Basic QM Question [Archive]

chris h fleming

Sep24-04, 08:08 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>Let\'s say we write down the Schroedinger equation for hydrogen. We\nthen solve it and get among other things the pdf for the electron\'s\norbit.\n\nMy question is: why didn\'t I use the pdf of the electron as a form\nfactor in the coulomb interaction. It seems like we start with a point\nparticle for the interaction and then derive a pdf which contradicts\nthe assumption.\n\nIs the calculation an apporoximation or is there some fundamental\nreason? (And can a reason be given without invoking QFT?)\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>Let's say we write down the Schroedinger equation for hydrogen. We
then solve it and get among other things the pdf for the electron's
orbit.

My question is: why didn't I use the pdf of the electron as a form
factor in the coulomb interaction. It seems like we start with a point
particle for the interaction and then derive a pdf which contradicts
the assumption.

Is the calculation an apporoximation or is there some fundamental
reason? (And can a reason be given without invoking QFT?)

Arnold Neumaier

Sep25-04, 05:00 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>chris h fleming wrote:\n> Let\'s say we write down the Schroedinger equation for hydrogen. We\n> then solve it and get among other things the pdf for the electron\'s\n> orbit.\n>\n> My question is: why didn\'t I use the pdf of the electron as a form\n> factor in the coulomb interaction. It seems like we start with a point\n> particle for the interaction and then derive a pdf which contradicts\n> the assumption.\n>\n> Is the calculation an apporoximation or is there some fundamental\n> reason? (And can a reason be given without invoking QFT?)\n\nThe form factor is _not_ a proability distribution but a factor in the\neffective Schroedinger or Dirac equation. Hence there is no contradiction.\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>chris h fleming wrote:
> Let's say we write down the Schroedinger equation for hydrogen. We
> then solve it and get among other things the pdf for the electron's
> orbit.
>
> My question is: why didn't I use the pdf of the electron as a form
> factor in the coulomb interaction. It seems like we start with a point
> particle for the interaction and then derive a pdf which contradicts
> the assumption.
>
> Is the calculation an apporoximation or is there some fundamental
> reason? (And can a reason be given without invoking QFT?)

The form factor is _not_ a proability distribution but a factor in the
effective Schroedinger or Dirac equation. Hence there is no contradiction.

Arnold Neumaier

urs@lfa222122.richmond.edu

Sep27-04, 03:31 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\n\nOn Fri, 24 Sep 2004 13:08:56 +0000, chris h fleming wrote:\n\n> Let\'s say we write down the Schroedinger equation for hydrogen. We then\n> solve it and get among other things the pdf for the electron\'s orbit.\n>\n> My question is: why didn\'t I use the pdf of the electron as a form factor\n> in the coulomb interaction. It seems like we start with a point particle\n> for the interaction and then derive a pdf which contradicts the\n> assumption.\n>\n> Is the calculation an apporoximation or is there some fundamental reason?\n> (And can a reason be given without invoking QFT?)\n\nPicture this:\nQuantization\nClassical System ------------------------------> Quantum System\n^ |\n+---------------------------------------------------+\nClassical Limit\n\nGoing around the arrows in the above diagram should take you where you\nback to where you started. We know what the classical limit for an\nelectron is, it\'s a point particle, not a wave like the electromagnetic\nfield. This observation obviously breaks down in a double slit experiment\nor in the hydrogen atom, but that\'s exactly where the systems are not\nclassical any more, so it need not be considered within the classical\nlimit.\n\nKnowing what the Classical System is does not give us a unique Quantum\nSystem that reduces to it in the classical limit. We can however exclude\nthe Quantum Systems that have the wrong classical limit. Above, you\npropose to start with a wave (or a field) description of the electron in\nthe classical limit and get a Quantum System through quantization, but if\nyou do your quantization right, quite clearly, this Quantum System will\nnot reproduce the observed properties of the electron and it\'s interaction\nwith other matter and fields in the classical limit.\n\nHope this helps.\n\nIgor\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 Fri, 24 Sep 2004 13:08:56 +0000, chris h fleming wrote:

> Let's say we write down the Schroedinger equation for hydrogen. We then
> solve it and get among other things the pdf for the electron's orbit.
>
> My question is: why didn't I use the pdf of the electron as a form factor
> in the coulomb interaction. It seems like we start with a point particle
> for the interaction and then derive a pdf which contradicts the
> assumption.
>
> Is the calculation an apporoximation or is there some fundamental reason?
> (And can a reason be given without invoking QFT?)

Picture this:
Quantization
Classical System ------------------------------> Quantum System
^ |
+---------------------------------------------------+
Classical Limit

Going around the arrows in the above diagram should take you where you
back to where you started. We know what the classical limit for an
electron is, it's a point particle, not a wave like the electromagnetic
field. This observation obviously breaks down in a double slit experiment
or in the hydrogen atom, but that's exactly where the systems are not
classical any more, so it need not be considered within the classical
limit.

Knowing what the Classical System is does not give us a unique Quantum
System that reduces to it in the classical limit. We can however exclude
the Quantum Systems that have the wrong classical limit. Above, you
propose to start with a wave (or a field) description of the electron in
the classical limit and get a Quantum System through quantization, but if
you do your quantization right, quite clearly, this Quantum System will
not reproduce the observed properties of the electron and it's interaction
with other matter and fields in the classical limit.

Hope this helps.

Igor