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Tensor Operators

  1. Feb 28, 2012 #1
    I'm taking a course that's taught out of Shankar, and I'm going to be tested on Tensor Operators, which is 15.3, p.417-421 in Shankar. I've never actually worked with tensors before (except the Maxwell stress tensor in EM), and I find that section too hard to understand.

    Does anyone know of a more accessible treatment of tensor operators in another textbook? Particularly one that covers basics about tensors instead of assuming some prior exposure. (Tensor operators are way beyond my go-to book, Griffiths.) Here is an example of a problem I'm supposed to be able to do but I'm absolutely lost:

    Construct a spherical tensor operator of rank 2 from the components of the Cartesian tensor operator Tjk=xjxk.

    I would really appreciate any suggestions, or even explanations if you feel generous enough to explain tensor operators.
     
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  3. Feb 28, 2012 #2

    atyy

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    Last edited by a moderator: May 5, 2017
  4. Mar 1, 2012 #3
    Sakurai explains how to do this, and you can get by with Griffith's quantum mechanics book to do this. In Sakurai they work your specific example of a dyad. To get some idea how this works all you need to know is how to expand the tensor product of two vectors in terms of a direct sum. That is formula 4.186 in Griffifths quantum mechanics book [tex] |j_1,m_1>\otimes|j_2,m_2>=\sum <j_1 j_2 ; m_1 m_2|j_1 j _2 ; j m>|j,m>[/tex]

    So we have something like [tex] U_iV_j|1,i>\otimes|1,j>[/tex]

    We know by the triangle rule of adding angular momentum that where going to get a 2, 1, and 0. That is a rank 2 tensor a vector and a scalar. This just means that the tensor product of two vectors breaks down into the direct sum of a scalar, vector, and a rank 2 tensor. Let's see how this works explicitly. Consider, [tex]U_1V_-1|1,1>\otimes|1,-1> [/tex] Using the Clebsch Gordon formula we get that [tex]U_1V_-1|1,1>\otimes|1,-1>=U_1V_-1(1/\sqrt{6}|2,0>+1/\sqrt{2}|1,0>+1/\sqrt{3}|0,0>) [/tex] Working through all the combos of U and V you can see what goes where and you can check your answer in Sakurai page 236
     
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