Photon polarization
Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.
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.
Hermitian operators then follow for infinitesimal transformations of a classical polarization state.
The connection with quantum mechanics is made through the identification of a minimum packet size, called a photon, for energy in the electromagnetic field.
In the case of circular polarization, the electric field vector of constant magnitude rotates in the x–y plane.
The general case in which the electric field rotates in the x–y plane and has variable magnitude is called elliptical polarization.
If only the traced out shape and the direction of the rotation of (x(t), y(t)) is considered when interpreting the polarization state, i.e. only
It can be seen that for a linearly polarized state, M will be a line in the xy plane, with length 2 and its middle in the origin, and whose slope equals to tan(θ).
Birefringent crystals therefore provide an ideal test bed for examining the conservative transformation of polarization states.
Even though this treatment is still purely classical, standard quantum tools such as unitary and Hermitian operators that evolve the state in time naturally emerge.
Light polarized parallel to the axis are called "extraordinary rays" or "extraordinary photons", while light polarized perpendicular to the axis are called "ordinary rays" or "ordinary photons".
Thus, energy conservation requires that infinitesimal transformations of a polarization state occur through the action of a Hermitian operator.
It is a testament, however, to the generality of Maxwell's equations for electrodynamics that the treatment can be made quantum mechanical with only a reinterpretation of classical quantities.
[citation needed] Einstein's conclusion from early experiments on the photoelectric effect is that electromagnetic radiation is composed of irreducible packets of energy, known as photons.
This has been demonstrated for spin angular momentum, but it is in general true for any observable quantity.
Dirac explains this in the context of the double-slit experiment: Some time before the discovery of quantum mechanics people realized that the connection between light waves and photons must be of a statistical character.
Suppose we have a beam of light consisting of a large number of photons split up into two components of equal intensity.
In general, the rules for combining probability amplitudes look very much like the classical rules for composition of probabilities: [The following quote is from Baym, Chapter 1][clarification needed] For any legal[clarification needed] operators the following inequality, a consequence of the Cauchy–Schwarz inequality, is true.
which means that angular momentum and the polarization angle cannot be measured simultaneously with infinite accuracy.
Much of the mathematical apparatus of quantum mechanics appears in the classical description of a polarized sinusoidal electromagnetic wave.
The right and left circular components of the Jones vector can be interpreted as probability amplitudes of spin states of the photon.
These conclusions are a natural consequence of the structure of Maxwell's equations for classical waves.
Quantum mechanics enters the picture when observed quantities are measured and found to be discrete rather than continuous.
In the case angular momentum, for instance, the allowed observable values are the eigenvalues of the spin operator.
These concepts have emerged naturally from Maxwell's equations and Planck's and Einstein's theories.
In fact, the typical program is to assume the concepts of this section and then to infer the unknown dynamics of a physical system.
In that case, working back from the principles in this section, the quantum dynamics of particles were inferred, leading to Schrödinger's equation, a departure from Newtonian mechanics.
This is not the only occasion[dubious – discuss] in which Maxwell's equations have forced a restructuring of Newtonian mechanics.
Special relativity resulted from attempts to make classical mechanics consistent with Maxwell's equations (see, for example, Moving magnet and conductor problem).
