View Full Version : Fourier expansion of an operator?
jason cooper
May6-04, 07:53 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>Suppose I express an operator \'O\' as a linear combination of\noperators \'O_(i)\':\n\nO = sum[i]{c^i O_(i)}\n\nIf I take the discrete Fourier transform of the coefficients\n\'c^i\' with respect to some parameter \'a\', I get amplitudes\n\'c^i(n)\' or some such thing. My question is, are these such\nthat (approximately, hopefully making the point a little\nclearer):\n\nO = sum[i,n]{c^i(n) exp(in) O_(i)}\n\nor such that:\n\nO = sum[i,n]{c^i(n) O_(i) exp(in)}\n\nor such that:\n\nO = sum[i,n]{c^i(n) (exp(in) O_(i) + O_(i) exp(in))/2}\n\nor something else entirely, where the exponential does not in\ngeneral commute with O_(i)?\n\n-----------------------------------------------------------------\n. . . Except when they don\'t,\nBecause sometimes they won\'t. - Dr. Seuss\n-----------------------------------------------------------------\nJason Cooper jcooper@acs.ucalgary.ca\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>Suppose I express an operator 'O' as a linear combination of
operators 'O_(i)':O = sum[i]{c^i O_(i)}
If I take the discrete Fourier transform of the coefficients
'c^i' with respect to some parameter 'a', I get amplitudes
'c^i(n)' or some such thing. My question is, are these such
that (approximately, hopefully making the point a little
clearer):
O = sum[i,n]{c^i(n) \exp(in) O_(i)}
or such that:
O = sum[i,n]{c^i(n) O_(i) \exp(in)}
or such that:
O = sum[i,n]{c^i(n) (\exp(in) O_(i) + O_(i) \exp(in))/2}
or something else entirely, where the exponential does not in
general commute with O_(i)?
-----------------------------------------------------------------
. . . Except when they don't,
Because sometimes they won't. - Dr. Seuss
-----------------------------------------------------------------
Jason Cooper jcooper@acs.ucalgary.ca
Charles Francis
May7-04, 06:41 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 message <c76e4h\\$ilb\\$1@news.ucalgary.ca>, jason cooper\n<jcooper@acs4.acs.ucalgary.ca> writes\n>Suppose I express an operator \'O\' as a linear combination of\n>operators \'O_(i)\':\n>\n> O = sum[i]{c^i O_(i)}\n>\n>If I take the discrete Fourier transform of the coefficients\n>\'c^i\' with respect to some parameter \'a\', I get amplitudes\n>\'c^i(n)\' or some such thing. My question is, are these such\n>that (approximately, hopefully making the point a little\n>clearer):\n>\n> O = sum[i,n]{c^i(n) exp(in) O_(i)}\n>\n>or such that:\n>\n> O = sum[i,n]{c^i(n) O_(i) exp(in)}\n>\n>or such that:\n>\n> O = sum[i,n]{c^i(n) (exp(in) O_(i) + O_(i) exp(in))/2}\n>\n>or something else entirely, where the exponential does not in\n>general commute with O_(i)?\n\nThe exponential is just a number, so it does commute with O\n--\nCharles Francis\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 message <c76e4h$ilb$1@news.ucalgary.ca>, jason cooper
<jcooper@acs4.acs.ucalgary.ca> writes
>Suppose I express an operator 'O' as a linear combination of
>operators 'O_(i)':
>
> O = sum[i]{c^i O_(i)}
>
>If I take the discrete Fourier transform of the coefficients
>'c^i' with respect to some parameter 'a', I get amplitudes
>'c^i(n)' or some such thing. My question is, are these such
>that (approximately, hopefully making the point a little
>clearer):
>
> O = sum[i,n]{c^i(n) \exp(in) O_(i)}
>
>or such that:
>
> O = sum[i,n]{c^i(n) O_(i) \exp(in)}
>
>or such that:
>
> O = sum[i,n]{c^i(n) (\exp(in) O_(i) + O_(i) \exp(in))/2}
>
>or something else entirely, where the exponential does not in
>general commute with O_(i)?
The exponential is just a number, so it does commute with O
--
Charles Francis
jason cooper
May11-04, 05:11 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>Charles Francis (charles@lluestfarmpoultry.co.uk) wrote:\n\n: The exponential is just a number, so it does commute with O\n\nMy mistake. Of course, since this is a Fourier expansion it has\nto be an expansion with respect to some variable. Where I had\nwritten "exp(in)", I should have written "exp(ina)", where \'a\' is\nan angle. So that the exponential does not, for instance,\ncommute with the corresponding angular momentum operator.\n\n-----------------------------------------------------------------\n. . . Except when they don\'t,\nBecause sometimes they won\'t. - Dr. Seuss\n-----------------------------------------------------------------\nJason Cooper jcooper@acs.ucalgary.ca\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>Charles Francis (charles@lluestfarmpoultry.co.uk) wrote:
: The exponential is just a number, so it does commute with O
My mistake. Of course, since this is a Fourier expansion it has
to be an expansion with respect to some variable. Where I had
written "\exp(in)", I should have written "\exp(ina)", where 'a' is
an angle. So that the exponential does not, for instance,
commute with the corresponding angular momentum operator.
-----------------------------------------------------------------
. . . Except when they don't,
Because sometimes they won't. - Dr. Seuss
-----------------------------------------------------------------
Jason Cooper jcooper@acs.ucalgary.ca
Charles Francis
May14-04, 04:08 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>In message <c7od2d\\$7so\\$1@news.ucalgary.ca>, jason cooper\n<jcooper@acs4.acs.ucalgary.ca> writes\n>Charles Francis (charles@lluestfarmpoultry.co.uk) wrote:\n>\n>: The exponential is just a number, so it does commute with O\n>\n>My mistake. Of course, since this is a Fourier expansion it has\n>to be an expansion with respect to some variable. Where I had\n>written "exp(in)", I should have written "exp(ina)", where \'a\' is\n>an angle. So that the exponential does not, for instance,\n>commute with the corresponding angular momentum operator.\n\na is a variable. Its still just a number, not an operator\n\n--\nCharles Francis\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 message <c7od2d$7so$1@news.ucalgary.ca>, jason cooper
<jcooper@acs4.acs.ucalgary.ca> writes
>Charles Francis (charles@lluestfarmpoultry.co.uk) wrote:
>
>: The exponential is just a number, so it does commute with O
>
>My mistake. Of course, since this is a Fourier expansion it has
>to be an expansion with respect to some variable. Where I had
>written "\exp(in)", I should have written "\exp(ina)", where 'a' is
>an angle. So that the exponential does not, for instance,
>commute with the corresponding angular momentum operator.
a is a variable. Its still just a number, not an operator
--
Charles Francis
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