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We would need to look at, e.g., psi an eigenstate of L_z, so t he right-hand side is sharp, and we know psi." }}{PARA 0 "" 0 "" {TEXT -1 32 "Given that psi we can calculate:" }}{PARA 0 "" 0 "" {TEXT -1 48 "a) the time evolution of the expectation values;" }} {PARA 0 "" 0 "" {TEXT -1 62 "b) the average value of L_x, and L_y, and their uncertainties;" }}{PARA 0 "" 0 "" {TEXT -1 54 "c) check the abo ve relationship for the uncertainties." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 61 "We need the spherical harmonics (eige nstates of L^2 and L_z)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "The a ngular parts are spherical harmonics.:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "restart; with(orthopoly);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(%\"GG%\"HG%\"LG%\"PG%\"TG%\"UG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 53 "Plm:=proc(theta,l::nonnegint,m::integer) local x,y,f;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "x:=cos(theta);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "if m>0 then f:=subs(y=x,diff(P(l,y),y$m)) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "else f:=subs(y=x,P(l,y)); fi; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "(-1)^m*sin(theta)^m*f; end:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Plm(theta,3,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&)-%$sinG6#%&thetaG\"\"#\"\"\"-%$cosGF(F+ \"#:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "For the spherical harmoni cs we don't need the Plm's with negative argument." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Y:=proc(theta,phi,l::nonnegint,m::integer) \+ local m1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "m1:=abs(m); if m1>l th en RETURN(\"|m\} has to be <= l for Y_lm\"); fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "exp(I*m*phi)*Plm(theta,l,m1)*(-1)^m*sqrt((2*l+1)*(l-m 1)!/(4*Pi*(l+m1)!)); end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Y(theta,phi,3,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&,&*$)-% $cosG6#%&thetaG\"\"$\"\"\"#\"\"&\"\"#*&#F-F1F.F)F.!\"\"F.-%%sqrtG6#\" \"(F.F.*$-F66#%#PiGF.F4#F.F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Y(theta,phi,3,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&**-%$ex pG6#*&^#\"\"\"F+%$phiGF+F+-%$sinG6#%&thetaGF+,&*$)-%$cosGF/\"\"#F+#\"# :F6#\"\"$F6!\"\"F+-%%sqrtG6#\"#@F+F+*$-F=6#%#PiGF+F;#F+\"#7" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Y(theta,phi,3,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&**-%$expG6#*&^#\"\"#\"\"\"%$phiGF,F,)-%$ sinG6#%&thetaGF+F,-%$cosGF1F,-%%sqrtG6#\"$5#F,F,*$-F66#%#PiGF,!\"\"#F, \"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Y(theta,phi,3,3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*(-%$expG6#*&^#\"\"$\"\"\"%$ph iGF,F,)-%$sinG6#%&thetaGF+F,-%%sqrtG6#\"#NF,F,*$-F46#%#PiGF,!\"\"#F,\" \")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "No:=(l,m)->int(int( evalc(Y(theta,phi,l,m)*conjugate(Y(theta,phi,l,m))),phi=0..2*Pi)*sin(t heta),theta=0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NoGR6$%\"lG% \"mG6\"6$%)operatorG%&arrowGF)-%$intG6$*&-F.6$-%&evalcG6#*&-%\"YG6&%&t hetaG%$phiG9$9%\"\"\"-%*conjugateG6#F7F>/F;;\"\"!,$%#PiG\"\"#F>-%$sinG 6#F:F>/F:;FDFFF)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "No(1 ,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "No(1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "No(3,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Y(theta,phi,0,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$*&\"\"\"F%*$-%%sqrtG6#%#PiGF%!\"\"#F%\"\"#" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 264 48 "Uncertainty in L_x and L_y for an L_z eigenstate" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 147 "Suppose \+ psi is the state with 5 units of h_ (h-bar) measured for L_z. Let us a lso specify that it is the state with L^2 eigenvalue of 5*(5+1) h_^2. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "psi:=Y(theta,phi,5,5); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiG,$*&*(-%$expG6#*&^#\"\"&\" \"\"%$phiGF.F.)-%$sinG6#%&thetaGF-F.-%%sqrtG6#\"#xF.F.*$-F66#%#PiGF.! \"\"#\"\"$\"#K" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "What are the ex pectation values of L_x and L_y?" }}{PARA 0 "" 0 "" {TEXT -1 207 "To a nswer this question we need to either express L_x and L_y in spherical polar coordinates, and calculate the expectation in these coordinates , or we switch the eigenfunction over to Cartesian coordinates." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "LzOP:=f->-I*h_*simplify(diff (f,phi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%LzOPGR6#%\"fG6\"6$%)op eratorG%&arrowGF(*(^#!\"\"\"\"\"%#h_GF/-%)simplifyG6#-%%diffG6$9$%$phi GF/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "LzOP(psi)/psi; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&%#h_G\"\"\"),&!\"\"F'*$)-%$c osG6#%&thetaG\"\"#F'F'F1F'F'*$)-%$sinGF/\"\"%F'F*\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(LzOP(psi)/psi,trig);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#h_G\"\"&" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 125 "We have verified that the state psi has eigenvalue 5 h -bar for the L_z operator. Let us define the operators for L_x and L_y :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "LxOP:=f->I*h_*simplify (sin(phi)*diff(f,theta)+cos(phi)/tan(theta)*diff(f,phi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%LxOPGR6#%\"fG6\"6$%)operatorG%&arrowGF(*(^# \"\"\"F.%#h_GF.-%)simplifyG6#,&*&-%$sinG6#%$phiGF.-%%diffG6$9$%&thetaG F.F.*&*&-%$cosGF7F.-F:6$F " 0 "" {MPLTEXT 1 0 80 "LyOP:=f->I*h_*simplify(-cos(phi)*di ff(f,theta)+sin(phi)/tan(theta)*diff(f,phi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%LyOPGR6#%\"fG6\"6$%)operatorG%&arrowGF(*(^#\"\"\"F.% #h_GF.-%)simplifyG6#,&*&-%$cosG6#%$phiGF.-%%diffG6$9$%&thetaGF.!\"\"*& *&-%$sinGF7F.-F:6$FF.F.F(F(F(" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 149 "These definitions follow from a straightforward a pplication of the transformation from Cartesian to spherical polar coo rdinates of L_x, L_y, and L_z." }}{PARA 0 "" 0 "" {TEXT -1 88 "We can \+ also define the angular momentum squared operator in spherical polar c oordinates:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "L2OP:=f->-h _^2*simplify(diff(sin(theta)*diff(f,theta),theta)/sin(theta)+diff(f,ph i$2)/sin(theta)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%L2OPGR6#%\"f G6\"6$%)operatorG%&arrowGF(,$*&)%#h_G\"\"#\"\"\"-%)simplifyG6#,&*&-%%d iffG6$*&-%$sinG6#%&thetaGF1-F86$9$F>F1F>F1F;!\"\"F1*&-F86$FA-%\"$G6$%$ phiGF0F1*$)F;F0F1FBF1F1FBF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "L2OP(psi)/psi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&)%#h_G \"\"#\"\"\"),&!\"\"F)*$)-%$cosG6#%&thetaGF(F)F)F(F)F)*$)-%$sinGF1\"\"% F)F,\"#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%#h_G\"\"#\"\"\"\"#I" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "This is consistent with 5*(5+1)*h_^2. We can verify that the result from L2OP is consistent with the result ca lculated from Lx^2+Ly^2+Lz^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "L2OPlong:=f->LxOP(LxOP(f))+LyOP(LyOP(f))+LzOP(LzOP(f));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%)L2OPlongGR6#%\"fG6\"6$%)operatorG%& arrowGF(,(-%%LxOPG6#-F.6#9$\"\"\"-%%LyOPG6#-F5F1F3-%%LzOPG6#-F9F1F3F(F (F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "L2OPlong(psi)/psi;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&*(,(*&*.^##\"#:\"#K\"\"\")%#h_G \"\"#F--%$expG6#*&^#\"\"&F-%$phiGF-F--%%sqrtG6#\"#xF-,4*&^#\"\"'F-)-%$ cosG6#%&thetaGF0F-F-*&^#!\"&F-)FA\"\"%F-F-*(^#!#5F-)-FB6#F7F0F-F@F-F-* (^#\"\"*F-FMF-FHF-F-^#!\"\"F-*&^#F-F-FMF-F-**\"#5F-FNF-F@F--%$sinGFOF- FT**FRF-FNF-FHF-FYF-F-*&FYF-FNF-F-F--FZFCF-F-*$-F96#%#PiGF-FTF-*&*.^## !#:F,F-F.F-F1F-F8F-,2FJF-FPF-FUF-**FXF-FNF-F@F-FYF-FT**FRF-FNF-FHF-FYF -F-FfnF-*&^#FIF-F@F-F-*&^#!\"%F-FHF-F-F-FgnF-F-*$-F96#F[oF-FTF-*&*.#\" #vF,F-F.F-F1F-),&FTF-*$F@F-F-F0F-FgnF-F8F-F-*$-F96#F[oF-FTF-F-F8F--F96 #F[oF-F-*&F1F-)FgnF6F-FT#F,\"$J#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$) %#h_G\"\"#\"\"\"\"#I" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Let us ve rify that the state psi is not an eigenstate of L_x, or L_y:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "LxOP(psi)/psi;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*&*,^#\"\"&\"\"\"%#h_GF'),&!\"\"F'*$)-%$cosG6#%& thetaG\"\"#F'F'F2F'F.F',&-%$sinG6#%$phiGF'*&^#F'F'-F/F6F'F'F'F'*$)-F5F 0F&F'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&**^#\"\"&\"\"\",&-%$sinG6#%$phiGF'*& ^#F'F'-%$cosGF+F'F'F'-F06#%&thetaGF'%#h_GF'F'-F*F2!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "Clearly this is a function of theta and \+ phi! Therefore L_x |psi> is not proportional to |psi>. A similar resul t is obtained for L_y." }}{PARA 0 "" 0 "" {TEXT -1 103 "What are the a verage values (expectation values) for L_x and L_y while the particle \+ is in state |psi> ?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "ExpV _Lx:=f->int(int(evalc(conjugate(f)*LxOP(f)),phi=0..2*Pi)*sin(theta),th eta=0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(ExpV_LxGR6#%\"fG6\"6 $%)operatorG%&arrowGF(-%$intG6$*&-F-6$-%&evalcG6#*&-%*conjugateG6#9$\" \"\"-%%LxOPGF8F:/%$phiG;\"\"!,$%#PiG\"\"#F:-%$sinG6#%&thetaGF:/FG;F@FB F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ExpV_Lx(psi);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "ExpV_Ly:=f->int(int(evalc(conjugate(f)*LyOP(f)),phi=0 ..2*Pi)*sin(theta),theta=0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% (ExpV_LyGR6#%\"fG6\"6$%)operatorG%&arrowGF(-%$intG6$*&-F-6$-%&evalcG6# *&-%*conjugateG6#9$\"\"\"-%%LyOPGF8F:/%$phiG;\"\"!,$%#PiG\"\"#F:-%$sin G6#%&thetaGF:/FG;F@FBF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ExpV_Ly(psi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "On average the particle has zero value in the components of L_x and L_y. " }}{PARA 0 "" 0 "" {TEXT -1 72 "Just to confirm our previous result for L_z obtained via the eigenvalue:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "ExpV_Lz:=f->int(int(evalc (conjugate(f)*LzOP(f)),phi=0..2*Pi)*sin(theta),theta=0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(ExpV_LzGR6#%\"fG6\"6$%)operatorG%&arrowGF (-%$intG6$*&-F-6$-%&evalcG6#*&-%*conjugateG6#9$\"\"\"-%%LzOPGF8F:/%$ph iG;\"\"!,$%#PiG\"\"#F:-%$sinG6#%&thetaGF:/FG;F@FBF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ExpV_Lz(psi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#h_G\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "No w we can define the uncertainties (via their squares):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "dLxsq:=f->int(int(evalc(conjugate( LxOP(f))*LxOP(f)),phi=0..2*Pi)*sin(theta),theta=0..Pi)-ExpV_Lx(f)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dLxsqGR6#%\"fG6\"6$%)operatorG%&a rrowGF(,&-%$intG6$*&-F.6$-%&evalcG6#*&-%*conjugateG6#-%%LxOPG6#9$\"\" \"F:F>/%$phiG;\"\"!,$%#PiG\"\"#F>-%$sinG6#%&thetaGF>/FI;FBFDF>*$)-%(Ex pV_LxGF!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " dA:=sqrt(dLxsq(psi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dAG,$*&-%% sqrtG6#\"#5\"\"\"-F(6#*$)%#h_G\"\"#F+F+#F+F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "dLysq:=f->int(int(evalc(conjugate(LyOP(f))*LyOP (f)),phi=0..2*Pi)*sin(theta),theta=0..Pi)-ExpV_Ly(f)^2;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&dLysqGR6#%\"fG6\"6$%)operatorG%&arrowGF(,&-%$ intG6$*&-F.6$-%&evalcG6#*&-%*conjugateG6#-%%LyOPG6#9$\"\"\"F:F>/%$phiG ;\"\"!,$%#PiG\"\"#F>-%$sinG6#%&thetaGF>/FI;FBFDF>*$)-%(ExpV_LyGF! \"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dB:=sqrt(dLys q(psi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dBG,$*&-%%sqrtG6#\"#5\" \"\"-F(6#*$)%#h_G\"\"#F+F+#F+F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "dA*dB;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%#h_G\"\"#\"\"\" #\"\"&F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 164 "This shows that the \+ state psi has the minimum uncertainty allowed under the uncertainty pr inciple which follows from the commutation relation [L_x, L_y] = I h_ \+ L_z." }}{PARA 0 "" 0 "" {TEXT -1 323 "The RHS of the uncertainty relat ion dA*dB >= h_/2 || where C = I*h_*L_z equals the LHS for our part icular eigenstate. We notice that the state psi (5 units of angular mo mentum in the z-direction) has an avaerage value of 0 for each of the \+ components that cannot be measured exactly with an uncertainty of sqr t(10)/2 h_ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "evalf(sqrt(1 0)/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+I)Q6e\"!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "What are the uncertainties for a state wh ose L_z eigenvalue is not maximal (i.e., for which " }{TEXT 267 1 "M" }{TEXT -1 14 " is less than " }{TEXT 268 1 "L" }{TEXT -1 2 ")?" }} {PARA 0 "" 0 "" {TEXT -1 179 "Before we proceed with the calculations \+ for the L=5 multiplet, let us verify that the uncertainty (deviation) \+ is zero for the L_z component, as the system is in an L_z eigenstate: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "dLzsq:=f->int(int(eval c(conjugate(LzOP(f))*LzOP(f)),phi=0..2*Pi)*sin(theta),theta=0..Pi)-Exp V_Lz(f)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dLzsqGR6#%\"fG6\"6$%) operatorG%&arrowGF(,&-%$intG6$*&-F.6$-%&evalcG6#*&-%*conjugateG6#-%%Lz OPG6#9$\"\"\"F:F>/%$phiG;\"\"!,$%#PiG\"\"#F>-%$sinG6#%&thetaGF>/FI;FBF DF>*$)-%(ExpV_LzGF!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "dLzsq(psi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"! " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Indeed we find that for an ei genstate the uncertainty in the measurement is zero." }}{PARA 0 "" 0 " " {TEXT -1 94 "We proceed with the check of the uncertainty relation f or eigenstates of L_z with eigenvalues " }{TEXT 266 1 "M" }{TEXT -1 1 "<" }{TEXT 265 1 "L" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "psi:=Y(theta,phi,5,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiG,$*&**-%$expG6#*&^#\"\"%\"\"\"%$phiGF.F.)-%$sinG6#%&theta GF-F.-%$cosGF3F.-%%sqrtG6#\"$q(F.F.*$-F86#%#PiGF.!\"\"#\"\"$\"#K" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dA:=sqrt(dLxsq(psi));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dAG*&-%%sqrtG6#\"\"(\"\"\"-F'6#*$)% #h_G\"\"#F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#*$)%#h_G\"\"#\"\"\"F,$ \"+68vXE!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dB:=sqrt(dL ysq(psi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dBG*&-%%sqrtG6#\"\"( \"\"\"-F'6#*$)%#h_G\"\"#F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "dA*dB;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%#h_G\"\"#\"\"\"\" \"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "This is to be compared wit h the right-hand side of:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "h_/2*4*h_;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%#h_G\"\"#\"\" \"F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 288 "We conclude that the unc ertainty is much larger than for maximal L_z eigenstate! In fact, the \+ uncertainty principle is now realized not with the minimal uncertainty , but the LHS grew larger, while the RHS was smaller than for the maxi mal L_z eigenstate! Can this behaviour be generalized?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "psi:=Y(theta,phi,5,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiG,$*&**-%$expG6#*&^#\"\"$\"\"\"%$phiGF.F. )-%$sinG6#%&thetaGF-F.,&*$)-%$cosGF3\"\"#F.#\"$X*F:#\"$0\"F:!\"\"F.-%% sqrtG6#\"$&QF.F.*$-FA6#%#PiGF.F?#F.\"%!o\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 34 "sqrt(dLxsq(psi)),sqrt(dLysq(psi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&-%%sqrtG6#\"#U\"\"\"-F&6#*$)%#h_G\"\"#F)F)#F)F /F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*$-%%sqrtG6#*$)%#h_G\"\"#\"\"\"F,$\"+\\.PS K!\"*F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "psi:=Y(theta,phi ,5,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiG,$*&**-%$expG6#*&^#\" \"#\"\"\"%$phiGF.F.)-%$sinG6#%&thetaGF-F.,&*$)-%$cosGF3\"\"$F.#\"$:$F- *&#\"$0\"F-F.F8F.!\"\"F.-%%sqrtG6#\"%5BF.F.*$-FB6#%#PiGF.F@#F.\"$S)" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sqrt(dLxsq(psi)),sqrt(dLys q(psi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*&-%%sqrtG6#\"#8\"\"\"-F%6 #*$)%#h_G\"\"#F(F(F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "eval f(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*$-%%sqrtG6#*$)%#h_G\"\"#\" \"\"F,$\"+w7b0O!\"*F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ps i:=Y(theta,phi,5,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiG,$*&**- %$expG6#*&^#\"\"\"F-%$phiGF-F--%$sinG6#%&thetaGF-,(*$)-%$cosGF1\"\"%F- #\"$:$\"\")*&#\"$0\"F8F-*$)F6\"\"#F-F-!\"\"#\"#:F;F-F--%%sqrtG6#\"$I$F -F-*$-FF6#%#PiGF-FB#F-\"#g" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sqrt(dLxsq(psi)),sqrt(dLysq(psi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&-%%sqrtG6#\"#e\"\"\"-F&6#*$)%#h_G\"\"#F)F)#F)F/F#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$,$*$-%%sqrtG6#*$)%#h_G\"\"#\"\"\"F,$\"+`l)y!Q!\"*F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "psi:=Y(theta,phi,5,0);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiG,$*&*&,(*$)-%$cosG6#%&thetaG\" \"&\"\"\"#\"#j\"\")*&#\"#N\"\"%F0*$)F+\"\"$F0F0!\"\"*&#\"#:F3F0F+F0F0F 0-%%sqrtG6#\"#6F0F0*$-F@6#%#PiGF0F;#F0\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sqrt(dLxsq(psi)),sqrt(dLysq(psi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*&-%%sqrtG6#\"#:\"\"\"-F%6#*$)%#h_G\"\"#F(F(F#" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$,$*$-%%sqrtG6#*$)%#h_G\"\"#\"\"\"F,$\"+YL)H(Q!\"*F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "psi:=Y(theta,phi,5,-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiG,$*&**-%$expG6#*&^#!\"\"\"\" \"%$phiGF.F.-%$sinG6#%&thetaGF.,(*$)-%$cosGF2\"\"%F.#\"$:$\"\")*&#\"$0 \"F9F.*$)F7\"\"#F.F.F-#\"#:F " 0 "" {MPLTEXT 1 0 34 "sqrt(dLxsq(psi)),s qrt(dLysq(psi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&-%%sqrtG6#\"#e \"\"\"-F&6#*$)%#h_G\"\"#F)F)#F)F/F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*$-%%sq rtG6#*$)%#h_G\"\"#\"\"\"F,$\"+`l)y!Q!\"*F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Indeed we do find that the product of the uncertainties f or L_x and L_y grows with decreasing " }{TEXT 269 1 "M" }{TEXT -1 75 " , while the RHS of the uncertainty principle gets smaller (it vanishes for " }{TEXT 270 1 "M" }{TEXT -1 64 "=0 eigenstates of L_z). Remarkab ly, for the states with maximal " }{TEXT 275 1 "z" }{TEXT -1 33 "-proj ection (the two states with " }{TEXT 274 1 "M" }{TEXT -1 1 "=" }{TEXT 273 1 "L" }{TEXT -1 5 " and " }{TEXT 272 1 "M" }{TEXT -1 2 "=-" } {TEXT 271 1 "L" }{TEXT -1 90 ") we find that they are minimum-uncertai nty eigenstates of the angular momentum operators." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 183 "These observations are c onsistent with the diagram that shows the multiplet of angular momentu m states displayed as vectors whose endpoints lie on a sphere of radiu s squared equal to " }{TEXT 277 1 "L" }{TEXT -1 1 "(" }{TEXT 276 1 "L " }{TEXT -1 48 "+1) h_^2, with L_z component sharply defined as " } {TEXT 278 1 "M" }{TEXT -1 128 " h_, and undetermined L_x and L_y: the \+ tip of the angular momentum vector lies on a circle that intersects th e sphere at height " }{TEXT 280 2 "z " }{TEXT -1 2 "= " }{TEXT 279 1 " M" }{TEXT -1 4 " h_." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 31 "Why do the states with maximal " }{TEXT 259 1 "z" } {TEXT -1 134 "-projection have the smallest uncertainty in Lx and Ly? \+ These are the states where most of the angular momentum vector resides in the " }{TEXT 260 1 "z" }{TEXT -1 68 "-component which is known exa ctly. On the other hand in the case of " }{TEXT 281 1 "M" }{TEXT -1 77 "=0 we have the largest uncertainty, as all of the angular momentum is in the " }{TEXT 262 1 "x" }{TEXT -1 6 "- and " }{TEXT 261 1 "y" } {TEXT -1 230 "- components. Ironically, the uncertainty principle woul d allow us the smallest (zero) uncertainty for these states in princip le. This result from the generalized uncertainty principle is required so that it remains valid for the " }{TEXT 283 1 "L" }{TEXT -1 1 "=" } {TEXT 282 1 "M" }{TEXT -1 329 "=0 eigenstates (whose eigenfunction is \+ a constant). For these particular states all three components L_x, L_y , and L_z are determined to have zero eigenvalue, i.e., are known exac tly. Note that this represents the only case when [L_x, L_y] = i h_ L_ z can be satisfied, and still L_x and L_y are compatible for these ei genstates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 11 "Exercise 1:" }}{PARA 0 "" 0 "" {TEXT -1 118 "Repeat the above \+ calculations for orbital angular momentum states of a different L-valu e. What happens in the case of " }{TEXT 285 1 "L" }{TEXT -1 1 "=" } {TEXT 284 1 "M" }{TEXT -1 3 "=0?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 305 58 "Verification of the angular momen tum commutation relations" }}{PARA 0 "" 0 "" {TEXT -1 195 "We can use \+ the operators to verify some commutation relations by acting on an uns pecified function of the angles theta and phi. Watch out for the order in which the operators act on the function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "CR1:=LxOP(LyOP(f(theta,phi)))-LyOP(LxOP(f(theta, phi)))=I*h_*LzOP(f(theta,phi)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Note how evalb does not carry out a simplification: one has to be \+ careful while using it." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " evalb(CR1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalb(simplify(CR1));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "simplify(lhs(CR1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#h_G \"\"#\"\"\"-%%diffG6$-%\"fG6$%&thetaG%$phiGF/F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "The amount of required simplifications is formidable ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "CR2:=L2OP(LxOP(f(theta ,phi)))-LxOP(L2OP(f(theta,phi))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify(CR2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 263 48 "Visualization of orbital angular momentum states " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "We ca n use " }{TEXT 19 10 "sphereplot" }{TEXT -1 265 " to visualize the sph erical harmonics. We look at the density, and keep in mind that Maple \+ follows the mathematics and not the physics convention for naming the \+ angles in spherical polar coordinates. For us physicists theta is the \+ polar and phi the azimuthal angle." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "psi:=Y(theta,phi,3,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiG,$*&**-%$expG6#*&^#\"\"\"F-%$phiGF-F--%$sinG6#%& thetaGF-,&*$)-%$cosGF1\"\"#F-#\"#:F8#\"\"$F8!\"\"F--%%sqrtG6#\"#@F-F-* $-F?6#%#PiGF-F=#F-\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "r ho:=evalc(conjugate(psi)*psi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$r hoG*(,&*&*()-%$cosG6#%$phiG\"\"#\"\"\"-%$sinG6#%&thetaGF/,&*$)-F+F2F.F /#\"#:F.#\"\"$F.!\"\"F/F/%#PiGF<#\"\"(\"#[*&**F>F/)-F1F,F.F/F0F/F4F/F/ F=F " 0 "" {MPLTEXT 1 0 144 "sphere plot(rho,phi=0..2*Pi,theta=0..Pi,style=patchcontour,color=phi,axes=box ed,numpoints=3000,scaling=constrained,style=wireframe,thickness=2);\n " }}{PARA 13 "" 1 "" {GLPLOT3D 799 352 352 {PLOTDATA 3 "6(-%%MESHG6$X. 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rFfrFfrFfrFfrFfrFfrFfrFfrFfrFfrFfrFfrFfrFfrFfrFfrFfrFfrFfrFfr$\"\"\"F/ FhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrF hrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFhrFh rFhrFhrFhrFhrFhrFhrFhrFhr-%*AXESSTYLEG6#%$BOXG-%&STYLEG6#%%LINEG-%*THI CKNESSG6#\"\"#-%+AXESLABELSG6%%\"xG%\"yG%\"zG-%(SCALINGG6#%,CONSTRAINE DG" 1 6 0 1 10 2 2 1 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 286 11 "Exercise 2:" }}{PARA 0 "" 0 " " {TEXT -1 146 "Look at the visualizations of other angular momentum e igenstates. Rotate the diagram to observe the projections onto particu lar planes (e.g., the " }{TEXT 288 1 "x" }{TEXT -1 1 "-" }{TEXT 287 1 "z" }{TEXT -1 151 " plane), and compare with textbook diagrams. What i s the meaning of these diagrams in the context of the probabilistic in terpretation of wavefunctions?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Can we graph the diagram for the v isualization of the angular momentum vector? The " }{TEXT 19 9 "plotto ols" }{TEXT -1 46 " package allows us to graph spheres and cones." }} {PARA 0 "" 0 "" {TEXT -1 115 "We pick units in which h_ equals unity. \+ To visualize the knowledge available for the angular momentum vector i n an " }{TEXT 289 1 "L" }{TEXT -1 90 ">0 eigenstate we use a cone: the height of the cone is used to indicate the sharply known " }{TEXT 290 1 "z" }{TEXT -1 38 "-projection. The indeterminacy of the " } {TEXT 292 1 "x" }{TEXT -1 6 "- and " }{TEXT 291 1 "y" }{TEXT -1 591 "- components is combined with the knowledge that the average value of e ither of these components is zero, and that the deviations (uncertaint ies) are the same in both components. This makes it obvious that the L x and Ly values are determined to be somewhere on a circle of a known \+ radius (determined by the uncertainty). Unlike in classical mechanics \+ where the complete angular momentum vector is known precisely (and con served for motion in central potentials) our knowledge of this vector \+ in quantum mechanics is incomplete. The mantle of the cone visualizes \+ the possible locations of the " }{TEXT 293 1 "L" }{TEXT -1 177 " vecto r in the sense of a bundle of vectors: given a measurement of L^2 and \+ L_z (after which we are in an |LM> eigenstate) we only know that Lx an d Ly have values such that the " }{TEXT 294 1 "L" }{TEXT -1 188 " vect or lies on the mantle of the cone. Of course, a measurement of either \+ L_x or L_y would destroy the |LM> eigenstate (a subsequent remeasureme nt of L_z would lead to any of the allowed " }{TEXT 295 1 "M" }{TEXT -1 71 "-values). Therefore, this picture has to be taken with a grain \+ of salt." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "L:=5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "r:=sqrt(L*(L+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"rG*$-%%sqrtG6#\"#I\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(plottools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "s :=sphere([0,0,0],r,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "M:=4: R:=sqrt(r^2-M^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c2a:=cone([0,0,0],R,M,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "M:=2: R:=sqrt(r^2-M^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "c2b:=cone([0,0,0],R,M,color=blue):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "M:=-3: R:=sqrt(r^2-M^2):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "c2c:=cone([0,0,0],R,M,color =magenta):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "C1:=display(s 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[el7%$!+1J`C8F[vF^\\dlFi]`lFe_el7&Fg[elFi[el7%$!+**3BO7F[vFb\\dlFd^`lF i_el7&Fi[elF[\\el7%$!+#pGz9\"F[vFh\\dlF__`lF]`el7&F[\\elF]\\el7%$!+&[E 'f5F[vF^dblFj_`lFa`el7&F]\\elF_\\el7%$!+yFC8(*F^rFbdblFe``lFe`el7&F_\\ elFa\\el7%$!+22AI))F^rFjdblF`a`lFi`el7&Fa\\elFc\\el7%$!+P')>ZzF^rFbebl F[b`lF]ael7&Fc\\elFe\\el7%$!+ml " 0 "" {MPLTEXT 1 0 23 "phi1:=Y(theta,phi,5,3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%phi1G,$*&**-%$expG6#*&^#\"\"$\"\"\" %$phiGF.F.)-%$sinG6#%&thetaGF-F.,&*$)-%$cosGF3\"\"#F.#\"$X*F:#\"$0\"F: !\"\"F.-%%sqrtG6#\"$&QF.F.*$-FA6#%#PiGF.F?#F.\"%!o\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "simplify(LxOP(phi1)+I*LyOP(phi1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*.%#h_G\"\"\"-%$expG6#*&^#\"\"$F'% $phiGF'F',&!\"\"F'*$)-%$cosG6#%&thetaG\"\"#F'F'F'-%%sqrtG6#\"$&QF'F3F' ,**(^#F'F'-%$sinG6#F.F'F2F'F'*&^#F0F'F?F'F'*&-F4FAF'F2F'F'FEF0F'F'*$-F 96#%#PiGF'F0#!\"*\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "si mplify(%/Y(theta,phi,5,4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**-%% sqrtG6#\"\"#\"\"\",&*&^#F)F)-%$sinG6#%$phiGF)F)-%$cosGF/F)F)-%$expG6#* &^#!\"\"F)F0F)F)%#h_GF)!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalc(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&-%%sqrtG6#\"\"# \"\"\")-%$cosG6#%$phiGF)F*!\"$*(\"\"$F*F&F*)-%$sinGF.F)F*!\"\"F*%#h_GF *" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify(%,trig);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%%sqrtG6#\"\"#\"\"\"%#h_GF)!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "Clearly the action of L_x+I*L_ y on the state |5,3> led to a state propotional to |5,4>. The factor c an be checked to be consistent with the analytical result:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "L:='L': M:='M': subs(L=5,M=3,h_*sqr t((L-M)*(L+M+1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#h_G\"\"\"-%% sqrtG6#\"#=F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "This result agre es (apart from a minus sign) with the ratio found above." }}{PARA 0 " " 0 "" {TEXT -1 63 "Let us check that L_x-I*L_y lowers the magnetic qu antum number:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "simplify(L xOP(phi1)-I*LyOP(phi1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*.%#h_ G\"\"\"-%$expG6#*&^#\"\"$F'%$phiGF'F',&!\"\"F'*$)-%$cosG6#%&thetaG\"\" #F'F'F'-%%sqrtG6#\"$&QF'F3F',**(F,F'-%$sinG6#F.F'F2F'F'*&^#F0F'F>F'F'* (F-F'-F4F@F'F2F'F0FDF'F'F'*$-F96#%#PiGF'F0#!\"$\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(%/Y(theta,phi,5,2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%#h_G\"\"\"-%$expG6#*&^#F&F&%$phiGF&F&,& *&F+F&-%$sinG6#F,F&F&-%$cosGF1!\"\"F&-%%sqrtG6#\"\"'F&\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalc(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&,&*&%#h_G\"\"\")-%$cosG6#%$phiG\"\"#F'!\"#*(F-F'F&F ')-%$sinGF+F-F'!\"\"F'-%%sqrtG6#\"\"'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify(%,trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$*&%#h_G\"\"\"-%%sqrtG6#\"\"'F&!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "L:='L': M:='M': subs(L=5,M=3,h_*sqrt((L+M)*(L-M+1))); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#h_G\"\"\"-%%sqrtG6#\"#CF%" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "As for the L(+) operator we find \+ that the lowering operator L(-) produces the lowered normalized state \+ up to a well-defined factor." }}{PARA 0 "" 0 "" {TEXT -1 177 "We have \+ verified by example that the combinations L(+) := L_x + i L_y and (L-) := L_x - i L_y act as ladder operators which allow us to step through the eigenstates in a given (" }{TEXT 303 2 "LM" }{TEXT -1 113 ")-mult iplet. One can look at the action of the products L(+) L(-) and L(-)L( +) to establish the allowed range of " }{TEXT 304 1 "M" }{TEXT -1 8 "- values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 455 "Our objective is now to generalize on the basis of the i deas behind orbital angular momentum. We are looking for a realization of the angular momentum algebra with substantially fewer degrees of f reedom: instead of the orbital angular momentum operators which act of functions of theta and phi (polar and azimuthal angle) we are interes ted in operators that act on so-called spinors. The spinors are in the simplest case column vectors of numbers of size " }{TEXT 311 1 "N" } {TEXT -1 145 " (such as the [1,0] and [0,1] basis vectors representing spin alignment and counter-alignment with a given axis), and the oper ators are given as " }{TEXT 313 1 "N" }{TEXT -1 4 "-by-" }{TEXT 312 1 "N" }{TEXT -1 66 " matrices that mix the components of spinors while a cting on them." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "We define the matrices that represent the components of a n J=1/2 angular momentum operator." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning , the protected names norm and trace have been redefined and unprotect ed\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "S_x:=matrix([[0,h_/ 2],[h_/2,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$S_xG-%'matrixG6#7 $7$\"\"!,$%#h_G#\"\"\"\"\"#7$F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "S_y:=matrix([[0,-I/2*h_],[I/2*h_,0]]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$S_yG-%'matrixG6#7$7$\"\"!*&^##!\"\"\"\"#\"\" \"%#h_GF07$*&^##F0F/F0F1F0F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "S_z:=matrix([[h_/2,0],[0,-h_/2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$S_zG-%'matrixG6#7$7$,$%#h_G#\"\"\"\"\"#\"\"!7$F/,$F+#!\"\"F. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "S2:=evalm(S_x &* S_x + \+ S_y &* S_y + S_z &* S_z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S2G-%' matrixG6#7$7$,$*$)%#h_G\"\"#\"\"\"#\"\"$\"\"%\"\"!7$F3F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalm(S_x &* S_y - S_y &* S_x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$*&^##\"\"\"\"\"#F+)%#h _GF,F+\"\"!7$F/*&^##!\"\"F,F+F-F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Clearly this agrees with I*h_ S_z." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 73 "The eigenvectors of S_z and S^2 are t he spinor solutions [1,0] and [0,1]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 306 11 "Exercise 4:" }}{PARA 0 "" 0 "" {TEXT -1 64 "Verify the commutation relations for the other permutations of \+ [" }{TEXT 309 1 "x" }{TEXT -1 2 ", " }{TEXT 308 1 "y" }{TEXT -1 2 ", \+ " }{TEXT 307 1 "z" }{TEXT -1 81 "], i.e., demonstrate that S_x, S_y, a nd S_z satisfy the angular momentum algebra." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 310 11 "Exercise 5:" }}{PARA 0 "" 0 " " {TEXT -1 226 "Verify the angular momentum algebra for higher spinor \+ representations (S=1, and S=3/2 matrices are defined below). Calculate the eigenvectors of S_z. Comment on the eigenvalues of S_z and S^2, a nd on the uncertainty principle." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "S1_x:=evalm(h_/sqrt(2)*matrix([[0,1,0],[1,0,1],[0,1,0 ]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%S1_xG-%'matrixG6#7%7%\"\"! ,$*&%#h_G\"\"\"-%%sqrtG6#\"\"#F.#F.F2F*7%F+F*F+F)" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 60 "S1_y:=evalm(h_/sqrt(2)*matrix([[0,-I,0],[I,0 ,-I],[0,I,0]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%S1_yG-%'matrixG 6#7%7%\"\"!*(^##!\"\"\"\"#\"\"\"%#h_GF0-%%sqrtG6#F/F0F*7%*(^##F0F/F0F1 F0F2F0F*F+7%F*F6F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "S1_z: =evalm(h_*matrix([[1,0,0],[0,0,0],[0,0,-1]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%S1_zG-%'matrixG6#7%7%%#h_G\"\"!F+7%F+F+F+7%F+F+,$F*! \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "S3o2_x:=evalm(h_/2* matrix([[0,sqrt(3),0,0],[sqrt(3),0,2,0],[0,2,0,sqrt(3)],[0,0,sqrt(3),0 ]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'S3o2_xG-%'matrixG6#7&7&\" \"!,$*&%#h_G\"\"\"-%%sqrtG6#\"\"$F.#F.\"\"#F*F*7&F+F*F-F*7&F*F-F*F+7&F *F*F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "S3o2_y:=evalm(h _/2*matrix([[0,-I*sqrt(3),0,0],[I*sqrt(3),0,-2*I,0],[0,2,0,-I*sqrt(3)] ,[0,0,I*sqrt(3),0]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'S3o2_yG-% 'matrixG6#7&7&\"\"!*(^##!\"\"\"\"#\"\"\"%#h_GF0-%%sqrtG6#\"\"$F0F*F*7& *(^##F0F/F0F1F0F2F0F**&^#F.F0F1F0F*7&F*F1F*F+7&F*F*F7F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "S3o2_x:=evalm(h_/2*matrix([[3,0,0,0 ],[0,1,0,0],[0,0,-1,0],[0,0,0,-3]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'S3o2_xG-%'matrixG6#7&7&,$%#h_G#\"\"$\"\"#\"\"!F/F/7&F/,$F+#\"\" \"F.F/F/7&F/F/,$F+#!\"\"F.F/7&F/F/F/,$F+#!\"$F." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "143" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }