Basic Harmonic Oscillator question [Archive]

Constantine

Aug2-04, 06:10 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>Hello All.\n\nA really basic QM question, but it drives me crazy. Suppose you have the\nusual harmonic oscillator, where\n\n|0>: the vacuum in the number representation (the one you build with a,\na^\\dag)\n\nand\n\n|p>: eigenstate of the momentum.\n\nNow\n\n|p> = Constant Exp[ (-1/4)p^2 + p a^\\dag - (1/2) a^\\dag a^\\dag ] |0>.\n\nThere is something similar relating |x> (eigenstate of position) with |0>.\n\nHow do you prove that? I know that it is something basic, but for some\nreason I can not see it. Any help will be much appreciated.\n\nFriendly, Kostas.\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>Hello All.

A really basic QM question, but it drives me crazy. Suppose you have the
usual harmonic oscillator, where

|0>:[/itex] the vacuum in the number representation (the one you build with a,
[itex]a^\dag)

and

|p>: eigenstate of the momentum.

Now

|p> = Constant \Exp[ (-1/4)p^2 + p a^\dag - (1/2) a^\dag a^\dag ] |0>.

There is something similar relating |x> (eigenstate of position) with |0>.

How do you prove that? I know that it is something basic, but for some
reason I can not see it. Any help will be much appreciated.

Friendly, Kostas.

Patrick Van Esch

Aug4-04, 01:22 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>"Constantine" <Konstantinos.Kyritsis@durham.ac.uk> wrote in message news:<cel7tn\\$3q4\\$1@heffalump.dur.ac.uk>...\n\n >\n> |p> = Constant Exp[ (-1/4)p^2 + p a^\\dag - (1/2) a^\\dag a^\\dag ] |0>.\n>\n> There is something similar relating |x> (eigenstate of position) with |0>.\n>\n> How do you prove that? I know that it is something basic, but for some\n> reason I can not see it. Any help will be much appreciated.\n\nI didn\'t do this, but I\'d propose to try to show that the above\nexpression for |p> is an eigenvector of the p operator, which can be\nwritten as i sqrt(m hbar omega/2)(a-dagger - a). You probably have to\nwrite out the exponential of the operator as a series, so you\'d\ncalculate:\n\nP |p> = i sqrt[m hbar omega / 2] (a-dagger - a) Sum[operators in\na-dagger] |0> and hope that you can work this out into something like\np x |p>.\n\njust a suggestion, I don\'t know if it works out.\n\ncheers,\nPatrick.\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>"Constantine" <Konstantinos.Kyritsis@durham.ac.uk> wrote in message news:<cel7tn$3q4$1@heffalump.dur.ac.uk>...

>
> |p> = Constant \Exp[ (-1/4)p^2 + p a^\dag - (1/2) a^\dag a^\dag ] |0>.
>
> There is something similar relating |x> (eigenstate of position) with |0>.
>
> How do you prove that? I know that it is something basic, but for some
> reason I can not see it. Any help will be much appreciated.

I didn't do this, but I'd propose to try to show that the above
expression for |p> is an eigenvector of the p operator, which can be
written as i \sqrt(m \hbar \omega/2)(a-dagger - a). You probably have to
write out the exponential of the operator as a series, so you'd
calculate:

P |p> = i \sqrt[m \hbar \omega / 2] (a-dagger - a) Sum[operators in
a-dagger] |0> and hope that you can work this out into something like
p x |p>.

just a suggestion, I don't know if it works out.

cheers,
Patrick.