Talk:Mathematical formulation of quantum mechanics/archive 1
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This archive covers the discussion up to 2004 November 19
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 Nov 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 Jul 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 Jul 2004 (UTC)
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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 Jul 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
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- 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
then S satisfies the Hamilton-Jacobi equation up to a term proportional to 
. - 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.
- — Miguel 06:14, 2004 Nov 19 (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 Nov 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.
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- 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 Nov 2004 (UTC)
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- 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. — Miguel 06:07, 2004 Nov 19 (UTC)
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- I have toned down the historical interpretation, though, by calling De Broglie radical and stating that Schrödinger successfully formulated wave-particle duality mathematically.
- —Miguel 05:59, 2004 Nov 19 (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 Nov 2004 (UTC)
