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Shankar 12.4.4 - the rotation matrix vs. a rotation matrix (tensor operators QM)

  1. Nov 22, 2011 #1
    Shankar 12.4.4 - "the" rotation matrix vs. "a" rotation matrix (tensor operators QM)

    1. The problem statement, all variables and given/known data

    My question comes up in the context of Shankar 12.4.4. See attached .pdf.

    2. Relevant equations

    See attached .pdf

    3. The attempt at a solution

    See attached .pdf

    I have this problem: on Shankar p. 313, they say:
    >>>>
    We call V a vector operator if V's components transform as components of a vector under a passive transformation generated by U[R]",

    [itex]{U^\dag }[R]{V_i}U[R] = {R_{ij}}{V_j}[/itex]

    where R[ij] is *the* 2x2 rotation matrix appearing in [12.2.1] (NOTE, below)...The same definition of a vector operator holds in 3D as well, with the obvious difference that R[ij] is a 3x3 matrix."
    <<<<<<<<<<<<<<<<
    But aren't there multiple 3x3 rotation matrices?

    Also: I have attached some work from Merzbacher, beginning of chapter 12. I did this work, and something isn't clicking in my thick skull.... :-p



    NOTE: Can't look up 12.2.1, because p. 306 of my .pdf book is gone, and I didn't bring my hard-copy book home, as I'm on Thanksgiving break).
     
  2. jcsd
  3. Nov 22, 2011 #2
    Re: Shankar 12.4.4 - "the" rotation matrix vs. "a" rotation matrix (tensor operators

    I'm sorry, I didn't make my question clear. Initially, I am asking: *what* R[ij] is being talked about as "The" R[ij]. I think this is somewhat-related to my "another another" try in the first attachment "320 - pr 4-4 - classical-esque...", where I try to reverse-engineer a matrix R that has the components I want. However, as my aim is to solve this problem correctly, that may not even be the right question to ask....
     
  4. Nov 22, 2011 #3
    Re: Shankar 12.4.4 - "the" rotation matrix vs. "a" rotation matrix (tensor operators

    ....It also looks like the .pdfs didn't go through -_-
     
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