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Vector and Tensor Operators in Quantum Mechanics Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: November 24

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Vector and Tensor Operators in Quantum Mechanics Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: November 24 Vector and Tensor operators in quantum mechanics Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: November 24. 2015) We discuss the properties of vector and tensor operators under rotations in quantum mechanics. We use the analogy from the classical physics, where vector and tensor of a quantity with three components that transforms by definition like Vi ' ijV j Tij ' ik jlTikl j k ,l under rotation. First I will present a brief review in the case of classical physics. After that the properties of vector and tensor operators under rotation in quantum mechanics. The irreducible spherical tensor operators are introduced based on the rotation operators. The Cartesian tensors are decomposed into irreducible spherical tensors. The Wigner – Eckart theorem will be discussed elsewhere. 1. Definition of vector in classical physics (i) 2D rotation Suppose that the vector r is rotated through (counter-clock wise) around the z axis. The position vector r is changed into r' in the same orthogonal basis {e1, e2}. y r' e 2 r e ' 2 x2 e1' f x2 x1 f x x1 e1 In this Fig, we have e e ' cos e e ' sin 1 1 , 2 1 e1 e2 ' sin e2 e2 ' cos Spherical Harmonics as rotator matrices 1 9/3/2017 We define r and r' as r' x 'e x 'e x e ' x e ' 1 1 2 2 1 1 2 2 , and r x e x e 1 1 2 2 . Using the relation e1 r' e1 (x1'e1 x2 'e2 ) e1 (x1e1'x2e2 ') e2 r' e2 (x1'e1 x2 'e2 ) e2 (x1e1'x2e2 ') we have x1' e1 (x1e1'x2e2 ') x1 cos x2 sin x2 ' e2 (x1e1'x2e2 ') x1 sin x2 cos or x1 ' x1 11 12 x1 cos sin x1 () x2 ' x2 21 22 x2 sin cos x2 where ij ei e j ' (ii) 3D rotation Next we discuss the three-dimensional (3D) case, 3 3 3 r x j e j , r' x j 'e j x j e j ' j1 j1 j1 Note that ei (ei e j ')e j ' ij e j ' j j ei ' (ei 'e j )e j (e j ei ')e j ji e j j j j where Spherical Harmonics as rotator matrices 2 9/3/2017 ij ei e j ' Then we get 3 3 3 r' xi 'ei x j e j ' x j ij ei ij x j ei i1 j1 ji1 i, j or xi ' ij x j j or x ' x 1 11 12 13 1 r' x2 ' z ()r 21 22 23 x2 x3 ' 31 32 33 x3 where cos sin 0 z () sin cos 0 0 0 1 Using the relation r' z ()r , we can write down 3 3 3 r' z ()r z ()( x j e j ) x jz ()e j x j e j ' j1 j1 j1 where z ()e j e j ' , ei z ()e j ei e j ' ij We note that ijik (ei e j ')(ei ek ') e j 'ek ' j,k i i since e j ' (ei e j ')ei (formula) i Spherical Harmonics as rotator matrices 3 9/3/2017 _____________________________________________________________ Now we consider more general case in order to get the definition of vector. x3 P A Q x2 x1 Suppose that OP yi ei , OP' yi 'ei yiei ' i i i OQ zi ei , OQ' zi 'ei zi ei ' i i i Then we have A PQ OQ OP (zi yi )ei i A' P'Q' OQ' OP' (zi ' yi ')ei i (zi yi )ei ' i Since e j ' (ei e j ')ei ij ei , i j Spherical Harmonics as rotator matrices 4 9/3/2017 the expression of A can be rewritten as (zi ' yi ')ei (z j y j )e j ' (zi yi )ij ei . i j i, j Therefore we get zi ' yi ' ij (z j y j ) . j Since the component of A is given by Ai zi yi , and Ai ' zi 'yi ' in the old and new co-ordinate systems, we can write Ai ' Rij Aj . j We note that Ai 'ei Aie j ' i i In summary, under the rotation of the co-ordinate system, the components of the vector are transformed through Ai ' ij Aj j where the vector is a tensor of rank 1. Similarly, the components of the tensor are transformed through , for the tensor of rank 2 Ti j ' ik jlTkl j and for the tensor of rank 3 Ti jk ' il jmknTlmn j ((Note)) The scalar (tensor of rank 0) is invariant under the rotation. Spherical Harmonics as rotator matrices 5 9/3/2017 A'B' Ai B j (ei 'e j ') Ai B j i, j Ai Bi A B i, j i, j i ________________________________________________________________________ 2. Vector operators in quantum mechanics The operators corresponding to various physical quantities will be characterized by their behavior under rotation as scalars, vectors, and tensors. Vi ijV j j Note that is the rotation matrix. For the rotation by the angle around the z axis cos sin 0 cos sin 0 z () sin cos 0 , , az ( ) z () sin cos 0 0 0 1 0 0 1 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 Vj j or ˆ ˆ ˆ ˆ R Vi R ijVj j or ˆ ˆ ˆ ˆ Vi RijV j R j ((Note)) We show that Spherical Harmonics as rotator matrices 6 9/3/2017 ˆ ˆ ˆ ˆ RV j R Viij i From the relation ˆ ˆ ˆ ˆ ˆ ˆ ˆ Vi RijV j R ij RV j R j j we get ˆ ˆ ˆ ˆ Viik ijik RV j R i i, j ˆ ˆ ˆ RV j R ijik j i ˆ ˆ ˆ jk RV j R j ˆ ˆ ˆ RVk R or ˆ ˆ ˆ ˆ RV j R Viij i _______________________________________________________________________ We now consider a special case, infinitesimal rotation. i Rˆ 1ˆ Jˆ n 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 where ˆ ˆ ˆ ˆ ˆ ˆ V1 Vx , V2 Vy , V3 Vz For n = ez, Spherical Harmonics as rotator matrices 7 9/3/2017 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 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 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, Spherical Harmonics as rotator matrices 8 9/3/2017 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. ((Example)) When Vˆ Jˆ, ˆ ˆ ˆ [Ji , J j ] iijk Jk When Vˆ rˆ and Aˆ rˆ, ˆ [xˆi ,Lj ] iijk xˆk Spherical Harmonics as rotator matrices 9 9/3/2017 When Aˆ pˆ,and Aˆ rˆ, ˆ [pˆi ,Lj ] iijk pˆk ((Mathematica)) Signature[i,j,k], which is equal to the Levi-Civita ijk . ((Note)) Speherical component We define the spherical components as 1 1 Vˆ (Vˆ iVˆ ) Vˆ , 1 2 x y 2 ˆ 1 ˆ ˆ 1 ˆ ˆ ˆ V (V iV ) V , V0 Vz 1 2 x y 2 We note that ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [Vx , J x ] 0 , [V y , J x ] iVz , [Vz , J x ] iV y ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [Vx , J y ] iVz , [Vy , J y ] 0, [Vz , J y ] iVx ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [Vx , J z ] iVy , [Vy , J z ] iVx , [Vz , J z ] 0 We introduce the operators as ˆ ˆ ˆ ˆ ˆ ˆ V Vx iVy , J J x iJ y Using the above relation, we get ˆ ˆ ˆ ˆ ˆ ˆ ˆ [V , J x ] [Vx iVy , J x ] i[Vy , J x ] Vz ˆ ˆ ˆ ˆ ˆ ˆ ˆ [V , J x ] [Vx iVy , J x ] i[Vy , J x ] Vz ˆ ˆ ˆ ˆ ˆ ˆ ˆ [V , J y ] [Vx iVy , J y ] [Vx , J y ] iVz ˆ ˆ ˆ ˆ ˆ ˆ ˆ [V , J y ] [Vx iVy , J y ] [Vx , J y ] iVz ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [V , J z ] [Vx iVy , J z ] [Vx , J z ] i[Vy , J z ] V Spherical Harmonics as rotator matrices 10 9/3/2017 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [V , J z ] [Vx iVy , J z ] [Vx , J z ] i[Vy , J z ] V These relations in turn can be shown to lead to ˆ ˆ ˆ ˆ [V , J z ] V , [V , J z ] V ˆ ˆ ˆ ˆ [V , J ] 0 , [V , J ] 0 , ˆ ˆ ˆ ˆ ˆ ˆ [V , J ] 2Vz , [V , J ] 2Vz , Using the above relations, we get ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [J z ,V1,] V1 , [J z ,V1] V1 , [J z ,V0 ] 0 ˆ ˆ ˆ ˆ [J ,V1] 0 , [J ,V1,] 0 , ˆ ˆ ˆ ˆ ˆ [J1,V1] 2V0 , [J ,V1] 2V0 , These satisfy the following relations ˆ ˆ ˆ [J z ,Vq ] qVq ˆ ˆ ˆ [J ,Vq ] 2 q(q 1)Vq1 where q = 0, 1.
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