Fine structure constant value [Archive]

Michel Deby

Nov17-04, 10:48 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>When we look at http://physics.nist.gov/cuu/Constants/, we see the\nvalue for the Fine Structure Constant : FSC = 1/137.03599911 +/-\n0.00000046.\n\nHowever, when we recompute it from the other fundamental constants\ninvolved, we find FSC = 1/137.03599937.\n\nOk, it falls within the limits but why the central value given by\nCodata is not the one computed from the central values of the involved\nconstants ?\n\nMichel Deby\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">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>When we look at http://physics.nist.gov/cuu/Constants/, we see the
value for the Fine Structure Constant : FSC = 1/137.03599911 +/-
.00000046.

However, when we recompute it from the other fundamental constants
involved, we find FSC = 1/137.03599937.

Ok, it falls within the limits but why the central value given by
Codata is not the one computed from the central values of the involved
constants ?

Michel Deby

Arnold Neumaier

Nov18-04, 12:52 PM

<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>\nMichel Deby wrote:\n> When we look at http://physics.nist.gov/cuu/Constants/, we see the\n> value for the Fine Structure Constant : FSC = 1/137.03599911 +/-\n> 0.00000046.\n>\n> However, when we recompute it from the other fundamental constants\n> involved, we find FSC = 1/137.03599937.\n>\n> Ok, it falls within the limits but why the central value given by\n> Codata is not the one computed from the central values of the involved\n> constants ?\n\nSince the numbers represent mean and standard deviation.\n\nThe mean of a nonlinear function of some random variables\nis generally not equal to the same function applied to the means.\n(Try the product of two uniformly distributed variables to\nconvince you of that!)\n\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">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Michel Deby wrote:
> When we look at http://physics.nist.gov/cuu/Constants/, we see the
> value for the Fine Structure Constant : FSC = 1/137.03599911 +/-
> .00000046.
>
> However, when we recompute it from the other fundamental constants
> involved, we find FSC = 1/137.03599937.
>
> Ok, it falls within the limits but why the central value given by
> Codata is not the one computed from the central values of the involved
> constants ?

Since the numbers represent mean and standard deviation.

The mean of a nonlinear function of some random variables
is generally not equal to the same function applied to the means.
(Try the product of two uniformly distributed variables to
convince you of that!)

Arnold Neumaier

ebunn@lfa221051.richmond.edu

Nov19-04, 01:32 PM

<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>In article <419B8D02.9020305@univie.ac.at>,\nArnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:\n\n>The mean of a nonlinear function of some random variables\n>is generally not equal to the same function applied to the means.\n>(Try the product of two uniformly distributed variables to\n>convince you of that!)\n\nOK, let\'s try it. Let x and y be independent random variables with\npdfs f and g, and let z = xy. Then the mean of z is\n\n<z> = \\int xy f(x) g(y) dx dy\n= \\int x f(x) dx \\int y g(y) dy\n= <x> <y>.\n\nHmm. Maybe that wasn\'t such a good example!\n\nI\'m just being picky, of course: your general point is right. For one\nthing, it wasn\'t really fair of me to assume the variables are\nindependent. The values for the physical constants are all derived\nfrom a simultaneous fit to many different measurements that depend on\nthe constants in different ways, so the uncertainty in, for instance e\nand hbar might be strongly correlated.\n\n(In fact, the NIST web site will tell you the correlation between the\ntwo, and it turns out that those two in particular have errors that\nare almost 100% correlated!)\n\nAlso, calculating the fine structure constant involves more than just\nmultiplying: it involves squaring and dividing. <x^2> is not <x>^2,\nand <x/y> is not <x> / <y>, even if x and y are independent.\n\n-Ted\n\n\n\n--\n[E-mail me at name@domain.edu, as opposed to name@machine.domain.edu.]\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">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <419B8D02.9020305@univie.ac.at>,
Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:

>The mean of a nonlinear function of some random variables
>is generally not equal to the same function applied to the means.
>(Try the product of two uniformly distributed variables to
>convince you of that!)

OK, let's try it. Let x and y be independent random variables with
pdfs f and g, and let z = xy. Then the mean of z is

<z> = \int xy f(x) g(y) dx dy= \int x f(x) dx \int y g(y) dy
= <x> <y>.

Hmm. Maybe that wasn't such a good example!

I'm just being picky, of course: your general point is right. For one
thing, it wasn't really fair of me to assume the variables are
independent. The values for the physical constants are all derived
from a simultaneous fit to many different measurements that depend on
the constants in different ways, so the uncertainty in, for instance e
and \hbar might be strongly correlated.

(In fact, the NIST web site will tell you the correlation between the
two, and it turns out that those two in particular have errors that
are almost 100% correlated!)

Also, calculating the fine structure constant involves more than just
multiplying: it involves squaring and dividing. <x^2> is not <x>^2,
and <x/y> is not <x> / <y>, even if x and y are independent.

-Ted

--
[E-mail me at name@domain.edu, as opposed to name@machine.domain.edu.]

robert bristow-johnson

Nov21-04, 02:50 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>in article af929f1.0411160642.e796f55@posting.google.com, Michel Deby at\nmicheldeby@gmail.com wrote on 11/17/2004 11:48:\n\n> When we look at http://physics.nist.gov/cuu/Constants/, we see the\n> value for the Fine Structure Constant : FSC = 1/137.03599911 +/-\n> 0.00000046.\n>\n> However, when we recompute it from the other fundamental constants\n> involved, we find FSC = 1/137.03599937.\n\ni got a little closer result when i computed\n\nalpha = (e/h_bar)^2 * h_bar/(c * 4*pi*epsilon_0) = 1/137.035999146\n\n\nin article cnl52q\\$140\\$1@lfa222122.richmond.edu, ebunn@lfa221051.richmond.edu\nat ebunn@lfa221051.richmond.edu wrote on 11/19/2004 14:32:\n\n> The values for the physical constants are all derived\n> from a simultaneous fit to many different measurements that depend on\n> the constants in different ways, so the uncertainty in, for instance e\n> and hbar might be strongly correlated.\n\nthey are.\n\ni think they can measure e/h_bar together\n( http://physics.nist.gov/cgi-bin/cuu/Value?esh|search_for=elecmag_in! )\nbetter than they measure either by themselves.\n\nr b-j\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">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>in article af929f1.0411160642.e796f55@posting.google.com, Michel Deby at
micheldeby@gmail.com wrote on 11/17/2004 11:48:

> When we look at http://physics.nist.gov/cuu/Constants/, we see the
> value for the Fine Structure Constant : FSC = 1/137.03599911 +/-
> .00000046.
>
> However, when we recompute it from the other fundamental constants
> involved, we find FSC = 1/137.03599937.

i got a little closer result when i computed

\alpha = (e/h_{bar})^2 * h_{bar}/(c * 4*\pi*\epsilon_0) = 1/137[/itex].035999146

in article cnl52q$140$1@lfa222122.richmond.edu, ebunn@lfa221051.richmond.edu
at ebunn@lfa221051.richmond.edu wrote on 11/19/2004 14:32:

> The values for the physical constants are all derived
> from a simultaneous fit to many different measurements that depend on
> the constants in different ways, so the uncertainty in, for instance e
> and [itex]\hbar might be strongly correlated.

they are.

i think they can measure e/h_{bar} together
( http://physics.nist.gov/cgi-bin/cuu/Value?esh|search_for=elecmag_in! )
better than they measure either by themselves.

r b-j

robert bristow-johnson

Nov21-04, 02:51 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>in article af929f1.0411160642.e796f55@posting.google.com, Michel Deby at\nmicheldeby@gmail.com wrote on 11/17/2004 11:48:\n\n> When we look at http://physics.nist.gov/cuu/Constants/, we see the\n> value for the Fine Structure Constant : FSC = 1/137.03599911 +/-\n> 0.00000046.\n>\n> However, when we recompute it from the other fundamental constants\n> involved, we find FSC = 1/137.03599937.\n\nactually, when i compute it directly, i get 1/137.03599906\n\n\n> Ok, it falls within the limits but why the central value given by\n> Codata is not the one computed from the central values of the involved\n> constants ?\n\nbecause of the way that errors might team up, you want to compute alpha (or\n1/alpha) from the most direct physically measured quantity that can be\nmanipulated to the quantity you want. i think it\'s the Conductance Quantum\n\nG0 = 2*e^2/h = e^2/(h_bar*pi)\n\nnote the similarity to the Fine-structure Constant, alpha, and that the\nother factors you need are all *defined* with no experimental error. both\nthe Conductance Quantum and alpha share the same relative standard deviation\nof error: 3.3e-9. so\n\nalpha = e^2/(h_bar*c*4*pi*epsilon_0) = G0/(c*4*epsilon_0)\n\n= G0 * Z0/4 = 0.00729735256826\n\n(where Z0 is the Characteristic Impedance of Free Space)\n\njust a hair higher than 1/137.03599911 , but if i compute 1/alpha from the\ninverse of the Conductance Quantum (which is measured in ohms if you\'re SI)\nas stated in CODATA, it\'s pretty good.\n\n1/alpha = 1/G0 *4/Z0 = 12906.403725 * 1/(c*pi*10^-7)\n\n= 137.0359991093\n\npretty dang close to 137.03599911 .\n\nhow they measure the Conductance Quantum or its reciprocal, i dunno, but\nthey do it much better than they measure Planck\'s constant or the Elementary\ncharge. at least they say they do. and i don\'t even know who "they" is.\n\nr b-j\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">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>in article af929f1.0411160642.e796f55@posting.google.com, Michel Deby at
micheldeby@gmail.com wrote on 11/17/2004 11:48:

> When we look at http://physics.nist.gov/cuu/Constants/, we see the
> value for the Fine Structure Constant : FSC = 1/137.03599911 +/-
> .00000046.
>
> However, when we recompute it from the other fundamental constants
> involved, we find FSC = 1/137.03599937.

actually, when i compute it directly, i get 1/137.03599906

> Ok, it falls within the limits but why the central value given by
> Codata is not the one computed from the central values of the involved
> constants ?

because of the way that errors might team up, you want to compute \alpha (or1/\alpha) from the most direct physically measured quantity that can be
manipulated to the quantity you want. i think it's the Conductance Quantum

G0 = 2*e^2/h = e^2/(h_{bar}*\pi)

note the similarity to the Fine-structure Constant, \alpha, and that the
other factors you need are all *defined* with no experimental error. both
the Conductance Quantum and \alpha share the same relative standard deviation
of error: 3.3e-9. so

\alpha = e^2/(h_{bar}*c*4*\pi*\epsilon_0) = G0/(c*4*\epsilon_0)= G0 * Z0/4 =[/itex] .00729735256826

(where Z0 is the Characteristic Impedance of Free Space)

just a hair higher than 1/137.03599911 , but if i compute 1/\alpha from the
inverse of the Conductance Quantum (which is measured in ohms if you're SI)
as stated in CODATA, it's pretty good.

1/\alpha = 1/G0 *4/Z0 = 12906.403725 [itex]* 1/(c*\pi*10^-7)

= 137.0359991093

pretty dang close to 137.03599911 .

how they measure the Conductance Quantum or its reciprocal, i dunno, but
they do it much better than they measure Planck's constant or the Elementary
charge. at least they say they do. and i don't even know who "they" is.r b-j

Arnold Neumaier

Nov22-04, 05:14 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\nebunn@lfa221051.richmond.edu wrote:\n> In article <419B8D02.9020305@univie.ac.at>,\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:\n>\n>>The mean of a nonlinear function of some random variables\n>>is generally not equal to the same function applied to the means.\n>>(Try the product of two uniformly distributed variables to\n>>convince you of that!)\n>\n> OK, let\'s try it. Let x and y be independent random variables with\n> pdfs f and g, and let z = xy. Then the mean of z is\n>\n> <z> = \\int xy f(x) g(y) dx dy\n> = \\int x f(x) dx \\int y g(y) dy\n> = <x> <y>.\n>\n> Hmm. Maybe that wasn\'t such a good example!\n\nTake instead y=1-x, then\n<xy>= integral x(1-x) dx = x^2/2-x^3/3 from 0 to 1 = 1/2-1/3=1/6\nwhile <x><y>=1/2*1/2=1/4.\n\nThus you just selected the exceptional case...\n\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">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>ebunn@lfa221051.richmond.edu wrote:
> In article <419B8D02.9020305@univie.ac.at>,
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
>
>>The mean of a nonlinear function of some random variables
>>is generally not equal to the same function applied to the means.
>>(Try the product of two uniformly distributed variables to
>>convince you of that!)
>
> OK, let's try it. Let x and y be independent random variables with
> pdfs f and g, and let z = xy. Then the mean of z is
>
> <z> = \int xy f(x) g(y) dx dy
> = \int x f(x) dx \int y g(y) dy
> = <x> <y>.
>
> Hmm. Maybe that wasn't such a good example!

Take instead y=1-x, then
<xy>= integral x(1-x) dx = x^2/2-x^3/3 from to 1 = 1/2-1/3=1/6
while <x><y>=1/2*1/2=1/4.

Thus you just selected the exceptional case...

Arnold Neumaier

Michel Deby

Dec3-04, 04:55 PM

<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> how they measure the Conductance Quantum or its reciprocal, i dunno, but\n> they do it much better than they measure Planck\'s constant or the Elementary\n> charge. at least they say they do. and i don\'t even know who "they" is.\n>\n\nThank you for the answers, especially this idea with the Conductance\nQuantun. It sounds logical.\n\nMichel Deby\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">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>>
> how they measure the Conductance Quantum or its reciprocal, i dunno, but
> they do it much better than they measure Planck's constant or the Elementary
> charge. at least they say they do. and i don't even know who "they" is.
>

Thank you for the answers, especially this idea with the Conductance
Quantun. It sounds logical.

Michel Deby

Hans de Vries

Dec6-04, 07:16 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>robert bristow-johnson <rbj@audioimagination.com> wrote in message news:<BDC44D5A.20B8%rbj@audioimagination.com>...\n \n> how they measure the Conductance Quantum or its reciprocal, i dunno, but\n> they do it much better than they measure Planck\'s constant or the Elementary\n> charge. at least they say they do. and i don\'t even know who "they" is.\n>\n> r b-j\n\nThe fine structure constant and the conductance quantum were\nmeasured in the same experiment using the Quantum Hall Effect:\n\nhttp://physics.nist.gov/cuu/Constants/alpha.html\n\n\n\nRegards, Hans\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">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>robert bristow-johnson <rbj@audioimagination.com> wrote in message news:<BDC44D5A.20B8%rbj@audioimagination.com>...

> how they measure the Conductance Quantum or its reciprocal, i dunno, but
> they do it much better than they measure Planck's constant or the Elementary
> charge. at least they say they do. and i don't even know who "they" is.
>
> r b-j

The fine structure constant and the conductance quantum were
measured in the same experiment using the Quantum Hall Effect:

http://physics.nist.gov/cuu/Constants/\alpha.html

Regards, Hans