( a ', jm lj - V l a, jm ) jm 'l ] Ijm ), ( ) Ii 2j ( j + 1 ) We will give applications of the theorem in subsequent sections.

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1 Theory of Angular Momentum 4 But the righthand side of 3144) is the same as a', jm lj V la, jm ) / a, jm lj 1 a, jm ) by 314) and 3143) Moreover, the lefthand side of 3143) is just j j 1 ) 1i So a ', jm lj V l a, jm ) jm 'l ] Ijm ), ) q Ii j j 1 ) o which proves the projection theorem a ', jm 'I Vq la ' jm ) We will give applications of the theorem in subsequent sections PROBLEMS 1 Find the eigenvalues and eigenvectors of ) electron is in the spin state ; If probability of the result Ii /? Consider the x matrix defined by U Sy y ) Suppose an is measured, what is the ao io a ao o a, l where a i s a real number and a is a threedimensional vector with real components a Prove that U is unitary and unimodular b In general, a X unitary unimodular matrix represents a rotation in three dimensions Find the axis and angle of rotation appropriate for U in terms of ao, a I ' a, and a3 3 The spindependent Hamiltonian of an electronpositron system in the presence of a uniform magnetic field in the zdirection can be written as H As e) s e) :! ) s}e ) Sze ) ) Suppose the spin function of the system is given by X ) X ) a Is this an eigenfunction of H in the limit A, eb/ me 1 O? If it is, what is the energy eigenvalue? If it is not, what is the expectation value of H? b Same problem when eb/ me, A 1 O 4 Consider a spin 1 particle Evaluate the matrix elements of Sz Sz 1i ) Sz Ii ) and Sx Sx Ii ) Sx Ii ) 5 Let the Hamiltonian of a rigid body be H where K IS ) 1 K I Ki K3 I; 1 1;, the angular momentum in the body frame From this

2 Problems 43 expression obtain the Heisenberg equation of motion for K and then find Euler's equation of motion in the correspondence limit 6 Let U e i G3 ae igf3e,g3y, where a, /3, y ) are the Eulerian angles In order that U represent a rotation a, /3, y ), what are the commutation rules satisfied by the G k? Relate G to the angular momentum operators 7 What is the meaning of the following equation: U 1AkU L R k,a " where the three components of A are matrices? From this equation show that matrix elements m I A k l n ) transform like vectors 8 Consider a sequence of Euler rotations represented by ) ) fijl 1 / ) a " /3 Y ) exp i 3 <X exp i /3 exp i 3Y ;;z; e i a y )/ cos /3 /3 ) e, a y )/ sm e i a y )j sin /3 e, a y )/ cos /3 Because of the group properties of rotations, we expect that this sequence of operations is equivalent to a single rotation about some axis by an angle O Find O 9 a Consider a pure ensemble of identically prepared spin t systems Suppose the expectation values Sx ) and Sz ) and the sign of S)I ) ' are known Show how we may determine the state vector Why is it unnecessary to know the magnitude of Sy )? b Consider a mixed ensemble of spin systems Suppose the ensemble averages [ Sx l, [ S)I )' and [ Sz ) are all known Show how we may construct the X density matrix that characterizes the ensemble 1 a Prove that the time evolution of the density operator p in the Schrodinger picture) is given by p t ) O// t, t o ) p t o ) O// t t, to ) b Suppose we have a pure ensemble at t O Prove that it cannot evolve into a mixed ensemble as long as the time evolution is governed by the Schrooinger equation 1 1 Consider an ensemble of spin 1 systems The density matrix is now a 3 X 3 matrix How many independent real) parameters are needed to characterize the density matrix? What must we know in addition to [ Sx)' [ Sy )' and [ Sz ) to characterize the ensemble completely? 1 An angularmomentum eigenstate I ), m m ) is rotated by an infinitesimal angle about the yaxis Without using the explicit form of the d!j, function, obtain an expression for the probability for the new rotated state to be found in the original state up to terms of order max

3 44 Theory of Angular Momentum 1 3 Show that the 3 x 3 matrices Gi i 1,, 3) whose elements are given by { GJ J k ihf/j k ' where j and k are the row and column indices, satisfy the angular momentum commutation relations What is the physical or geometric) significance of the transformation matrix that connects Gi to the more usual 3 X 3 representations of the angularmomentum operator Ji with J3 taken to be diagonal? Relate your result to V V fi 8 c[> x V under infinitesimal rotations Note: This problem may be helpful in understanding the photon spin) 14 a Let J be angular momentum It may stand for orbital L, spin S, or Jtotal') Using the fact that Jx' Jy, Jz J ± Jx ijy ) satisfy the usual angularmomentum commutation relations, prove J J/ J J Mz b Using a) or otherwise), derive the " famous" expression for the coefficient c that appears in J l/;j m c l/;j m t 1 5 The wave function of a particle subjected to a spherically symmetrical potential V r ) is given by l/; {x} { x y 3z }f{ r } a Is l/; an eigenfunction of L? If so, what is the Ivalue? If not, what are the possible values of I we may obtain when L is measured? b What are the probabilities for the particle to be found in various m I states? c Suppose it is known somehow that l/; x) is an energy eigenfunction with eigenvalue E Indicate how we may find V r ) 16 A particle in a spherically symmetrical potential is known to be in an eigenstate of L and Lz with eigenvalues h 1 1 1) and mh, respec tively Prove that the expectation values between 1 1 m ) states satisfy [/i 1 } /i m /i ] < L; ) < L ) < Lx ) < Ly ), Interpret this result semiclassically 17 Suppose a halfinteger Ivalue, say t, were allowed for orbital angular momentum From we may deduce, as usual,

4 45 Problems Now try to construct Yl /, 1/ fj, </» ; by a) applying L to Y1/, 1/ fj, </» ; and b) using L Y1/, 1/ fj, </» Show that the two procedures lead to contradictory results This gives an argument against halfinteger Ivalues for orbital angular momentum) 1 8 Consider an orbital angularmomentum eigenstate 11, m ) Sup pose this state is rotated by an angle P about the yaxis Find the probability for the new state to be found in m, 1, and The spherical harmonics for 1, 1, and given in Appendix A may be useful) 1 9 What is the physical significance of the operators K a t a and K a a in Schwinger' s scheme for angular momentum? Give the nonvanishing matrix elements of K ± We are to add angular momenta il 1 and i 1 to form i, 1, and states Using either the ladder operator method or the recursion relation, express all nine) { i, m } eigenkets in terms of l il i ; m 1 m ) Write your answer as I } 1, m 1) v' I, ) 1 1 v' 1, ),, where and stand for m 1 1,, respectively 1 a Evaluate j L I d, p ) 1 m m j for any i integer or halfinteger); then check your answer for i t b Prove, for any i, t m J m 1 d Jl ) 1 j j 1 ) sinjl m, 3 cos Jl 1) [ Hint: This can be proved in many ways You may, for instance, examine the rotational properties of J/ using the spherical irreduci ble) tensor language] a Consider a system with i 1 Explicitly write < i 1, m 'I!v I i 1, m ) in 3 X 3 matrix form / b Show that for i 1 only, it is legitimate to replace e 1 Jy/3 11 by Jy Jy 1 i h sin P h 1 cos P ) ) )

5 46 Theory of Angular Momentum d {j I ) f3 ) c Using b), prove ) 1 cos f3 ) ) sin p ) cos f3 ) cos f3 ) sin p ) sin P ) 1 cos f3 ) ) sin p ) 1 cos f3 ) 1 a f3y IJ? lal f3 IYI ) in terms of a series in fj naf3 y ) af3y l imn ) 3 Express the matrix element 4 Consider a system made up of two spin particles Observer A specializes in measuring the spin components of one of the particles S l z, S l x and so on), while observer B measures the spin components of the other particle Suppose the system is known to be in a spinsinglet state, that is, Stota) O a What is the probability for observer A to obtain Slz h / when observer B makes no measurement? Same problem for S I x h / b Observer B determines the spin of particle to be in the S z Ii / state with certainty What can we then conclude about the outcome of observer A' s measurement if i) A measures Slz and ii) A measures S l ) Justify your answer 5 Consider a spherical tensor of rank 1 that is, a vector) V ± l )l fi Vx i Vy l V;O ) Vz ' Using the expression for d{j l) given in Problem, evaluate L di' f3 ) Vq\l ) q' and show that your results are just what you expect from the transfor mation properties of Vx, y, z under rotations about the yaxis 6 a Construct a spherical tensor of rank l out of two different vectors U Ux ' '!v' Uz ) and V Vx' Vy ' ) Explicitly write TILo in terms of Ux, y, z and Vx y, b Construct a spherical tensor of rank out of two different vectors U and V Write down explicitly T, ± I, O in terms of Ux, y, z and Vx, y, z ' 7 Consider a spinless particle bound to a fixed center by a central force potential, z '

6 47 Problems a Relate, as much as possible, the matrix elements n ', l ', m 'l x iy ) I n, l, m ) and n ', l ', m 'l z l n, I, m ) using only the WignerEckart theorem Make sure to state under what conditions the matrix elements are nonvanishing b Do the same problem using wave functions tj; x) R n ' r ) yt, 8 a Write xy, xz, and x y ) as components of a spherical irreduci ble) tensor of rank b The expectation value <J» Q e a, j, m j l 3 z r ) l a, j, m j) is known as the quadrupole mom ent Evaluate e a, j, m ' l x y ) l a, j, m j ), where m ' j, j 1 j, ) in terms of Q and appropriate ClebschGordan coefficients 9 A spin nucleus situated at the origin is subjected to an external inhomogeneous electric field The basic electric quadrupole interaction may by taken to be, e Hi n, S S l W [ :: Ls;!: los; :: Ls}] where <J> is the electrostatic potential satisfying Laplace' s equation and the coordinate axes are so chosen that a<j» ayaz a<j» aa<j» x az ax a y Show that the interaction energy can be written as A 3Sz S) B S S ), and express A and B in terms of a <J>/ ax ) and so on Determine the energy eigenkets in terms of 1 m ), where m, and the corre sponding energy eigenvalues Is there any degeneracy? n