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

  • Thread starter bjnartowt
  • Start date
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).
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....
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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