{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 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helv etica" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "T imes" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } } {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 256 19 "Spherical Harmonics" }} {PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 74 "In th is worksheet, you will examine the graphs of the spherical harmonics \+ " }{XPPEDIT 18 0 "Y[l,m](theta,phi)" "6#-&%\"YG6$%\"lG%\"mG6$%&thetaG% $phiG" }{TEXT -1 110 ". Since the spherical harmonics are complex fun ctions of the angular coordinates, it is not possible to plot " } {XPPEDIT 18 0 "Y[l,m](theta,phi)" "6#-&%\"YG6$%\"lG%\"mG6$%&thetaG%$ph iG" }{TEXT -1 32 " itself. Instead, we will plot " }{XPPEDIT 18 0 "ab s(Y[l,m](theta,phi))^2 " "6#*$-%$absG6#-&%\"YG6$%\"lG%\"mG6$%&thetaG%$ phiG\"\"#" }{TEXT -1 324 "as a function of the angular coordinates. T he square of the norm give the probability density in quantum mechanic s. NOTE: By looking only at the square of the norm in this initial w orksheet, we are throwing away information about the phase. We will e xplore the physical importance of the phase in a subsequent worksheet. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:with(plots):with(orthop oly):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "numpts:=8000;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "assume(phi,real);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Choose a value of " }{XPPEDIT 18 0 "l" "6#%\"lG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "m" "6#%\"mG" } {TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "el:=1;m:=0 ;" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 72 "Each spherical harmonic is the product of a real normalization constant " }{XPPEDIT 18 0 "n[l,m] " "6#&%\"nG6$%\"lG%\"mG" }{TEXT -1 40 ", a (real) Associated Legendre function " }{XPPEDIT 18 0 "P[l,m](theta)" "6#-&%\"PG6$%\"lG%\"mG6#%&t hetaG" }{TEXT -1 28 ", and a complex exponential " }{XPPEDIT 18 0 "exp (i*m*phi)" "6#-%$expG6#*(%\"iG\"\"\"%\"mGF(%$phiGF(" }{TEXT -1 28 ". \+ (What will happen to the " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 80 " dependence when we take the square of the norm of a single spheri cal harmonic?)" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 58 "First, find the value of the Associated Legendre f unction:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "p[el,m]:=(1-z^2)^(abs(m )/2)\n*diff(P(el,z),[z$abs(m)]);" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 42 "Now change variables from z to cos(theta):" }}{PARA 259 "> " 0 "" {MPLTEXT 1 0 42 "Ptheta[el,m]:=subs(z=cos(theta),\np[el,m]);" }}} {EXCHG {PARA 259 "" 0 "" {TEXT -1 43 "What does this Legendre function look like?" }}{PARA 259 "> " 0 "" {MPLTEXT 1 0 43 "plot(Ptheta[el,m], theta=0..Pi,\naxes=BOXED);" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 34 "N ormalize the spherical harmonics:" }}{PARA 259 "> " 0 "" {MPLTEXT 1 0 98 "nrm[el,m]:=(-1)^((m+abs(m))/2)\n*sqrt((2*el+1)\n*factorial(el-abs( m))\n/(4*Pi*factorial(el+abs(m))));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "\nThe spherical harmonic " }{XPPEDIT 18 0 "Y[l,m](theta,phi);" "6#-&%\"YG6$%\"lG%\"mG6$%&thetaG%$phiG" }{TEXT -1 9 " is then:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Y[el,m]:=nrm[el,m]*Ptheta[el,m]*exp (I*m*phi);" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 51 "\nThe square of t he norm of the spherical harmonic, " }{XPPEDIT 18 0 "abs(Y[l,m](theta, phi))^2;" "6#*$-%$absG6#-&%\"YG6$%\"lG%\"mG6$%&thetaG%$phiG\"\"#" } {TEXT -1 4 " is:" }}{PARA 259 "> " 0 "" {MPLTEXT 1 0 45 "Ysq[el,m]:=ev alc(conjugate(Y[el,m])*Y[el,m]);" }}}{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Plot the square of the norm of the spherical harmonic " } {XPPEDIT 18 0 "abs(Y[l,m](theta, phi))^2" "6#*$-%$absG6#-&%\"YG6$%\"lG %\"mG6$%&thetaG%$phiG\"\"#" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 108 "plot3d(Ysq[el,m],\nphi=0..2*Pi,theta=0..Pi,\ncolor =Ysq[el,m],\nstyle=patchnogrid,\nnumpoints=numpts,axes=boxed);" }} {PARA 0 "" 0 "" {TEXT -1 111 "Notice the color function. The color is determined by the value of the function (blue is large, red is small) ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "Now let's look at a \"flat\" graph, where the value of the functi on is shown just with color, not with \"height\"." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 118 "plot3d(1,phi=0..2*Pi,theta=0..Pi,\ncolor=Ys q[el,m],\nnumpoints=numpts,axes=boxed,\nstyle=patchnogrid,orientation= [90,0]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "But we do not want to think of " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 188 " as Cartesian coordinate s, but rather as angular coordinates on a sphere, so let's wrap up (\" morph\") this color plot of the square of the norm of the spherical ha rmonic onto a sphere plot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "sphereplot(1, \nphi=0..2*Pi, theta=0..Pi, \ncolor=Ysq[el,m],\nnum points=numpts,style=patchnogrid,\nscaling=constrained, axes=boxed);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 266 "Often, in textbooks, you will s ee the spherical harmonics (or the squares of the norms of the spheric al harmonics plotted on a sphereplot where the radial direction is use d to plot the value of the function. The resulting pictures look much like electron orbitals, " }{TEXT 257 59 "BUT THESE PICTURES (BELOW) A RE NOT THE ELECTRON ORBITALS. " }{TEXT -1 411 "Not until we have incl uded the radial dependence in the wave function will you truly be able to see the probability distribution as a function of all three spatia l coordinates. You should think of the spherical harmonics as functio ns on the sphere. However, the code below will show you the squares o f the norms of the spherical harmonics plotted as a sphereplot, just s o you can see what many textbooks show." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "sphereplot(Ysq[el,m], \nphi=0..2*Pi, theta=0..Pi, \n color=Ysq[el,m],\nnumpoints=numpts,style=patchnogrid,\nscaling=constra ined, axes=boxed);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Now try other values of " } {XPPEDIT 18 0 "l" "6#%\"lG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "m" "6# %\"mG" }{TEXT -1 1 "." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }