Show that for a polarization state Jpsi delta cos psi ei de

Show that for a polarization state J(psi, delta) = [cos psi e^i delta sin psi] the orthogonal state is [sin psi e^i(pi + delta) cos psi]

Solution

Polarization (also polarisation) is a parameter applying to waves that specifies the geometrical orientation of the oscillation.Electromagnetic waves such as light exhibit multiple polarizations, as do many other types of waves such as gravitational waves^[1] and sound waves in solids. On the other hand, sound waves in a gas or liquid only oscillate in the wave\'s direction of propagation, and the oscillation of ocean waves is always in the vertical direction. In these cases one doesn\'t normally speak of \"polarization\" since the oscillation\'s direction is not in question.

In an electromagnetic wave, both the electric field and magnetic field are oscillating but in different directions; by convention the \"polarization\" of light refers to the polarization of the electric field. Light which can be approximated as a plane wave in free space or in anisotropic medium propagates as a transverse wave—both the electric and magnetic fields are perpendicular to the wave\'s direction of travel. The oscillation of these fields may be in a single direction (linear polarization), or the field may rotate at the optical frequency (circular or elliptical polarization). In that case the direction of the fields\' rotation, and thus the specified polarization, may be either clockwise or counter clockwise; this is referred to as the wave\'s chirality or handedness.

The most common optical materials (such as glass) are isotropic and simply preserve the polarization of a wave but do not differentiate between polarization states. However, there are important classes of lossless materials classified as birefringent or optically active in which this is not the case and a wave\'s polarization will generally be modified or will affect propagation through it. In linear dichroism and circular dichroism, attenuation in propagation is dependent on the wave\'s polarization. One familiar example is the polarizer, an optical filter that transmits only one polarization.

Polarization is an important parameter in areas of science dealing with transverse wave propagation, such as optics, seismology, radio, and microwaves. Especially impacted are technologies such as lasers, wireless and optical fiber telecommunications, and radar.

According to quantum mechanics, the energy, linear momentum, and angular momentum of an electromagnetic wave are quantized in the form of photons. Then there is an identification between the electromagnetic polarization of the wave and polarization operators which determine the probability of a photon to be found in a given polarization state. In particular, the spin operator is shown to correspond to the basis states of circular polarization as described below in terms of electromagnetic fields. This is described in detail at Photon polarization.