Consider a perfectly isolated [[photon]], it may be [polarized](matter%20waves.md#Matter%20Wave%20Polarization) and this polarization comes about as a quantized form of the [polarization](Optical%20Polarization.md) observed in optics, where we describe a classical model of an emergent wave-propagation that happens with many photons. # [[Linear polarization]] The quantum [[State vector]] for a linearly polarized photon has two components, corresponding to the $x$ and $y$ axes of a cartesian plane, thus it is given as $|\psi\rangle = \begin{pmatrix}\psi_x\\\psi_y\end{pmatrix}=e^{i\alpha}\begin{pmatrix}\cos{(\theta)}\\\sin{(\theta)}\end{pmatrix}$ And the horizontal and vertical polarization states are, up to a phase factor $e^{i\alpha}$, given as $|\psi\rangle=\begin{pmatrix}1\\0\end{pmatrix}=|\leftrightarrow\rangle$ $|\psi\rangle=\begin{pmatrix}0\\1\end{pmatrix}=|\updownarrow\rangle$ This forms an [orthonormal basis](State%20vector.md#Decomposing%20state%20vectors%20into%20sets%20of%20orthonormal%20eigenstates), $\{|\rightarrow\rangle,|\updownarrow\rangle\}$, and thus: # [[Circular polarization]] Here we refer to _right_ and _left_ polarized states as $|R\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\i\end{pmatrix}$ $|L\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\-i\end{pmatrix}$ This forms an [orthonormal basis](State%20vector.md#Decomposing%20state%20vectors%20into%20sets%20of%20orthonormal%20eigenstates), $\{|R\rangle,||L\rangle\}.$ Left and right polarized photons are equivalently [Spin](Spin.md#Spin%20quantum%20numbers) _up_ and spin _down_ photons respectively. ## [[Elliptical polarization]] This is simply a generalization from the circular form, as it is in classical optics. # Switching between linear and circular bases Each basis can be represented as a [[Quantum superposition]] of the other basis. # Probabilities The probability that a photon is measured at a particular spin state follows straight from the [[Born rule]]. # Observables We can use the eigenbases to construct to correspond observables represented as the: [[polarization operator]]s. --- # Recommended reading Photon polarization is a common example introduced in the first chapter of many general introductions to quantum mechanics. This is because it is an example that clearly motivates an introduction to [two-level systems](Stationary%20two-level%20systems.md) and can be done so without introducing time-dependence. See for example: * [Baym, G., _Lectures on Quantum Mechanics_, Westview Press, 1990](Baym,%20G.,%20Lectures%20on%20Quantum%20Mechanics,%20Westview%20Press,%201990.md) pgs. 1-25. #QuantumMechanics/QuantumOptics #QuantumMechanics/StationaryStateQuantumSystems #QuantumMechanics/TwoLevelSystems