Determine how the linear combinations εR=-√((1)/(2))(ε1+i ε2) εL=√((1)/(2))(ε1-i ε2)

Step 1: Understand the rotation matrix for a rotation about the $z$-axis. A rotation by an angle

Question

Determine how the linear combinations $$ \begin{aligned} & \boldsymbol{\varepsilon}_R=-\sqrt{\frac{1}{2}}\left(\varepsilon_1+i \varepsilon_2\right) \\ & \boldsymbol{\varepsilon}_L=\sqrt{\frac{1}{2}}\left(\varepsilon_1-i \varepsilon_2\right) \end{aligned} $$ transform under a rotation $\theta$ about the $z$ axis. Hence, show that $\varepsilon_R$ and $\varepsilon_L$ describe a photon of helicity +1 and -1 , respectively; $\varepsilon_{R, L}$ are called circular polarization vectors.

    Determine how the linear combinations
$$
\begin{aligned}
& \boldsymbol{\varepsilon}_R=-\sqrt{\frac{1}{2}}\left(\varepsilon_1+i \varepsilon_2\right) \\
& \boldsymbol{\varepsilon}_L=\sqrt{\frac{1}{2}}\left(\varepsilon_1-i \varepsilon_2\right)
\end{aligned}
$$
transform under a rotation $\theta$ about the $z$ axis. Hence, show that $\varepsilon_R$ and $\varepsilon_L$ describe a photon of helicity +1 and -1 , respectively; $\varepsilon_{R, L}$ are called circular polarization vectors.

Quarks and leptons: introductory course in modern particle physics

Francis Halzen, Alan… 1st Edition

Chapter 6, Problem 11

Instant Answer

Step 1

A rotation by an angle $\theta$ about the $z$-axis in a right-handed coordinate system can be represented by the rotation matrix: $$ R_z(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix}. $$ Show more…

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Determine how the linear combinations $$ \begin{aligned} & \boldsymbol{\varepsilon}_R=-\sqrt{\frac{1}{2}}\left(\varepsilon_1+i \varepsilon_2\right) \\ & \boldsymbol{\varepsilon}_L=\sqrt{\frac{1}{2}}\left(\varepsilon_1-i \varepsilon_2\right) \end{aligned} $$ transform under a rotation $\theta$ about the $z$ axis. Hence, show that $\varepsilon_R$ and $\varepsilon_L$ describe a photon of helicity +1 and -1 , respectively; $\varepsilon_{R, L}$ are called circular polarization vectors.

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