# Linear Polarization Given that the [photon linear polarization states](photon%20polarization.md#linear%20polarization) are constructed with vectors $|\leftrightarrow\rangle=\begin{pmatrix}1\\0\end{pmatrix}$ and, $|\updownarrow\rangle=\begin{pmatrix}0\\1\end{pmatrix}$ we [construct](Observable.md#Constructing%20observables) the corresponding [observable](Observables%20of%20two%20level%20systems.md) as follows: $\hat{\varepsilon}=\hbar(|\leftrightarrow\rangle\langle\leftrightarrow|-|\updownarrow\rangle\langle\updownarrow|)=\hbar\hat{\sigma}_3$ Notice that this is just the the third $(z)$ [Pauli matrix](Observables%20of%20two%20level%20systems.md#^d36fb7), where the [eigenvalues](Observables%20of%20two%20level%20systems.md#Construction%20of%20two-level%20system%20observables) of $\sigma_3$ are $+1$ and $-1$ of respectively for the horizontal and vertical polarizations, and thus the possible measured quantities are $+\hbar$ and $-\hbar$ for a single photon polarized horizontally and vertically respectively. # Circular Polarization The [observable](Observables%20of%20two%20level%20systems.md) that gives the polarization direction for a circularly polarized photon is called the [helicity](Helicity%20(optics).md) operator. _This is equivalent to the [spin one](Spin%20one%20operator.md) operator_. The helicity is encoded in the corresponding eigenvalues ($+1$ helicity for $|R\rangle$ and $-1$ helicity for $|L\rangle$), where the measured quantities thus $\pm \hbar$, we simply rewrite the observable in the $|R\rangle$, $|L\rangle$ [basis](photon%20polarization.md#circular%20polarization). $\hat{\varepsilon}=\hbar(|R\rangle\langle R|-|L\rangle\langle L|)=\hbar\hat{\sigma}_2$ where here instead we obtain the 2nd $(y)$, [Pauli matrix](Observables%20of%20two%20level%20systems.md#^d36fb7) in the matrix representation from evaluating in this new basis. #QuantumMechanics/QuantumOptics #QuantumMechanics/QuantumMeasurement/QuantumObservables