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Most op­er­a­tors in quan­tum me­chan­ics are of a spe­cial kind called Her­mit­ian. This sec­tion lists their most im­por­tant prop­er­ties.

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The first says that you can swap  and  if you take the com­plex con­ju­gate. (It is sim­ply a re­flec­tion of the fact that if you change the sides in an in­ner prod­uct, you turn it into its com­plex con­ju­gate. Nor­mally, that puts the op­er­a­tor at the other side, but for a Her­mit­ian op­er­a­tor, it does not make a dif­fer­ence.) The sec­ond is im­por­tant be­cause or­di­nary real num­bers typ­i­cally oc­cupy a spe­cial place in the grand scheme of things. (The fact that the in­ner prod­uct is real merely re­flects the fact that if a num­ber is equal to its com­plex con­ju­gate, it must be real; if there was an  in it, the num­ber would change by a com­plex con­ju­gate.)

In a given medium, both the phase velocity and group velocity depend on the frequency and on the medium's material properties (refractive index)

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In the lin­ear al­ge­bra of real ma­tri­ces, Her­mit­ian op­er­a­tors are sim­ply sym­met­ric ma­tri­ces. A ba­sic ex­am­ple is the in­er­tia ma­trix of a solid body in New­ton­ian dy­nam­ics. The or­tho­nor­mal eigen­vec­tors of the in­er­tia ma­trix give the di­rec­tions of the prin­ci­pal axes of in­er­tia of the body.

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Noting, the equation indicates that the group speed is equal to the phase velocity only when the refractive index is a constant (in dependent to the frequency) or the frequency is a constant (independent to the wavenumber). In this case both phase and group velocity are independent of frequency and equal to c/n.

An or­tho­nor­mal com­plete set of eigen­vec­tors or eigen­func­tions is an ex­am­ple of a so-called “ba­sis.” In gen­eral, a ba­sis is a min­i­mal set of vec­tors or func­tions that you can write all other vec­tors or func­tions in terms of. For ex­am­ple, the unit vec­tors , , and  are a ba­sis for nor­mal three-di­men­sion­al space. Every three-di­men­sion­al vec­tor can be writ­ten as a lin­ear com­bi­na­tion of the three.

S&P polarization refers to the plane in which the electric field of a light wave is oscillating. S-Polarization is the plane of polarization perpendicular to the page (coming out of the monitor screen). P-polarization is the plane of polarization parallel to the page (in the plane of the monitor screen). See figure below:

When referring to polarization states, the p-polarization refers to the polarization plane parallel to the polarization axis of the polarizer being used ("p" is for "parallel"). The s-polarization refers to the polarization plane perpendicular to the polarization axis of the polarizer. A linear polarizer, by design, polarizes light in the p-polarization.

Otherwise, if they vary with freuqency, the medium is called dispersive, the relation w=w(k) is known as the dispersion relation of the medium.