��<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0//EN"> <HEAD> <TITLE> The Asymptotic Limits to the Quantum Theoretic Probability Distributions for a Harmonic Oscillator </TITLE> <SCRIPT language=javascript src=""></SCRIPT> <STYLE>BODY { BACKGROUND: #aaffdd; MARGIN-LEFT: 0.2in; COLOR: #000000; MARGIN-RIGHT: 0.2in } H2 { FONT-SIZE: 18pt; COLOR: #ff0000; TEXT-ALIGN: center } H4 {margin-left:2in; margin-right:2in; text-align:center; font-size:20pt; color:#000000; background:#FFFF00; padding:20px; border-style:outset; border-color:#FF0000; border-width:1px;} LI { BORDER-RIGHT: white 5px outset; PADDING-RIGHT: 20px; BORDER-TOP: white 5px outset; PADDING-LEFT: 20px; FONT-SIZE: 18pt; BACKGROUND: #ddffff; PADDING-BOTTOM: 20px; BORDER-LEFT: white 5px outset; TEXT-INDENT: 0.2in; LINE-HEIGHT: 1.2em; PADDING-TOP: 20px; BORDER-BOTTOM: white 5px outset; text-weight: 400 } UL { BORDER-TOP-WIDTH: 20px; PADDING-RIGHT: 20px; PADDING-LEFT: 20px; BORDER-LEFT-WIDTH: 20px; BORDER-LEFT-COLOR: black; BACKGROUND: #aaffff; BORDER-BOTTOM-WIDTH: 20px; BORDER-BOTTOM-COLOR: black; PADDING-BOTTOM: 20px; BORDER-TOP-COLOR: black; PADDING-TOP: 20px; BORDER-RIGHT-WIDTH: 20px; BORDER-RIGHT-COLOR: black; spacing: 20px } TH { FONT-SIZE: 16pt } TD { FONT-SIZE: 16pt; TEXT-ALIGN: LEFT; spacing: 20px } P { FONT-SIZE: 22pt } BLOCKQUOTE { FONT-SIZE: 24pt; COLOR: #ff00ff } </STYLE> <META content="MSHTML 6.00.2800.1601" name=GENERATOR></HEAD> <BODY vLink=#000000 link=#ff0000 bgColor=#9acd32> <CENTER> <TABLE cellPadding=15 bgColor=#0000aa border=9 frame=box> <TBODY> <TR> <TH><FONT color=#ffff00 size=5>San Jos&eacute; State University</FONT> </B></TH></TR></TBODY></TABLE> </CENTER><p> <CENTER> <TABLE cellPadding=15 bgColor=#00ffff border=9 frame=box> <TBODY> <TR> <TH><FONT color=#880000 size=5>applet-magic.com<BR><FONT color=#880000 size=4><I>Thayer Watkins</I><BR>Silicon Valley<BR>&amp; Tornado Alley<BR>USA</FONT></B></FONT></TH></TR></TBODY></TABLE></CENTER> <P><CENTER><TABLE BORDER=9 FRAME=box BGCOLOR=#0000aa CELLPADDING=10><th> <CENTER><FONT SIZE="5" COLOR=#FFFFFF><b> The Asymptotic Limits to the Quantum Theoretic<br> Probability Distributions for a Harmonic Oscillator </b></FONT></CENTER> </th></TABLE> </CENTER> <b><p> A harmonic oscillator is a device for which the restoring force on a particle mass is proportional to its displacement from equilibrium; i.e., <h4> F = &minus;kx <br> and thus<br> m(d&sup2;x/dt&sup2;) = &minus;kx </h4> <p>where m is the mass of the particle and k is a constant, usually called the stiffness coefficient. <p>The potential energy function is then V(x)=&frac12;kx&sup2;. <p>The Hamiltonian H for the harmonic oscillator is then <h4> H = &frac12;p&sup2/m + &frac12;kx&sup2; </h4> <p>where p is the momentum of the particle. <p>This means that the Hamiltonian operator for a harmonic oscillator is <h4> H^&phi; = &minus;(<s>h</s>&sup2;/2m)(d&sup2&phi;/dt&sup2;) + &frac12;kx&sup2;&phi; <br>and thus the <br>time independent<br>Schr&ouml;dinger equation is<br> &minus;(<s>h</s>&sup2;/2m)(d&sup2&phi;/dt&sup2;) + &frac12;kx&sup2;&phi; = E&phi; </h4> <p>where &phi; is the wave function and <s>h</s> is Planck's constant. <p>The energy E is an eigenvalue of the equation and is equal to (n+&frac12;)<s>h</s>. <p>The wave function is a complex-valued function such that its squared value is the probability density. <p>The solutions give the probability density functions in terms of the dimensionless variable &zeta;=x/&sigma; <h4> &phi;<sub>n</sub>&sup2;(&zeta;) = (1/(2<sup>n</sup>n!&radic;&pi;)<i>H</i><sub>n</sub>&sup2;(&zeta;)exp(&minus;&zeta;&sup2;) </h4> <p>where <i>H</i><sub>n</sub>(&zeta;) is the Hermite polynomial of order n. <p>The probability density function in terms of the displacement x is then given by <h4> P<sub>n</sub>(x) = &phi;<sub>n</sub>&sup2;(x/&sigma;)/&sigma; </h4> <p>where <h4> &sigma;&sup2; = <strike>h</strike>&omega;/k = <strike>h</strike>/(m&omega;) <br>where the frequency &omega; is<br> &omega; = (k/m)<sup>&frac12;</sup> </h4> <p>It can be shown that in the limit as n&rarr;&infin; the squared values of the Hermite polynomials <i>H</i><sub>n</sub>&sup2; approach <h4> 2(2n/e)<sup>n</sup>cos&sup2;(x(2n)<sup>&frac12;</sup>&minus;n&pi;/2)exp(x&sup2;)/(1&minus;x&sup2;/2n)<sup>&frac12;</sup> </h4> <p>where e=2.7218.... <p>This means that the probability density functions for the harmonic oscillators asymptotically approach <h4> (&phi;(x))&sup2; = 2(1/(2<sup>n</sup>n!&radic;&pi;))(2n/e)<sup>n</sup>cos&sup2;(x(2n)<sup>&frac12;</sup>&minus;n&pi;/2)/(1&minus;x&sup2;/2n)<sup>&frac12;</sup> </h4> <p>The average of cos&sup2;(z) is 1/2. So the spatial average of (&phi;(x))&sup2; is essentially <h4> (&Phi;(x))&sup2; = (1/(2<sup>n</sup>n!&radic;&pi;))(2n/e)<sup>n</sup>/(1&minus;x&sup2;/2n)<sup>&frac12;</sup> </h4> <p>The classical time-spent probability density function for a hamrmonic oscillator is <h4> P(x) = 1/(&pi;(x<sub>max</sub>&sup2; &minus; x&sup2;)<sup>&frac12;</sup><br> = (x<sub>max</sub>&pi;)/(1 &minus; &frac12;(x/x<sub>max</sub>)&sup2;)<sup>&frac12;</sup> </h4> <p>where x<sub>max</sub> is the maximum deviation. The energy of the oscillator is equal to &frac12;kx<sub>max</sub>&sup2;. Therefore x<sub>max</sub>=(2E/k)<sup>&frac12;</sup>. By the appropriate choice of units P(x) can be made proportional to (&Phi;(x))&sup2;. For probability distributions constant factors do not matter because they cancel out in the normalization process. <p>The conclusion is then therefore that the spatial average of the quantum theoretic probability distribution for a harmonic oscillator is asymptotically equal to the classical time-spent probability distribution for such an oscillator. ` ` <HR> <CENTER> <TABLE cellPadding=20 bgColor=#00ff00 border=9 frame=box> <TBODY> <TR> <TD> <CENTER><A href="http://www.applet-magic.com/index.html">HOME PAGE OF <I>applet-magic</I> <BR><A href="http://www.sjsu.edu/faculty/watkins/watkins.htm">HOME PAGE OF <I>Thayer Watkins</I></A>, </CENTER></TD></TR></TBODY></TABLE></CENTER> </body> </html> <!-- text below generated by server. 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