Transformation of a vector operator

In summary, the problem involves calculating the result of the transformation of the vector operator \hat{V_{y}} by rotation \hat{R_{x}} around an angle \alpha. The corresponding operator for the rotation is \hat{R_{x}} = \begin{pmatrix} 1& 0& 0\\ 0& cos\alpha & -sin\alpha \\ 0 & sin\alpha & cos\alpha \end{pmatrix}. The main confusion is how the rotation matrix corresponds to the operator, and it is believed that the transformation would involve transforming the ket \mid \psi \rangle in this way: \langle \psi \mid V_i \mid \psi \rangle \to R_{
  • #1
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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.
 
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  • #2
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
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].
 

1. What is a vector operator?

A vector operator is a mathematical operation that acts on vector quantities, such as displacement, velocity, or force. It is used to describe physical quantities that have both magnitude and direction.

2. How is a vector operator transformed?

A vector operator is transformed by applying a mathematical operation, known as a transformation operator, to the original vector. This transformation operator can be a rotation, translation, or reflection, depending on the desired transformation.

3. Why is the transformation of a vector operator important?

The transformation of a vector operator is important because it allows us to describe physical quantities in different coordinate systems. This is crucial in many scientific fields, such as physics and engineering, where different coordinate systems are used to analyze and understand various phenomena.

4. What are some common transformation rules for vector operators?

Some common transformation rules for vector operators include the commutative and associative properties, as well as the distributive property. These rules allow us to manipulate and simplify vector equations, making them easier to solve.

5. How does the transformation of a vector operator affect its components?

The transformation of a vector operator can affect its components in various ways, depending on the type of transformation applied. For example, a rotation transformation will change the direction of the vector, while a translation transformation will change its magnitude. The components of a vector operator can also be affected by a combination of transformations.

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