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

bjnartowt

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

1. Homework Statement

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

2. Homework 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]",

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

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.... 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).

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bjnartowt

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....

bjnartowt

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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