{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 0 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 18 0 0 0 0 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 18 0 0 0 0 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 3" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 0 1 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 4 " -1 259 1 {CSTYLE "" -1 -1 "Times" 1 24 0 0 0 0 1 1 2 0 0 0 0 0 0 1 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 260 "" 0 "" {TEXT 256 20 "Radial Wavefunctions" } }{PARA 261 "" 0 "" {TEXT -1 0 "" }{TEXT 257 35 "by Corinne Manogue and Kerry Browne" }}{PARA 262 "" 0 "" {TEXT -1 30 "Copyright 2001 Corinne Manogue" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "In this worksheet, you will explore the graphs of the radial wavef unctions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 36 "restart:with(plots):with(orthopoly);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "The orthopoly package contains commands \+ involving systems of orthogonal polynomials, including the Laguerre po lynomials " }{XPPEDIT 18 0 "L[q](x)" "6#-&%\"LG6#%\"qG6#%\"xG" }{TEXT -1 2 ".\n" }}{PARA 0 "" 0 "" {TEXT -1 23 "There is a factor of Z/" } {XPPEDIT 18 0 "a[0]" "6#&%\"aG6#\"\"!" }{TEXT -1 109 " in most of the \+ terms defining the radial function. \nWe'll start by setting this to \+ some constant, B, below:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "B:=1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "We ca n now define the orbital quantum number, el, and the radial quantum nu mber, n:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:=1;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "el:=0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Before we calculate the radial wavefunction, \nlet's see what the Laguerre factor in it looks like:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 68 "Laguerre[n+el,2*el+1]:=\nsimplify(diff(n!*L(n+ el,rho),rho$(2*el+1)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " plot(Laguerre[n+el,2*el+1],rho=0..10*n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "This is written with the variable " }{XPPEDIT 18 0 "rho; " "6#%$rhoG" }{TEXT -1 123 ",\nwhich is the dimensionless parameter we used in class. \nWe need to transform back to obtain the correct dep endence on r:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Lr[n+el,2* el+1]:=subs(rho=2*B*r/n,\nLaguerre[n+el,2*el+1]);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 30 "The normalization is given by:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "N[n+el,2*el+1]:=-sqrt((2*B/n)^3*(n-el-1)!/ \n(2*n*((n+el)!)^3));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Now com es the time to put it all into the radial wavefunction, normalizations , asymptotic values and all:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "R[n+el,2*el+1]:=\nN[n+el,2*el+1]*(2*B*r/n)^el*\nexp(-B*r/n)*Lr[n +el,2*el+1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(R[n+el ,2*el+1],r=0..12*n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "The proba bility density is given by " }{XPPEDIT 18 0 "abs(R[n,el])^2;" "6#*$-%$ absG6#&%\"RG6$%\"nG%#elG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot((abs(R[n+el,2*el+1]))^2,r=0..12*n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "\nFor a given value of " }{XPPEDIT 18 0 "n;" "6#%\"n G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "l;" "6#%\"lG" }{TEXT -1 24 ", b ut for all values of " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 177 ", \+ it turns out that the\nangular dependence of an energy eigenstate is c onstant. Therefore, if we\nwant to know the probability of finding th e particle in a thin shell of radius " }{XPPEDIT 18 0 "r;" "6#%\"rG" } {TEXT -1 79 ", \nwe can just multiply the probability density above by the area of the shell:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 " plot(4*Pi*r^2*(abs(R[n+el,2*el+1]))^2,r=0..12*n);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Noti ce what the extra factors of " }{XPPEDIT 18 0 "r;" "6#%\"rG" }{TEXT -1 18 " do at the origin." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 61 "Go back to the top of the worksheet and v ary the parameters, " }{XPPEDIT 18 0 "l;" "6#%\"lG" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 23 ", \nto see what happens. " }}}}{MARK "20 0 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }