page_title: Photon polarization
section_title: Hermitian operators and energy conservation
hierarchical_section_title: Photon polarization / Optical filters and crystals / Example of energy conservation: Passage of a classical wave through a birefringent crystal / Hermitian operators and energy conservation
caption_attribution_description: English: Large crystal of Calcite on display at the National Museum of Natural History in Washington, DC. Also known as Icelandic Spar Русский: Кристалл исландского шпата
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: If the crystal is very thin, the final state will be only slightly different from the initial state. The unitary operator will be close to the identity operator. We can define the operator H by
  and the adjoint by
  Energy conservation then requires
  This requires that
  Operators like this that are equal to their adjoints are called Hermitian or self-adjoint.
  The infinitesimal transition of the polarization state is
  Thus, energy conservation requires that infinitesimal transformations of a polarization state occur through the action of a Hermitian operator.
caption: Doubly refracting Calcite from Iceberg claim, Dixon, New Mexico. This 35 pound (16 kg) crystal, on display at the National Museum of Natural History, is one of the largest single crystals in the United States.
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