Quantum propagation from a Dirac initial point [Archive]

Perfectly Innocent

Jun7-04, 04:54 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\nI saw the solution to the one-dimensional Schrodinger equation for\n"free particles" long ago but have forgotten it. No one at sci.physics\ncould answer the question. What\'s the wave function of a particle in\none spatial dimension for t>0 assuming that at t=0 the particle is at\nx=0 with 100% probability. I require that the scalar potential V(x)=0.\n\nEugene Shubert\nhttp://www.everythingimportant.org\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>I saw the solution to the one-dimensional Schrodinger equation for
"free particles" long ago but have forgotten it. No one at sci.physics
could answer the question. What's the wave function of a particle in
one spatial dimension for t>0 assuming that at t=0 the particle is at
x=0 with 100% probability. I require that the scalar potential V(x)=0.

Eugene Shubert
http://www.everythingimportant.org

C. M. Heard

Jun8-04, 04:03 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>\nOn 7 Jun 2004, Eugene Shubert wrote:\n> I saw the solution to the one-dimensional Schrodinger equation for\n> "free particles" long ago but have forgotten it. No one at sci.physics\n> could answer the question. What\'s the wave function of a particle in\n> one spatial dimension for t>0 assuming that at t=0 the particle is at\n> x=0 with 100% probability. I require that the scalar potential V(x)=0.\n\nThe answer is exp(i m x^2 / 2 h_cross t) / sqrt(2 pi h_cross t / i m)\n\nOne place this is discussed is in Feynam & Hibbs, _Path Integrals\nand Quantum Mechanics_, McGraw-Hill, 1965.\n\nMike Heard\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 7 Jun 2004, Eugene Shubert wrote:
> I saw the solution to the one-dimensional Schrodinger equation for
> "free particles" long ago but have forgotten it. No one at sci.physics
> could answer the question. What's the wave function of a particle in
> one spatial dimension for t>0 assuming that at t=0 the particle is at
> x=0 with 100% probability. I require that the scalar potential V(x)=0.

The answer is \exp(i m x^2 / 2 h_{cross} t) / \sqrt(2 \pi h_{cross} t / i m)

One place this is discussed is in Feynam & Hibbs, _Path Integrals
and Quantum Mechanics_, McGraw-Hill, 1965.

Mike Heard

Arkadiusz Jadczyk

Jun8-04, 04:03 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>\nOn 7 Jun 2004 05:54:29 -0400, perfectlyInnocent@as-if.com (Perfectly\nInnocent) wrote:\n\n>I saw the solution to the one-dimensional Schrodinger equation for\n>"free particles" long ago but have forgotten it. No one at sci.physics\n>could answer the question. What\'s the wave function of a particle in\n>one spatial dimension for t>0 assuming that at t=0 the particle is at\n>x=0 with 100% probability. I require that the scalar potential V(x)=0.\n\nTake the Fourier transform in x. Laplacian becomes p^2, delta becomes a\nconstant. Solve your equation (exponentiate ip^2t), take inverse\nFourier transform. Play with integral transform formulas to get a nice\nform (or check Feynman-Hibbs).\n\nThere is one subtlety there. Dirac delta is not square integrable, even\nas a distribution.\n\nark\n--\n\nArkadiusz Jadczyk\nhttp://www.cassiopaea.org/quantum_future/homepage.htm\n\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 7 Jun 2004 05:54:29 -0400, perfectlyInnocent@as-if.com (Perfectly
Innocent) wrote:

>I saw the solution to the one-dimensional Schrodinger equation for
>"free particles" long ago but have forgotten it. No one at sci.physics
>could answer the question. What's the wave function of a particle in
>one spatial dimension for t>0 assuming that at t=0 the particle is at
>x=0 with 100% probability. I require that the scalar potential V(x)=0.

Take the Fourier transform in x. Laplacian becomes p^2, \delta becomes a
constant. Solve your equation (exponentiate ip^2t), take inverse
Fourier transform. Play with integral transform formulas to get a nice
form (or check Feynman-Hibbs).

There is one subtlety there. Dirac \delta is not square integrable, even
as a distribution.

ark
--

Arkadiusz Jadczyk
http://www.cassiopaea.org/quantum_future/homepage.htm

--

Perfectly Innocent

Jun12-04, 07:19 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>Perfectly Innocent wrote in message news:<c45b45b3.0406041857.704fed0d@posting.google. com>...\n> What\'s the wave function of a particle in\n> one spatial dimension for t>0 assuming that at t=0 the particle is at\n> x=0 with 100% probability. I require that the scalar potential V(x)=0.\n\nGentleman,\n\nThank you for your kind and authoritative responses. I hope you don\'t\nmind answering a follow-up question. If I had presupposed initially\nthat 3-space was a hypersphere and that I was interested in the same\none-dimensional problem defined on a great circle, what would the wave\nfunction be?\n\nEugene Shubert\nhttp://www.everythingimportant.org\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>Perfectly Innocent wrote in message news:<c45b45b3.0406041857.704fed0d@posting.google.com>...
> What's the wave function of a particle in
> one spatial dimension for t>0 assuming that at t=0 the particle is at
> x=0 with 100% probability. I require that the scalar potential V(x)=0.

Gentleman,

Thank you for your kind and authoritative responses. I hope you don't
mind answering a follow-up question. If I had presupposed initially
that 3-space was a hypersphere and that I was interested in the same
one-dimensional problem defined on a great circle, what would the wave
function be?

Eugene Shubert
http://www.everythingimportant.org