<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN"> <HTML><HEAD> <META http-equiv=Content-Type content="text/html; charset=utf-8"> <META content="MSHTML 6.00.6000.16788" name=GENERATOR> <STYLE></STYLE> </HEAD> <BODY style="WORD-WRAP: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space" bgColor=#ffffff> <DIV><FONT face="Courier New" size=2></FONT> </DIV> <BLOCKQUOTE style="PADDING-RIGHT: 0px; PADDING-LEFT: 5px; MARGIN-LEFT: 5px; BORDER-LEFT: #000000 2px solid; MARGIN-RIGHT: 0px"> <DIV style="FONT: 10pt arial">----- Original Message ----- </DIV> <DIV style="BACKGROUND: #e4e4e4; FONT: 10pt arial; font-color: black"><B>From:</B> <A title=gregory.woodhouse@gmail.com href="mailto:gregory.woodhouse@gmail.com">Gregory Woodhouse</A> </DIV> <DIV style="FONT: 10pt arial"><B>To:</B> <A title=plt-scheme@list.cs.brown.edu href="mailto:plt-scheme@list.cs.brown.edu">PLT List</A> </DIV> <DIV style="FONT: 10pt arial"><B>Sent:</B> Thursday, January 01, 2009 8:33 PM</DIV> <DIV style="FONT: 10pt arial"><B>Subject:</B> [plt-scheme] Currying and physics</DIV> <DIV><BR></DIV> <DIV>It seems to me that operators actually provide a more direct analog to currying than the (classical) fields discussed in that note. In elementary quantum mechanics, a particle is represented by a wave function (a complex valuied function of time and position). Observable quantities (or just observables) correspond to (linear) operators on the space of so-called wave functions. For example, in one dimension, position corresponds to i h bar (the imaginary unit times Planck's constant divided 2 pi) times differentiation with respect to x). In LaTeX, that's</DIV></BLOCKQUOTE> <DIV><FONT face="Courier New" size=2>You probably mean (linear) momentum. Position can be represented by an operator (function, functional) Ψ -> xΨ. The probability to find the particle at position x at time t is:</FONT></DIV> <DIV><FONT face="Courier New" size=2>integral over x of Ψ*(x,t) x Ψ(x,t) divided by</FONT></DIV> <DIV><FONT face="Courier New" size=2>the integral over x of Ψ*(x,t) Ψ(x,t)</FONT><FONT face="Courier New"><FONT size=2>,</FONT></FONT></DIV> <DIV><FONT face="Courier New"><FONT size=2>where Ψ*</FONT></FONT><FONT face="Courier New"><FONT size=2> is the complex conjugate of Ψ.</DIV></FONT></FONT> <DIV><FONT face="Courier New" size=2>In this case momentum is represented by the function Ψ -> (iħ/2π)(δΨ/δx).</FONT></DIV> <DIV><FONT face="Courier New" size=2>You could have choosen Ψ(x)=1/(1+x^2) as a function with finite norm,</FONT></DIV> <DIV><FONT face="Courier New" size=2>or in three dimensions 1/(1+x^2+y^2+z^2)</FONT></DIV> <DIV><FONT face="Courier New" size=2>In practice wave functions are often represented by time independent vectors (called kets) in a Hilbert space.</FONT></DIV> <DIV><FONT face="Courier New" size=2>Which functions Ψ</FONT><FONT face="Courier New"><FONT size=2> are to be included in this space is determined by the law of conservation of energy. In quantum mechanics this law says: HΨ=EΨ, where H is the so called Hamiltonian (an operator representing energy) and E a real number. The equation must be solved for both Ψ and E (and the solution usually consists of an infinite number of pairs Ψ and E) By using the symmetry properties of the system being studied, many parts of the integrals can be simplified to summations.</DIV></FONT></FONT> <DIV><FONT face="Courier New" size=2>Jos</FONT></DIV> <DIV><FONT face="Courier New"><FONT size=2><FONT face="Courier New" size=2></FONT></FONT></FONT> </DIV> <DIV><FONT face="Courier New"><FONT size=2><FONT face="Courier New" size=2></FONT></FONT></FONT> </DIV> <DIV><FONT face="Courier New"><FONT size=2><FONT face="Courier New" size=2></FONT></FONT></FONT> </DIV> <DIV><FONT face="Courier New"><FONT size=2><FONT face="Courier New" size=2></FONT></FONT></FONT> </DIV> <BLOCKQUOTE style="PADDING-RIGHT: 0px; PADDING-LEFT: 5px; MARGIN-LEFT: 5px; BORDER-LEFT: #000000 2px solid; MARGIN-RIGHT: 0px"> <DIV> </DIV> <DIV><BR></DIV> <DIV><SPAN class=Apple-style-span style="FONT-SIZE: 11px; FONT-FAMILY: 'Lucida Grande'; -webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px">P = i\hbar \frac{\partial}{\partial x}</SPAN></DIV> <DIV><FONT class=Apple-style-span face="'Lucida Grande'" size=3><SPAN class=Apple-style-span style="FONT-SIZE: 11px; -webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px"><BR></SPAN></FONT></DIV> <DIV><FONT class=Apple-style-span face="'Lucida Grande'" size=3><SPAN class=Apple-style-span style="FONT-SIZE: 11px; -webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px">So, if psi (the letter traditionally used to represent wave functions) is x^2, then Px is 2i \hbar x (never mind the fact that it isn't square integrable). So, if you think of the probability density for position being function of both the observable (in this case, position) and the quantum state, you take the first input variable (the observable) and generate a function (or, as some people like to say, functional) that can be applied to to the wave function to give you a new function (this time of the interval over which you are integrating), then you take the integral (another function!) to get the expected position. Without that last step, you get yet another function the norm of which is the probability density of position.</SPAN></FONT></DIV> <P> <HR> <P></P>_________________________________________________<BR>  For list-related administrative tasks:<BR>  http://list.cs.brown.edu/mailman/listinfo/plt-scheme<BR></BLOCKQUOTE></BODY></HTML>