page_title: Photon polarization
section_title: Dual of the final state
hierarchical_section_title: Photon polarization / Optical filters and crystals / Example of energy conservation: Passage of a classical wave through a birefringent crystal / Dual of the final state
context_page_description: Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon
  can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.
  The description of photon polarization contains many of the physical concepts and much of the mathematical machinery of more involved quantum descriptions, such as the quantum mechanics of an electron in a potential well. Polarization is an example of a qubit degree of freedom, which forms a fundamental basis for an understanding of more complicated quantum phenomena. Much of the mathematical machinery of quantum mechanics, such as state vectors, probability amplitudes, unitary operators, and Hermitian operators, emerge naturally from the classical Maxwell's equations in the description. The quantum polarization state vector for the photon, for instance, is identical with the Jones vector, usually used to describe the polarization of a classical wave.
context_section_description: The initial polarization state is transformed into the final state with the operator U. The dual of the final state is given by
  where  is the adjoint of U, the complex conjugate transpose of the matrix.
caption: A calcite crystal laid upon a paper with some letters showing the double refraction
exif: {}