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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);" }}}{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);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "For the spherical harmonics 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 then 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-m1)!/(4*Pi *(l+m1)!)); end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Y(theta ,phi,3,0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Y(theta,phi,3 ,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Y(theta,phi,3,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Y(theta,phi,3,3);" }}} {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(theta),th eta=0..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "No(1,0);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "No(1,1);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "No(3,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Y(theta,phi,0,0);" }}}{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 th e state with 5 units of h_ (h-bar) measured for L_z. Let us also speci fy 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);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "What are the expectation values of L_x and L_y?" }}{PARA 0 "" 0 "" {TEXT -1 207 "To answer 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));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "LzOP(psi)/psi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(LzOP(psi)/psi,trig);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "We have verified that the state p si has eigenvalue 5 h-bar for the L_z operator. Let us define the oper ators for L_x and L_y:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "L xOP:=f->I*h_*simplify(sin(phi)*diff(f,theta)+cos(phi)/tan(theta)*diff( f,phi));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "LyOP:=f->I*h_*s implify(-cos(phi)*diff(f,theta)+sin(phi)/tan(theta)*diff(f,phi));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 149 "These definitions follow from a s traightforward application of the transformation from Cartesian to sph erical polar coordinates 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 coordinates:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "L2OP:=f->-h_^2*simplify(diff(sin(theta)*diff(f,theta),theta)/ sin(theta)+diff(f,phi$2)/sin(theta)^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "L2OP(psi)/psi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{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 calculated from Lx^2+Ly^2+Lz^2:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "L2OPlong:=f->LxOP(LxOP(f))+L yOP(LyOP(f))+LzOP(LzOP(f));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "L2OPlong(psi)/psi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "s implify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Let us verify 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;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{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 result is obtained for L_y." } }{PARA 0 "" 0 "" {TEXT -1 103 "What are the average values (expectatio n 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(co njugate(f)*LxOP(f)),phi=0..2*Pi)*sin(theta),theta=0..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ExpV_Lx(psi);" }}}{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);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ExpV_Ly(psi);" }}}{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 res ult 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);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ExpV_Lz(psi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Now 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;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dA:=sqrt(dLxsq(psi));" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "dLysq:=f->int(int(evalc(co njugate(LyOP(f))*LyOP(f)),phi=0..2*Pi)*sin(theta),theta=0..Pi)-ExpV_Ly (f)^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dB:=sqrt(dLysq(ps i));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "dA*dB;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 164 "This shows that the state psi has the mi nimum uncertainty allowed under the uncertainty principle which follow s from the commutation relation [L_x, L_y] = I h_ L_z." }}{PARA 0 "" 0 "" {TEXT -1 323 "The RHS of the uncertainty relation dA*dB >= h_/2 | | where C = I*h_*L_z equals the LHS for our particular eigenstate. \+ We notice that the state psi (5 units of angular momentum in the z-dir ection) has an avaerage value of 0 for each of the components that can not be measured exactly with an uncertainty of sqrt(10)/2 h_ " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "evalf(sqrt(10)/2);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "What are the uncertainties for a s tate whose 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 calcula tions for the L=5 multiplet, let us verify that the uncertainty (devia tion) is zero for the L_z component, as the system is in an L_z eigens tate:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "dLzsq:=f->int(int (evalc(conjugate(LzOP(f))*LzOP(f)),phi=0..2*Pi)*sin(theta),theta=0..Pi )-ExpV_Lz(f)^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "dLzsq(ps i);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Indeed we find that for an eigenstate the uncertainty in the measurement is zero." }}{PARA 0 "" 0 "" {TEXT -1 94 "We proceed with the check of the uncertainty relatio n for 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);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dA:=sqrt(dLxsq(psi));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dB:=sqrt(dLysq(psi));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "dA*dB;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "This is to be compared with the right-hand side of:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "h_/2*4*h_;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 288 "We conclude \+ that the uncertainty is much larger than for maximal L_z eigenstate! I n 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 maximal L_z eigenstate! Can this behaviour be generalized?" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "psi:=Y(theta,phi,5,3);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sqrt(dLxsq(psi)),sqrt(dLysq( psi));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "psi:=Y(theta,phi,5,2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sqrt(dLxsq(psi)),sqrt(dLysq( psi));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "psi:=Y(theta,phi,5,1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sqrt(dLxsq(psi)),sqrt(dLysq( psi));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "psi:=Y(theta,phi,5,0);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sqrt(dLxsq(psi)),sqrt(dLysq( psi));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "psi:=Y(theta,phi,5,-1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sqrt(dLxsq(psi)),sqrt(dLysq( psi));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Indeed we do find that the product of the uncertainties for L_x and L_y grows with decreasing " }{TEXT 269 1 "M" }{TEXT -1 75 ", while the RHS of the uncertainty principle g ets smaller (it vanishes for " }{TEXT 270 1 "M" }{TEXT -1 64 "=0 eigen states of L_z). Remarkably, for the states with maximal " }{TEXT 275 1 "z" }{TEXT -1 33 "-projection (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-uncertainty eigenstates of the angular momentum operators. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 183 "Thes e observations are consistent with the diagram that shows the multiple t of angular momentum states displayed as vectors whose endpoints lie \+ on a sphere of radius squared equal to " }{TEXT 277 1 "L" }{TEXT -1 1 "(" }{TEXT 276 1 "L" }{TEXT -1 48 "+1) h_^2, with L_z component sharpl y 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 th at intersects the 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 uncertain ty in Lx and Ly? These are the states where most of the angular moment um vector resides in the " }{TEXT 260 1 "z" }{TEXT -1 68 "-component w hich is known exactly. 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 uncertain ty principle would allow us the smallest (zero) uncertainty for these \+ states in principle. This result from the generalized uncertainty prin ciple 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 co mponents L_x, L_y, and L_z are determined to have zero eigenvalue, i.e ., are known exactly. 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 compat ible for these eigenstates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 11 "Exercise 1:" }}{PARA 0 "" 0 "" {TEXT -1 118 "R epeat the above calculations for orbital angular momentum states of a \+ different L-value. 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 momentum commutation relations" }}{PARA 0 "" 0 "" {TEXT -1 195 "We can use the operators to verify some commutation relations \+ by acting on an unspecified function of the angles theta and phi. Watc h 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 simplificat ion: one has to be careful while using it." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "evalb(CR1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalb(simplify(CR1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "simplify(lhs(CR1));" }}}{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(th eta,phi))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify(CR2 );" }}}{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):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "psi:=Y(theta,phi,3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "rho:=evalc(conjugate(psi)*psi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "sphereplot(rho,phi=0..2*Pi,theta=0..Pi,style=patchco ntour,color=phi,axes=boxed,numpoints=3000,scaling=constrained,style=wi reframe,thickness=2);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 286 11 "Exerc ise 2:" }}{PARA 0 "" 0 "" {TEXT -1 146 "Look at the visualizations of \+ other angular momentum eigenstates. Rotate the diagram to observe the \+ projections onto particular planes (e.g., the " }{TEXT 288 1 "x" } {TEXT -1 1 "-" }{TEXT 287 1 "z" }{TEXT -1 151 " plane), and compare wi th textbook diagrams. What is the meaning of these diagrams in the con text of the probabilistic interpretation of wavefunctions?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Can we gr aph the diagram for the visualization of the angular momentum vector? \+ The " }{TEXT 19 9 "plottools" }{TEXT -1 46 " package allows us to grap h spheres and cones." }}{PARA 0 "" 0 "" {TEXT -1 115 "We pick units in which h_ equals unity. To visualize the knowledge available for the a ngular momentum vector in an " }{TEXT 289 1 "L" }{TEXT -1 90 ">0 eigen state we use a cone: the height of the cone is used to indicate the sh arply known " }{TEXT 290 1 "z" }{TEXT -1 38 "-projection. The indeterm inacy 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 av erage value of either of these components is zero, and that the deviat ions (uncertainties) are the same in both components. This makes it ob vious that the Lx and Ly values are determined to be somewhere on a ci rcle of a known radius (determined by the uncertainty). Unlike in clas sical mechanics where the complete angular momentum vector is known pr ecisely (and conserved 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 " vector in the sense of a bundle of vectors: given a mea surement of L^2 and L_z (after which we are in an |LM> eigenstate) we \+ only know that Lx and Ly have values such that the " }{TEXT 294 1 "L" }{TEXT -1 188 " vector lies on the mantle of the cone. Of course, a me asurement of either L_x or L_y would destroy the |LM> eigenstate (a su bsequent remeasurement 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;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "r:=sqrt(L*( L+1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(plottools): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "s:=sphere([0,0,0],r,col or=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,style=wireframe,scali ng=constrained,axes=boxed):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "C2a:=display(c2a,style=patchnogrid,scaling=constrained,axes=boxed) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "C2b:=display(c2b,style =patchnogrid,scaling=constrained,axes=boxed):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "C2c:=display(c2c,style=patchnogrid,scaling=const rained,axes=boxed):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "disp lay([C1,C2a,C2b,C2c],labels=[x,y,z],orientation=[150,65]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 298 11 "Exercise 3:" }}{PARA 0 "" 0 "" {TEXT -1 61 "Look at the graphical representation for different values of " } {TEXT 297 1 "L" }{TEXT -1 5 " and " }{TEXT 296 1 "M" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 53 "Matrix representation of the angular momentum algebra" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "We can ask the fo llowing question: given that the spherical harmonics are realizations \+ of the angular momentum algebra for " }{TEXT 299 1 "L" }{TEXT -1 1 ", " }{TEXT 300 1 "M" }{TEXT -1 47 " eigenstates in the coordinate repres entation (" }{TEXT 301 1 "L" }{TEXT -1 1 "," }{TEXT 302 1 "M" }{TEXT -1 58 " being integers), are there other representations as well?" }} {PARA 0 "" 0 "" {TEXT -1 301 "The problem is to construct matrices tha t represent the L_x, L_y, L_z observables; find representations in whi ch L_z and L^2 are diagonal, while the other two are represented by no n-diagonal, i.e., non-commuting matrices. A systematic approach based \+ on ladder operators can be developed (commut2.mws)." }}{PARA 0 "" 0 " " {TEXT -1 158 "It turns out that two linear combinations of L_x and L _y play the role of raising and lowering operators. Let us verify this in the coordinate representation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "phi1:=Y(theta,phi,5,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "simplify(LxOP(phi1)+I*LyOP(phi1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(%/Y(theta,phi,5,4));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalc(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify(%,trig);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "Clearly the action of L_x+I*L_y on the state |5,3> led t o a state propotional to |5,4>. The factor can be checked to be consis tent with the analytical result:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "L:='L': M:='M': subs(L=5,M=3,h_*sqrt((L-M)*(L+M+1))); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "This result agrees (apart fro m 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 quantum number: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "simplify(LxOP(phi1)-I*L yOP(phi1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(%/Y (theta,phi,5,2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalc(% );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify(%,trig);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "L:='L': M:='M': subs(L=5,M= 3,h_*sqrt((L+M)*(L-M+1)));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "As for the L(+) operator we find that the lowering operator L(-) produce s the lowered normalized state up to a well-defined factor." }}{PARA 0 "" 0 "" {TEXT -1 177 "We have verified by example that the combinati ons L(+) := L_x + i L_y and (L-) := L_x - i L_y act as ladder operator s which allow us to step through the eigenstates in a given (" }{TEXT 303 2 "LM" }{TEXT -1 113 ")-multiplet. One can look at the action of t he 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 g eneralize on the basis of the ideas behind orbital angular momentum. W e are looking for a realization of the angular momentum algebra with s ubstantially fewer degrees of freedom: instead of the orbital angular \+ momentum operators which act of functions of theta and phi (polar and \+ azimuthal angle) we are interested in operators that act on so-called \+ spinors. The spinors are in the simplest case column vectors of number s 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 wi th a given axis), and the operators are given as " }{TEXT 313 1 "N" } {TEXT -1 4 "-by-" }{TEXT 312 1 "N" }{TEXT -1 66 " matrices that mix th e components of spinors while acting on them." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "We define the matrices th at represent the components of an J=1/2 angular momentum operator." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "S_x:=matrix([[0,h_/2],[h_/2,0]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "S_y:=matrix([[0,-I/2*h_], [I/2*h_,0]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "S_z:=matri x([[h_/2,0],[0,-h_/2]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "S2:=evalm(S_x &* S_x + S_y &* S_y + S_z &* S_z);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 31 "evalm(S_x &* S_y - S_y &* S_x);" }}}{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 eigen vectors of S_z and S^2 are the 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 f or 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, and S_z satisfy the angular momentum alg ebra." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 310 11 "Exercise 5:" }}{PARA 0 "" 0 "" {TEXT -1 226 "Verify the angular momen tum 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, and on the uncertainty principle." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "S1_x:=evalm(h_/sqrt(2)*matri x([[0,1,0],[1,0,1],[0,1,0]]));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "S1_y:=evalm(h_/sqrt(2)*matrix([[0,-I,0],[I,0,-I],[0,I,0]]));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "S1_z:=evalm(h_*matrix([[1 ,0,0],[0,0,0],[0,0,-1]]));" }}}{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]]));" }}}{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]]));" }}}{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]]));" }}}{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 }