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10-5 Tensor Operator Tensors Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: December 04, 2010) 1 Vector operators The operators corresponding to various physical quantities will be characterized by their behavior under rotation as scalars, vectors, and tensors. Vi ijV j . j We assume that the state vector changes from the old state to the new state ' . ' Rˆ , or ' Rˆ . A vector operator Vˆ for the system is defined as an operator whose expectation is a vector that rotates together with the physical system. ˆ ˆ 'Vi ' ij V j j or ˆ ˆ ˆ ˆ R Vi R ijV j j or ˆ ˆ ˆ ˆ Vi RijV j R j We now consider a special case, infinitesimal rotation. i Rˆ 1ˆ Jˆ n 1 i Rˆ 1ˆ Jˆ n ˆ i ˆ ˆ ˆ i ˆ ˆ (1 J n)Vi (1 J n) ijV j j ˆ i ˆ ˆ ˆ Vi [Vi ,J n] ijV j j For n = ez, 1 0 z ( ) 1 0 0 0 1 ˆ i ˆ ˆ ˆ ˆ ˆ ˆ ˆ V1 [V1, J z ] 11V1 12V2 13V3 V1 V2 ˆ i ˆ ˆ ˆ ˆ ˆ ˆ ˆ V2 [V2 , J z ] 21V1 22V2 23V3 V1 V2 ˆ i ˆ ˆ ˆ ˆ ˆ ˆ V3 [V3, J z ] 31V1 32V2 33V3 V3 or ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [V1, J z ] iV2 , [V2 , J z ] iV1 , [V3, J z ] 0 For For n = ex, 1 0 0 x ( ) 0 1 0 1 ˆ i ˆ ˆ ˆ ˆ ˆ ˆ V1 [V1, J x ] 11V1 12V2 13V3 V1 ˆ i ˆ ˆ ˆ ˆ ˆ ˆ ˆ V2 [V2 , J x ] 21V1 22V2 23V3 V2 V3 2 ˆ i ˆ ˆ ˆ ˆ ˆ ˆ ˆ V3 [V3, J x ] 31V1 32V2 33V3 V2 V3 or ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [V1, J x ] 0 , [V2 , J x ] iV3 , [V3, J x ] iV2 For For n = ey, 1 0 y ( ) 0 1 0 0 1 ˆ i ˆ ˆ ˆ ˆ ˆ ˆ ˆ V1 [V1, J y ] 11V1 12V2 13V3 V1 V3 ˆ i ˆ ˆ ˆ ˆ ˆ ˆ V2 [V2 , J y ] 21V1 22V2 23V3 V2 ˆ i ˆ ˆ ˆ ˆ ˆ ˆ ˆ V3 [V3, Jyx ] 31V1 32V2 33V3 V1 V3 or ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [V1, J y ] iV3 , [V2 , J y ] 0, [V3, J y ] iV1 Using the Levi-Civita symbol, we have ˆ ˆ ˆ [Vi , J j ] iijkVk We can use this expression as the defining property of a vector operator. Levi-Civita symbol:ijk 123=231=312=1 132=213=321=-1 all other ijk. =0. 2 Example 3 (a) When Vˆ Jˆ, ˆ ˆ ˆ [Ji , J j ] iijk J k (b) When Vˆ rˆ and Jˆ Lˆ , ˆ [xˆi , L j ] i ijk xˆk (c) When Aˆ pˆ, and Jˆ Lˆ , ˆ [ pˆi , Lj ] iijk pˆ k 3 Cartesian tensor operators The standard definition of a Cartesian tensor is that each of its suffix transforms under the rotation as do the components of an ordinary 3D vector, The Cartesian tensor operator is defined by ˆ ' Tij ' ik jl Tkl k ,l under the rotation specified by the 3x3 orthogonal matrix . ˆ ˆ ˆ ˆ Tij Rik jlTkl R k ,l The simplest example of a Cartesian tensor of rank 2 is a dyadic formed out of two vectors Uˆ and Vˆ . ˆ ˆ ˆ Tij UiV j ˆ ˆ where Ui and Vi are the components of ordinary 3D vector operators. There are nine ˆ ˆ ˆ components: 1+3+5 = 9. The trouble with a Cartesian tensor like Tij UiV j is that it is reducible. It can be decomposed into the three parts. Uˆ Vˆ Uˆ Vˆ Uˆ Vˆ Uˆ Vˆ Uˆ Vˆ Uˆ Vˆ Uˆ Vˆ i j j i ( i j j i ) i j 3 ij 2 2 3 ij The first term on the right-hand side, Uˆ Vˆ is a scalar product invariant under the rotation. The second is an anti-symmetric tensor which can be written as 4 ˆ ˆ ijk (U V)k . There are 3 independent components. 0 T T a12 a13 Ta Ta12 0 Ta23 Ta13 Ta23 0 The third term is a 3x3 symmetric traceless tensor with 5 independent components (=6-1, where 1 comes from the traceless condition. T T T s11 s12 s13 Ts Ts12 Ts22 Ts23 Ts13 Ts23 Ts33 ˆ ˆ ˆ with Ts11 Ts22 Ts33 0 . In conculsion, the tensor Tij UiV j can be decomposed into spherical tensors of rank 0, 1, and 2. 4 Spherical tensor: definition Notice the numbers of elements of these irreducible subgroups: 1, 3, and 5. These are exactly the numbers of elements of angular momentum representations for j = 0, 1, and 2. The first term is trivial: the scalar by definition is not affected by rotation, and neither is an j = 0 state. To deal with the second and third terms, we introduce tensor operators having three and five components, such that under rotation these sets of components transform among themselves just as do the sets of eigenkets of angular momentum in the j = 1 and j = 2 representation, respectively. m m Suppose we take a spherical harmonics Yl (,) Yl (n) , where the orientation of the unit vector n is characterized by and . We now replace n by some vector Vˆ . Then we have a spherical tensor of rank k (in place of l) with magnetic quantum number (in place of m). (k ) mq ˆ Tq Yl k (V) . The quantity l m Pl,m (x, y, z) r Yl (,) is a homogeneous polynomial of order l. 5 Spherical tensor of rank 1 5 4 The quantity P (x, y, z) rY q (,) is a first order homogeneous polynomial in 1,q 3 1 x, y, and z. (i) 4 x iy P (x, y, z) rY 1(,) 1,1 3 1 2 which leads to Vˆ iVˆ T (1) x y . 1 2 (ii) 4 P (x, y, z) rY 0 (,) z , 1,0 3 1 which leads to (1) ˆ T0 Vz (iii) 4 x iy P (x, y, z) rY 1(,) 1,1 3 1 2 which leads to Vˆ iVˆ T (1) x y . 1 2 6 Spherical tensor of rank 2 (i) 5 P (x, y, z) r 2Y 0 (,) (3z2 r 2 ) 2,0 2 16 (x iy)(x iy) (x iy)(x iy) [2z 2 ] 15 2 2 8 6 which leads to 6 2 15 Tˆ (1)Tˆ (1) 2Tˆ (1) Tˆ (1)Tˆ (1) Tˆ (2) 1 1 0 1 1 . 0 8 6 (iii) 15 x iy P (x, y, z) r 2Y 2 (,) ( )2 2,2 2 16 2 which leads to ˆ (2) 15 ˆ (1) 2 T2 T1 16 (iv) 15 z(x iy) (x iy)z P (x, y, z) r 2Y 1(,) 2,1 2 8 2 2 which leads to 15 Tˆ (1)Tˆ (1) Tˆ (1)Tˆ (1) Tˆ (2) 0 1 1 0 1 8 2 (v) 15 z(x iy) (x iy)z P (x, y, z) r 2Y 1(,) 2,1 2 8 2 2 which leads to 15 Tˆ (1)Tˆ (1) Tˆ (1)Tˆ (1) Tˆ (2) 0 1 1 0 1 8 2 The above example is the simplest nontrivial example to illustrate the reduction of a Cartesian tensor into irreducible spherical tensors. 7 Spherical tensor under rotation We consider how (k ) mq ˆ Tq Yl k (V) transforms under rotation. Using the relations 7 n' Rˆ n , n' n Rˆ Rˆ l,m l,m' l,m' Rˆ l,m m' n' l,m n Rˆ l,m n l,m' l,m' Rˆ l,m m' we have m m' ˆ * Yl (n') Yl (n) l,m R l,m' m' where (l) ˆ ˆ Dm,m' (R) l,m R l,m' m ˆ If there is an operator that acts like Yl (V) , it is then reasonable to expect ˆ m ˆ ˆ m' ˆ ˆ * m' ˆ (l) * ˆ * R Yl (V)R Yl (V) l,m R l,m' Yl (V)Dmm' (R) m' m' ˆ (k ) We define a spherical tensor operator of rank k as a set of 2k+1, Tq , q = k, k-1,…, -k such that under rotation they transform like a set of angular momentum eigenkets, k ˆ ˆ (k ) ˆ (k )* ˆ ˆ (k ) R Tq R Dqq' (R)Tq' , (1) q'k where ˆ (k ) mq ˆ Tq Yl k (V) (k ) ˆ ˆ Dq,q' (R) k,q R k,q' or (k ) * ˆ ˆ * ˆ Dq,q' (R) k,q R k,q' k,q' R k,q with q = k, k-1,…, -k. The switching of Rˆ Rˆ in Eq.(1) leads to another expression 8 k k ˆ ˆ (k ) ˆ (k )* ˆ ˆ (k ) (k ) ˆ ˆ (k ) RTq R Dqq' (R )Tq' Dqq' (R)Tq' (2) q'k q'k Considering the infinitesimal form of the expression (1), we have ˆ ˆ k ˆ ˆ iJ n ˆ (k ) ˆ iJ n ˆ (k ) ˆ iJ n (1 )Tq (1 ) Tq' k,q'1 k,q q'k or k ˆ (k ) i ˆ ˆ (k ) ˆ (k ) i ˆ (k ) ˆ Tq [J n,Tq ] Tq Tq' k,q' J n k,q q'k or k ˆ (k ) ˆ (k ) ˆ [J n,Tq ] Tq' k,q' J n k,q q'k For n = ex, ey, ez k ˆ (k ) ˆ (k ) ˆ ˆ (k ) [J z ,Tq ] Tq' k,q' J z k,q qTq (3) q'k k ˆ (k ) ˆ (k ) ˆ [J x ,Tq ] Tq' k,q' J x k,q q'k k ˆ (k ) ˆ (k ) ˆ [J y ,Tq ] Tq' k,q' J y k,q q'k k ˆ (k ) ˆ (k ) ˆ ˆ (k ) [J ,Tq ] Tq' k,q' J k,q (k q)(k q 1)Tq1 (4) q'k These two commutation relations can also be taken as a definition of a spherical tensor of rank k.
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