Transformation of a vector operator

[andre220]

Homework Statement

Calculate the result of the transformation of the vector operator $$\hat{V_{y}}$$ by rotation $$\hat{R_{x}}$$ around an angle $$\alpha$$.

Homework Equations

I believe that $$\hat{R_{x}} = \begin{pmatrix} 1& 0& 0\\ 0& cos\alpha & -sin\alpha \\ 0 & sin\alpha & cos\alpha \end{pmatrix}$$

Not sure if the fact that it is an operator makes any difference here...

The Attempt at a Solution

So at first glance it seems that the solution should be something like the calculation of $$\hat{V_y} \hat{R_x}$$, however I am not sure since they are operators. If this is correct then would the solution be calculated by using arbitrary components of $$\hat{V_y}$$? If this is completely wrong, what is a better way to look at this problem?

Thank you for any help.

 

[George Jones]

Let's start with a more basic question: given a rotation

$$R_{x} = \begin{pmatrix} 1& 0& 0\\ 0& cos\alpha & -sin\alpha \\ 0 & sin\alpha & cos\alpha \end{pmatrix},$$

what is the corresponding operator $\hat{R_{x}}$ that acts on states (kets or wave functions)?

 

[andre220]

I guess that is what I am most confused about here. Is how does $$R_x$$ come correspond to $$\hat{R_x}$$.

So some thing like this: $$V_i \to R_{ij} V_j$$


If I had to make an educated guess here I would say it would transform on the ket $$\mid \psi \rangle$$ in this way: $$\langle \psi \mid V_i \mid \psi \rangle \to R_{ij} \langle \psi \mid V_j \mid \psi \rangle$$.