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**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]",

[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....

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).