = ψ(x) is just the component of the vector | ψ > in the | x > direction. The ket | ψ > 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 < ψ | P | ψ > . Note that usually one may write

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

where δ 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) .

[edit] bra-ket notation

bra-ket notation (a convention for brackets, denoted "<bra | ket>", later to be re-used in http's html)

I refer specifically to the HTML element#Links and anchors "A" form

<A URL> Text displayed for the HTTP link</A>

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)

[edit] 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) .

[edit] 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 {{subst:unsigned|username}}. -lethe ^{talk +} 13:25, 3 April 2006 (UTC)

[edit] 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)

[edit] 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)

[edit] 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)