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View Full Version : d/dhbar (physical system) (was classical em field)


Danny Ross Lunsford
Apr16-04, 05:51 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nArnold Neumaier wrote:\n\n&gt; What really happens is that hbar is a free parameter in a family\n&gt; of quantum theories. One particular choice of them corresponds to\n&gt; physical reality, and the limit hbar to zero corresponds to a classical\n&gt; world. Of course, since the real world is known not to be classical,\n&gt; one cannot take the limit in reality, where hbar is fixed.\n\nThis brings up a strange idea...\n\nFor finite h, a sane interpretation of QM says - the WF describes the\nexperimental setup (what we know about the system we are going to study)\nand the possible outcomes of experiments are distributed in some\nspectral representation of it. Now classical mechanics (h=0) has 1\noutcome for any experiment. Can we imagine setting up a relationship\nbetween the number of possible outcomes and the value of h? There would\nthen be a meaning to varying h and seeing how the spectrum changes. As\nh-&gt;0 then N-&gt;1 for any experiment. So the question is - how does the\nspectrum vary with h?\n\n&gt; Don\'t think the many excellent people were dudes!\n&gt; Whenever there is a discrepancy between tradition and one\'s own\n&gt; understanding, the reason lies in the vast majority of cases\n&gt; in the latter.\n\nWell said!\n\n-drl\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>Arnold Neumaier wrote:

> What really happens is that \hbar is a free parameter in a family
> of quantum theories. One particular choice of them corresponds to
> physical reality, and the limit \hbar to zero corresponds to a classical
> world. Of course, since the real world is known not to be classical,
> one cannot take the limit in reality, where \hbar is fixed.

This brings up a strange idea...

For finite h, a sane interpretation of QM says - the WF describes the
experimental setup (what we know about the system we are going to study)
and the possible outcomes of experiments are distributed in some
spectral representation of it. Now classical mechanics (h=0) has 1
outcome for any experiment. Can we imagine setting up a relationship
between the number of possible outcomes and the value of h? There would
then be a meaning to varying h and seeing how the spectrum changes. As
h->0 then N->1 for any experiment. So the question is - how does the
spectrum vary with h?

> Don't think the many excellent people were dudes!
> Whenever there is a discrepancy between tradition and one's own
> understanding, the reason lies in the vast majority of cases
> in the latter.

Well said!

-drl

Arnold Neumaier
Apr17-04, 05:08 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Danny Ross Lunsford wrote:\n&gt; Arnold Neumaier wrote:\n&gt;\n&gt;\n&gt;&gt;What really happens is that hbar is a free parameter in a family\n&gt;&gt;of quantum theories. One particular choice of them corresponds to\n&gt;&gt;physical reality, and the limit hbar to zero corresponds to a classical\n&gt;&gt;world. Of course, since the real world is known not to be classical,\n&gt;&gt;one cannot take the limit in reality, where hbar is fixed.\n&gt;\n&gt;\n&gt; This brings up a strange idea...\n&gt;\n&gt; For finite h, a sane interpretation of QM says - the WF describes the\n&gt; experimental setup (what we know about the system we are going to study)\n&gt; and the possible outcomes of experiments are distributed in some\n&gt; spectral representation of it. Now classical mechanics (h=0) has 1\n&gt; outcome for any experiment. Can we imagine setting up a relationship\n&gt; between the number of possible outcomes and the value of h? There would\n&gt; then be a meaning to varying h and seeing how the spectrum changes. As\n&gt; h-&gt;0 then N-&gt;1 for any experiment. So the question is - how does the\n&gt; spectrum vary with h?\n\nI don\'t think this can be made to work. If you talk about the\nspectrum of the Hamiltonian, it simply gets denser as hbar -&gt; 0,\nultimately being a continuum. You can check it with the harmonic\noscillator. So quantization becomes observable at smaller and smaller\nscales only, ultimately disappearing. Spin also simply disappears.\n\nThe right limit for accuracy questions is the thermodynamic limit\nN -&gt; infinity, which Charles Francis mentioned (in disguise).\nWhat happens is that standard deviations typically scale with 1/sqrt(N)\nby the law of large numbers, so deviations from mean field behavior\nbecome highly suppressed. There are exceptions, though (near\ncritical points).\n\nArnold Neumaier\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>Danny Ross Lunsford wrote:
> Arnold Neumaier wrote:
>
>
>>What really happens is that \hbar is a free parameter in a family
>>of quantum theories. One particular choice of them corresponds to
>>physical reality, and the limit \hbar to zero corresponds to a classical
>>world. Of course, since the real world is known not to be classical,
>>one cannot take the limit in reality, where \hbar is fixed.
>
>
> This brings up a strange idea...
>
> For finite h, a sane interpretation of QM says - the WF describes the
> experimental setup (what we know about the system we are going to study)
> and the possible outcomes of experiments are distributed in some
> spectral representation of it. Now classical mechanics (h=0) has 1
> outcome for any experiment. Can we imagine setting up a relationship
> between the number of possible outcomes and the value of h? There would
> then be a meaning to varying h and seeing how the spectrum changes. As
> h->0 then N->1 for any experiment. So the question is - how does the
> spectrum vary with h?

I don't think this can be made to work. If you talk about the
spectrum of the Hamiltonian, it simply gets denser as \hbar -> 0,
ultimately being a continuum. You can check it with the harmonic
oscillator. So quantization becomes observable at smaller and smaller
scales only, ultimately disappearing. Spin also simply disappears.

The right limit for accuracy questions is the thermodynamic limit
N -> infinity, which Charles Francis mentioned (in disguise).
What happens is that standard deviations typically scale with 1/\sqrt(N)
by the law of large numbers, so deviations from mean field behavior
become highly suppressed. There are exceptions, though (near
critical points).

Arnold Neumaier