Talk:Wave function/Archive 8: Difference between revisions

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:::::::::::... the matter wave

:::::::::::::::<math>\psi (\boldmathbf r_2, \tau_2) = \int K (\boldmathbf r_2 - \boldmathbf r_1; t_2 - t_1) \psi (\boldmathbf r_1, t_1) \, \mathrm d \boldmathbf r_1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (1)</math>

:::::::::::where

:::::::::::::::<math>K(\boldmathbf r, \tau) \,\,\,= (2\pi \hbar)^{{-3}} \int \exp \left [\frac {\mathrm i}{\hbar}(\boldmathbf p \cdot \boldmathbf r - E \tau)\right] \mathrm d \boldmathbf p.</math><ref>[[Albert Messiah|Messiah, A.]] (1958/1961). ''Quantum Mechanics'', volume 1, translated by G.M. Temmer from the French ''Mécanique Quantique'', North-Holland, Amsterdam, p. 75.</ref>

::::::Messiah does not feel a need for the ordinary-language word 'explicit' here because the situation seems obvious to him from what he has written, and he is not using the Dirac notation at that point, so as to need a contrast. I think for clarity for our purpose here an ordinary language word is needed to distinguish the two forms of expression. I did not invent the more technical term 'symbolic'; that is Dirac's. A symbolic expression is, for example, in Dirac's notation, |{{math|''z''}}〉 = {{math|''ζ''₁}}|{{math|''z''₁}}〉 + {{math|''ζ''₂}}|{{math|''z''₂}}〉 .

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::::::Weinberg in his ''Lectures'', on page 34, explicitly expresses a wave function as a function thus:

::::::::::::::::<math>\psi (\boldmathbf x) \,\,\,\,\,= R(r)Y(\theta,\phi) ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (2.1.21)</math><ref>[[Steven Weinberg|Weinberg, S.]] (2013). ''Lectures on Quantum Mechanics'', Cambridge University Press, Cambridge UK, {{ISBN |978-1-107-02872-2}}, p. 34.</ref>

'''References'''

{{Reflist}}

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::::::The kinds of experiments we shall consider are limited in the following respects: they are repeatable; there is a clear distinction between the state preparation device and the detection and registration device; and - this the most important limitation - we assume that for a given state-preparation device, preparing the system to be measured in a definite initial state, the state can be resolved into ''channels'', each of which can be independently blocked, in such a way that when only one channel is open the outcome of the experiment is ''deterministic'' - in the sense that if there is any registered outcome at all (on repetition of the experiment) it is always the ''same'' outcome.<ref>[[Simon Saunders|Saunders, S.]] (2004). 'Derivation of the Born rule from

operational assumptions', ''Proc. Roy. Soc. A'', '''460''': 1-18.</ref>
{{Reflist}}

::[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 10:49, 7 February 2016 (UTC)

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:Likely I am missing the main point here.

[[File:Quixo-panza.jpg|Don Quixote and Sancho Panza|right|thumb|250px|Don Quixote, his horse Rocinante and his squire Sancho Panza after an unsuccessful attack on a windmill. By [[Gustave Doré]].]]

:::::<small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/194.68.82.241|194.68.82.241]] ([[User talk:194.68.82.241|talk]]) 13:55, 10 February 2016 (UTC)</small><!-- Template:Unsigned IP --> <!--Autosigned by SineBot--> Re-posted by [[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 14:19, 10 February 2016 (UTC)

:But anyway, here's a start. "How do you characterize a "maximal commuting set of observables"?" I think this is standard phrasing, at least in some places. One starts with some choice of observable. Then one chooses another. If they commute, it stays; if they don't, it's out. Repeat until one can't find any more that commute. I suppose that seems rather rough and ready, and hardly convincing. I will forthwith have a look to check this. Or is this utterly missing the point?[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 11:05, 10 February 2016 (UTC)

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:::::::German original (1932/1996), ''Mathematische Grundlagen der Quantenmechanik'', Springer, Berlin, ISBN-13: 978-3-642-64828-1, p. 79: "Einen Operator, der keine echten Fortsetzungen besitzt — der also an allen Stellen, wo er vernünftigerweise, d. h. ohne Durchbrechung des Hermiteschen Charakters, definiert werden könnte, auch schon definiert ist — nennen wir maximal. Wir haben also gesehen: nur zu maximalen Operatoren kann eine Zerlegung der Einheit gehören."[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 04:41, 11 February 2016 (UTC)

:::::::Newton, R.G. (2002), in ''Quantum Physics: a Text for Graduate Students'', Springer, New York, {{ISBN |0-387-95473-2}}, writes on page 317: "Suppose that <math>\mathfrak V</math> is such that there is a maximal number of linearly independent vectors in it, i.e., given any set of non-zero vectors with more than <math>n</math> members, they must be linearly dependent. The number <math>n</math> is then called the dimension of <math>\mathfrak V</math>." He doesn't use it in that sense elsewhere in that book.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 16:31, 10 February 2016 (UTC)

:::::::Bransden, B.H., Joachain, C.J. (1989/2000), ''Quantum Mechanics'', second edition, Pearson–Prentice–Hall, Harlow UK, {{ISBN |978-0-582-35169-7}}, p. 641: "Until now we have considered quantum systems which can be described by a ''single'' wave function (state vector). Such systems are said to be in a ''pure state''. They are prepared in a specific way, their state vector being obtained by performing a ''maximal measurement'' in which all values of a complete set of commuting observables have been ascertained. In this chapter we shall study quantum systems such that the measurement made on them is not maximal. These systems, whose state is incompletely known, are said to be in ''mixed states''."[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 20:32, 10 February 2016 (UTC)

:::::::[[Gennaro Auletta|Auletta, G.]], Fortunato, M., [[Giorgio Parisi|Parisi, G.]] (2009), ''Quantum Mechanics'', Cambridge University Press, Cambridge UK, {{ISBN |978-0-521-86963-8}}, p. 174: "From Sec. 1.3 and Subsec. 2.3.3 we know that the state vector |{{math|''ψ''}}〉 contains the maximal information about a quantum system."[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 20:48, 10 February 2016 (UTC)

:::::::I Googled the phrase 'maximal set of commuting observables', and found [https://books.google.com.au/books?id=4ZwKCAAAQBAJ&pg=PA16&lpg=PA16&dq=maximal+set+of+commuting+observables+quantum+mechanics&source=bl&ots=ihL-ouMAoN&sig=iXg5Sm2CzApf0ZEKLZC0OsNO4Xk&hl=en&sa=X&ved=0ahUKEwiQg8GC6u7KAhVEkZQKHQ4qB_IQ6AEIMTAG#v=onepage&q=maximal%20set%20of%20commuting%20observables%20quantum%20mechanics&f=false this], and [https://books.google.com.au/books?id=lJaX2PsTxNoC&pg=PT94&lpg=PT94&dq=maximal+set+of+commuting+observables+quantum+mechanics&source=bl&ots=N7mHtcmXS9&sig=lIFMSkB-3JiMp3XG_yMjMaGqBAg&hl=en&sa=X&ved=0ahUKEwiQg8GC6u7KAhVEkZQKHQ4qB_IQ6AEILjAF#v=onepage&q=maximal%20set%20of%20commuting%20observables%20quantum%20mechanics&f=false also this], and [https://books.google.com.au/books?id=Xg2NZD73b4cC&pg=PA107&lpg=PA107&dq=maximal+set+of+commuting+observables+quantum+mechanics&source=bl&ots=XNpZbW3V-K&sig=hjtTtngz9OkXfbzrTtv2A4EeZA4&hl=en&sa=X&ved=0ahUKEwiQg8GC6u7KAhVEkZQKHQ4qB_IQ6AEIKjAE#v=onepage&q=maximal%20set%20of%20commuting%20observables%20quantum%20mechanics&f=false moreover this], and yet [https://books.google.com.au/books?id=eWdDAAAAQBAJ&pg=PA131&lpg=PA131&dq=maximal+set+of+commuting+observables+quantum+mechanics&source=bl&ots=5ICKajbc5X&sig=UHUeWc7xUNMAAFvJnAQlfR9MhMM&hl=en&sa=X&ved=0ahUKEwiQg8GC6u7KAhVEkZQKHQ4qB_IQ6AEIOTAI#v=onepage&q=maximal%20set%20of%20commuting%20observables%20quantum%20mechanics&f=false again this], and [https://books.google.com.au/books?id=v1owGsfiJcoC&pg=PA4&lpg=PA4&dq=maximal+set+of+commuting+observables+quantum+mechanics&source=bl&ots=k9hKAHG4-4&sig=wGW8ASjUSj6u_x9XrRjxzHZpvts&hl=en&sa=X&ved=0ahUKEwiQg8GC6u7KAhVEkZQKHQ4qB_IQ6AEIPDAJ#v=onepage&q=maximal%20set%20of%20commuting%20observables%20quantum%20mechanics&f=false now this].[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 04:18, 11 February 2016 (UTC)

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:"And how is "a state vector is an [[equivalence class]] of wave functions"?" I would have thought that was a standard way of expressing the situation. I learnt it when I studied algebra. It seems to be assumed as common mathematical parlance by the writer of this sentence: "Assuming that the unchanging reading of an ideal thermometer is a valid "tagging" system for the equivalence classes of a set of equilibrated thermodynamic systems, then if a thermometer gives the same reading for two systems, those two systems are in thermal equilibrium, and if we thermally connect the two systems, there will be no subsequent change in the state of either one." The sentence was posted in [https://en.wikipedia.org/w/index.php?title=Zeroth_law_of_thermodynamics&diff=next&oldid=667219209 this] edit by respected Editor [[User:PAR|PAR]]. My usage intends that all the wave functions that belong to a particular state are interconvertible by a group of one-to-one mathematical transformations. That makes them members of an equivalence class. (The equivalence class has the structure of a Hilbert space, more or less.) I find this form of expression helpful to show the relation between wave functions and state vectors. It may or may not be so for others.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 09:39, 14 February 2016 (UTC)

:Well, it seems that I have led myself astray by looking in Wikipedia and Google. Looking at a textbook on my shelves that I forgot I had, I find that indeed, as you say, a component is there defined as a scalar. Bloom, D.M. (1979), ''Linear Algebra and Geometry'', Cambridge University Press, Cambridge UK, {{ISBN |0-521-21959-0}}, p. 98. I hardly need say this makes me look silly. I am sorry. I can only say I misled myself by looking in Wikipedia and Google. That's a lesson. Well, I can only say I am sorry. My only excuse can be that I wrote "I would suggest adjusting it slightly, to make it agree with Wikipedia definitions as follows: .... I am suggesting to use the term [[scalar projection]]." Evidently that was a mistake. Now checking more in Wikipedia, I find at [[Basis (linear algebra)]] that I did not look in right place in Wikipedia. Just for clarity here, I will repeat, I now agree that 'component' is suitable. I guess a link to [[Basis (linear algebra)]] might be a good idea.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 12:37, 14 February 2016 (UTC)

:See [[Talk:Scalar projection#This article has gravely misled me, and helped to make me look foolish, because I thought that on such a simple matter, an article like this could be trusted.]][[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 12:59, 14 February 2016 (UTC)

:Also [[Talk:Basis (linear algebra)/Archive 1#customary terminology not clear in Wikipedia; local editors, heads up]].[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 18:17, 14 February 2016 (UTC)

:Perhaps I went overboard with the ''mea culpa''. Looking a bit further, I get the impression that customs vary.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 19:26, 14 February 2016 (UTC)