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Problem 2.) Cubic Crystal Field: Partial Quenching of Orbital Angular Momentum in t2g multiplet Consider a d-shell of an ion, whose five states 1 = 2,1) where l* = +2, +1,0, are split in a cubic crystal field into a tag triplet and an e, doublet. A particular orthonormal basis for the states of the triplet, denoted by m)) where m= 0, +1, is defined as follows | + 1)) = 2, -1), 10) = 13 (12,+2) 12 2)), 1 - 1) = -2, +1). (4) Consider now also the orthonormal basis of an ordinary angular momentum L = 1 triplet, |m) = \L = 1, m), m= 0, +1. (5) Show that the matrix elements of the orbital angular momentum operators = ?, + = * + i" taken between the basis states of the tz, triplet defined in Eq. (4), are related to the matrix elements of the same operators taken between the states of the ordinary angular momentum L = 1 triplet in Eq. (5) simply by a minus sign: {{m' Mm)) = (-1){m'|Mm), m, m', M = 0, +1. (6) Since such a relationship obviously then also holds for the Cartesian components of orbital angular momentum ", where a = , y, z = 1, 2, 3, one often writes Eq. (6) in the form ((m'|24|m)) = (-1) (4)m,m (7) where the 3 x 3 matrices (24)m'.m appearing on the right hand side are the (standard and familiar) matrices representing , ' and on an angular momentum L = 1 triplet. Denoting by the projection operator on the three-dimensional vector space spanned by the three states of the tag triplet, Eq. (4), and :=E, determine the eigenvalues of the two operators . = l'4 and . = le ca when acting on the d-shell (i.e. on the vector space spanned by the five kets 11 = 2, 1.)). [The calliographic symbol for the vector of angular momentum operators is to indicate that its appearance is unexpected here in the context of a d-shell. In particular, the L = 2 orbital angular momentum has effectively been quenched to the smaller orbital angular momentum 1 = 1, when projected onto the states of the tag triplet.] Hint: Use the familiar relations for the action of the angular momentum raising and lowering operators [*]L, M) = VL(L+ 1) M(M+1) |L, M+1). (8) a=r.y. a=r.y. =