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| May31-05, 11:42 PM | #1 |
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Matrix Representation
<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>Dear Members,\n\nOn p. 30 of Sakurai\'s Modern Quantum Mechanics, why in (1.4.28),\n<a"|B|a\'> represents Matrix elements of B?\n\nCheers,\nAli\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>Dear Members,
On p. 30 of Sakurai's Modern Quantum Mechanics, why in (1.4.28), <a"[itex]|B|a'>[/itex] represents Matrix elements of B? Cheers, Ali |
| Jun2-05, 12:41 AM | #2 |
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<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>On 2005-06-01, Ali <ph_question@yahoo.com> wrote:\n> Dear Members,\n>\n> On p. 30 of Sakurai\'s Modern Quantum Mechanics, why in (1.4.28),\n><a"|B|a\'> represents Matrix elements of B?\n\nThe short answer answer is: By definition.\n\nLonger answer. Let {|1>, |2>, ...} be an orthonormal basis for the\nHilbert space. B is a linear operator that maps the Hilbert space into\nitself. Let |a> be a vector and |b> = H|a>. Expand both vectors in the\nchosen basis: |a> = a_1|1> + a_2|2> + ..., |b> = b_1|1> + b_2|2> + ...\nRelate the coefficients (by orthonormality):\n\nb_i = <i|b> = <i|B|a> = sum_j <i|B|j> a_j.\n\nIf you consider b_i to be entries in a column vector, then this column\nvector is obtained by multiplication of a matrix with elements <i|B|j>\nwith another column vector with entries a_j. That is why the numbers\n<i|B|j> are called matrix elements of the operator B.\n\nHow are you doing with that supplementary book on linear algebra?\n\nIgor\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>On [itex]2005-06-01,[/itex] Ali <ph_question@yahoo.com> wrote:
> Dear Members, > > On p. 30 of Sakurai's Modern Quantum Mechanics, why in (1.4.28), ><a"[itex]|B|a'>[/itex] represents Matrix elements of B? The short answer answer is: By definition. Longer answer. Let [itex]{|1>, |2>, ...}[/itex] be an orthonormal basis for the Hilbert space. B is a linear operator that maps the Hilbert space into itself. Let |a> be a vector and [itex]|b> = H|a>[/itex]. Expand both vectors in the chosen basis: [itex]|a> = a_1|1> + a_2|2> + ..., |b> = b_1|1> + b_2|2> + ...[/itex] Relate the coefficients (by orthonormality): [itex]b_i = <i|b> = <i|B|a> = sum_j <i|B|j> a_j[/itex]. If you consider [itex]b_i[/itex] to be entries in a column vector, then this column vector is obtained by multiplication of a matrix with elements [itex]<i|B|j>[/itex] with another column vector with entries [itex]a_j[/itex]. That is why the numbers [itex]<i|B|j>[/itex] are called matrix elements of the operator B. How are you doing with that supplementary book on linear algebra? Igor |
| Jun2-05, 12:42 AM | #3 |
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<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>"Ali" <ph_question@yahoo.com> wrote in message\nnews:1117598940.983572.265940@z14g2000cwz.googlegroups.com...\ n> Dear Members,\n>\n> On p. 30 of Sakurai\'s Modern Quantum Mechanics, why in (1.4.28),\n> <a"|B|a\'> represents Matrix elements of B?\n>\n> Cheers,\n> Ali\n>\nUse the facts that <a\'\'|A = a\'\'<a\'\'| and A|a\'> = a\'|a\'> along with the\ndefinition of the commutator.\n\nJohn Lowry\nFlight Physics\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>"Ali" <ph_question@yahoo.com> wrote in message
news:1117598940.983572.265940@z14g20...egroups.com... > Dear Members, > > On p. 30 of Sakurai's Modern Quantum Mechanics, why in (1.4.28), > <a"[itex]|B|a'>[/itex] represents Matrix elements of B? > > Cheers, > Ali > Use the facts that [itex]<a''|A = a''<a''|[/itex] and [itex]A|a'> = a'|a'>[/itex] along with the definition of the commutator. John Lowry Flight Physics |
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