# Transformation of a vector operator

1. Sep 25, 2013

### andre220

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

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

2. Relevant 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...

3. 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.

2. Sep 26, 2013

### George Jones

Staff Emeritus

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

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

3. Sep 26, 2013

### 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: $$\langle \psi \mid V_i \mid \psi \rangle \to R_{ij} \langle \psi \mid V_j \mid \psi \rangle$$.