Transformation of a vector operator

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Homework Statement



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



Homework Equations



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

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 [tex] \hat{V_y} \hat{R_x} [/tex], however I am not sure since they are operators. If this is correct then would the solution be calculated by using arbitrary components of [tex] \hat{V_y} [/tex]? If this is completely wrong, what is a better way to look at this problem?

Thank you for any help.
 

Answers and Replies

  • #2
George Jones
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Let's start with a more basic question: given a rotation

[tex]R_{x} = \begin{pmatrix} 1& 0& 0\\ 0& cos\alpha & -sin\alpha \\ 0 & sin\alpha & cos\alpha \end{pmatrix},[/tex]

what is the corresponding operator [itex]\hat{R_{x}}[/itex] that acts on states (kets or wave functions)?
 
  • #3
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I guess that is what I am most confused about here. Is how does [tex] R_x [/tex] come correspond to [tex] \hat{R_x} [/tex].

So some thing like this: [tex] V_i \to R_{ij} V_j [/tex]


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

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