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'''Photon polarization''' is the [[Quantum mechanics|quantum mechanical]] description of the [[Classical physics|classical]] [[polarized]] [[sinusoidal]] [[plane wave|plane]] [[electromagnetic wave]]. Individual [[photon]] eigenstates have either right or left [[circular polarization]]. A photon that is in a superposition of eigenstates can have [[Linear polarization|linear]], [[Circular polarization|circular]], or [[elliptical polarization]].
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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]], and forms a fundamental basis for an understanding of more complicated quantum phenomena. Much of the mathematical machinery of quantum mechanics, such as [[Quantum state|state vectors]], [[probability amplitude]]s, [[unitary operator]]s, and [[Hermitian operator]]s, 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]]. Unitary operators emerge from the classical requirement of the [[conservation of energy]] of a classical wave propagating through media that alter the polarization state of the wave. Hermitian operators then follow for infinitesimal transformations of a classical polarization state.  
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Many of the implications of the mathematical machinery are easily verified experimentally. In fact, many of the experiments can be performed with two pairs (or one broken pair) of [[Polaroid (polarizer)|polaroid]] sunglasses.
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The connection with quantum mechanics is made through the identification of a minimum packet size, called a [[photon]], for energy in the electromagnetic field. The identification is based on the theories of [[Max Planck|Planck]] and the interpretation of those theories by [[Albert Einstein|Einstein]]. The [[correspondence principle]] then allows the identification of momentum and angular momentum (called [[Spin (physics)|spin]]), as well as energy, with the photon.
 
==Polarization of classical electromagnetic waves==
 
{{Main|Sinusoidal plane-wave solutions of the electromagnetic wave equation}}
 
===Polarization states===
====Linear polarization====
 
{{Main|Linear polarization}}
 
[[Image:Mudflats-polariser.jpg|right|thumb|400px|Effect of a polarizer on reflection from mud flats. In the first picture, the polarizer is rotated to minimize the effect; in the second it is rotated 90° to maximize it: almost all reflected sunlight is eliminated.]]
 
The wave is linearly polarized (or plane polarized) when the phase angles <math> \alpha_x^{ } , \alpha_y </math> are equal,
 
:<math>    \alpha_x =  \alpha_y \ \stackrel{\mathrm{def}}{=}\    \alpha.  </math>
 
This represents a wave with [[phase (waves)|phase]] <math>\alpha</math> polarized at an angle <math> \theta    </math> with respect to the x axis. In that case the Jones vector can be written
 
:<math>  |\psi\rangle  =  \begin{pmatrix} \cos\theta    \\ \sin\theta  \end{pmatrix} \exp \left ( i \alpha \right )  .</math>
 
The state vectors for linear polarization in x or y are special cases of this state vector.
 
If unit vectors are defined such that
 
:<math>  |x\rangle  \ \stackrel{\mathrm{def}}{=}\    \begin{pmatrix} 1    \\ 0  \end{pmatrix}    </math>
 
and
 
:<math>  |y\rangle  \ \stackrel{\mathrm{def}}{=}\    \begin{pmatrix} 0    \\ 1  \end{pmatrix}    </math>
 
then the linearly polarized polarization state can written in the "x-y basis" as
 
:<math>  |\psi\rangle  =  \cos\theta \exp \left ( i \alpha \right ) |x\rangle + \sin\theta \exp \left ( i \alpha \right ) |y\rangle = \psi_x |x\rangle + \psi_y |y\rangle. </math>
 
====Circular polarization====
{{Main|Circular polarization}}
If the phase angles <math>\alpha_x</math> and <math>\alpha_y</math> differs by exactly <math>\pi / 2</math> and the x amplitude equals the y amplitude the wave is [[Circular polarization|circularly polarized]]. The Jones vector then becomes
 
:<math>  |\psi\rangle  =  \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ \pm i  \end{pmatrix} \exp \left ( i \alpha_x \right )  </math>
 
where the plus sign indicates right circular polarization and the minus sign indicates left circular polarization. In the case of circular polarization, the electric field vector of constant magnitude rotates in the x-y plane.
 
If unit vectors are defined such that
 
:<math>  |R\rangle  \ \stackrel{\mathrm{def}}{=}\    {1 \over \sqrt{2}} \begin{pmatrix} 1    \\ i  \end{pmatrix}    </math>
 
and
 
:<math>  |L\rangle  \ \stackrel{\mathrm{def}}{=}\    {1 \over \sqrt{2}} \begin{pmatrix} 1    \\ -i  \end{pmatrix}    </math>
 
then an arbitrary polarization state can written in the "R-L basis" as
 
:<math>  |\psi\rangle  = \psi_R |R\rangle + \psi_L |L\rangle  </math>
 
where
 
:<math>\psi_R = \langle R|\psi\rangle = \frac{1}{\sqrt{2}}(\cos\theta\exp(i\alpha_x) - i\sin\theta\exp(i\alpha_y))</math>
 
and
 
:<math>\psi_L = \langle L|\psi\rangle = \frac{1}{\sqrt{2}}(\cos\theta\exp(i\alpha_x) + i\sin\theta\exp(i\alpha_y)).</math>
 
We can see that
 
:<math> 1 = |\psi_R|^2 + |\psi_L|^2 .  </math>
 
====Elliptical polarization====
 
{{Main|Elliptical polarization}}
 
The general case in which the electric field rotates in the x-y plane and has variable magnitude is called [[elliptical polarization]]. The state vector is given by
 
:<math>  |\psi\rangle  \ \stackrel{\mathrm{def}}{=}\  \begin{pmatrix} \psi_x  \\ \psi_y  \end{pmatrix} =  \begin{pmatrix} \cos\theta \exp \left ( i \alpha_x \right )  \\ \sin\theta \exp \left ( i \alpha_y \right )  \end{pmatrix}. </math>
 
====Geometric visualization of an arbitrary polarization state====
To get an understanding of what a polarization state looks like, one can observe the orbit that is made if the polarization state is multiplied by a phase factor of <math>e^{i\omega t}</math> and then having the real parts of its components interpreted as x and y coordinates respectively. That is:
 
:<math>\begin{pmatrix}x(t)\\y(t)\end{pmatrix} = \begin{pmatrix}\Re(e^{i\omega t}\psi_x)\\ \Re(e^{i\omega t}\psi_y)\end{pmatrix} = \Re\left[e^{i\omega t}\begin{pmatrix}\psi_x\\ \psi_y\end{pmatrix}\right] = \Re\left(e^{i\omega t}|\psi\rangle\right).</math>
 
If only the traced out shape and the direction of the rotation of {{math|(<var>x</var>(<var>t</var>), <var>y</var>(<var>t</var>))}} is considered when interpreting the polarization state, i.e. only
 
:<math>M(|\psi\rangle) = \left.\left\{\Big( x(t),\,y(t) \Big)\,\right|\,\forall\,t \right\}</math>
 
(where {{math|<var>x</var>(<var>t</var>)}} and {{math|<var>y</var>(<var>t</var>)}} are defined as above) and whether it is overall more right circularly or left circularly polarized (i.e. whether {{math|{{!}}<var>{{unicode|ψ}}<sub><sub>R</sub></sub></var>{{!}} > {{!}}<var>{{unicode|ψ}}<sub><sub>L</sub></sub></var>{{!}}}} or vice versa), it can be seen that the physical interpretation will be the same even if the state is multiplied by an arbitrary phase factor, since
 
:<math>M(e^{i\alpha}|\psi\rangle) = M(|\psi\rangle),\ \alpha\in\mathbb{R}</math>
 
and the direction of rotation will remain the same. In other words, there is no physical difference between two polarization states <math>|\psi\rangle</math> and <math>e^{i\alpha}|\psi\rangle</math>, between which only a phase factor differs.
 
It can be seen that for a linearly polarized state, <var>M</var> will be a line in the xy plane, with length 2 and its middle in the origin, and whose slope equals to {{math|tan(<var>{{unicode|θ}}</var>)}}. For a circularly polarized state, <var>M</var> will be a circle with radius {{math|1/{{sqrt|2}}}} and with the middle in the origin.
 
==Energy, momentum, and angular momentum of a classical electromagnetic wave==
===Energy density of classical electromagnetic waves===
====Energy in a plane wave====
 
{{Main|Energy density}}
 
The [[Energy density|energy per unit volume]] in classical electromagnetic fields is (cgs units)
 
:<math> \mathcal{E}_c = \frac{1}{8\pi} \left [ \mathbf{E}^2( \mathbf{r} , t ) + \mathbf{B}^2( \mathbf{r} , t ) \right ] .</math>
 
For a plane wave, this becomes
 
:<math> \mathcal{E}_c = \frac{\mid \mathbf{E} \mid^2}{8\pi}    </math>
 
where the energy has been averaged over a wavelength of the wave.
 
====Fraction of energy in each component====
 
The fraction of energy in the x component of the plane wave is
 
:<math> f_x = \frac{ \mid \mathbf{E} \mid^2 \cos^2\theta }{ \mid \mathbf{E} \mid^2 } = \psi_x^*\psi_x = \cos^2 \theta </math>
 
with a similar expression for the y component resulting in <math>f_y=\sin^2\theta</math>.
 
The fraction in both components is
 
:<math> \psi_x^*\psi_x + \psi_y^*\psi_y = \langle \psi  | \psi\rangle = 1. </math>
 
===Momentum density of classical electromagnetic waves===
 
The momentum density is given by the [[Poynting vector]]
 
:<math> \boldsymbol { \mathcal{P}} = {1 \over 4\pi c } \mathbf{E}( \mathbf{r}, t ) \times \mathbf{B}( \mathbf{r}, t ).  </math>
 
For a sinusoidal plane wave traveling in the z direction, the momentum is in the z direction and is related to the energy density:
 
:<math> \mathcal{P}_z c = \mathcal{E}_c.  </math>
 
The momentum density has been averaged over a wavelength.
 
===Angular momentum density of classical electromagnetic waves===
 
Electromagnetic waves can have both [[Light orbital angular momentum|orbital]] and [[Light spin angular momentum|spin]] angular momentum.<ref name= "Orbital">{{cite journal |author=Allen, L.; Beijersbergen, M.W.; Spreeuw, R.J.C.; Woerdman, J.P. |title=Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes |journal=Physical Review A |volume=45 |issue=11 |pages=8186–9 |date=June 1992 |doi=10.1103/PhysRevA.45.8185 |url=http://pra.aps.org/abstract/PRA/v45/i11/p8185_1 |pmid=9906912|bibcode = 1992PhRvA..45.8185A }}</ref> The total angular momentum density is  
 
:<math> \boldsymbol { \mathcal{L} } = \mathbf{r} \times \boldsymbol { \mathcal{P} } = {1 \over 4\pi c } \mathbf{r} \times \left [ \mathbf{E}( \mathbf{r}, t ) \times \mathbf{B}( \mathbf{r}, t ) \right ]. </math>
 
For a sinusoidal plane wave propagating along <math>z</math> axis the orbital angular momentum density vanishes. The spin angular momentum density is in the <math>z</math> direction and is given by
 
:<math> \mathcal{L} = { {\mid \mathbf{E} \mid^2}  \over {8\pi\omega} } \left ( \mid \langle R  | \psi\rangle \mid^2 - \mid \langle L  | \psi\rangle \mid^2 \right ) = { 1  \over \omega } \mathcal{E}_c \left ( \mid \psi_R \mid^2 - \mid \psi_L \mid^2 \right ) </math>
 
where again the density is averaged over a wavelength.
 
==Optical filters and crystals==
===Passage of a classical wave through a polaroid filter===
 
[[Image:060815 polaroid.svg|250px|thumb|right|Linear polarization]]
 
A [[linear filter]] transmits one component of a plane wave and absorbs the perpendicular component. In that case, if the filter is polarized in the x direction, the fraction of energy passing through the filter is
 
:<math> f_x  = \psi_x^*\psi_x = \cos^2\theta.\,  </math>
 
===Example of energy conservation: Passage of a classical wave through a birefringent crystal===
 
An ideal [[Birefringence|birefringent]] crystal transforms the polarization state of an electromagnetic wave without loss of wave energy. 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.
 
====Initial and final states====
 
A birefringent crystal is a material that has an '''optic axis''' with the property that the light has a different [[index of refraction]] for light polarized parallel to the axis than it has for light polarized perpendicular to the axis. 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''". If a linearly polarized wave impinges on the crystal, the extraordinary component of the wave will emerge from the crystal with a different phase than the ordinary component. In mathematical language, if the incident wave is linearly polarized at an angle <math> \theta  </math> with respect to the optic axis, the incident state vector can be written
 
:<math>  |\psi\rangle  =  \begin{pmatrix} \cos\theta    \\ \sin\theta    \end{pmatrix}  </math>
 
and the state vector for the emerging wave can be written
 
:<math>  |\psi '\rangle  =  \begin{pmatrix} \cos\theta \exp \left ( i \alpha_x \right )  \\ \sin\theta \exp \left ( i \alpha_y \right )  \end{pmatrix}  = \begin{pmatrix}  \exp \left ( i \alpha_x \right ) & 0  \\ 0 &  \exp \left ( i \alpha_y \right )  \end{pmatrix} \begin{pmatrix} \cos\theta    \\ \sin\theta    \end{pmatrix} \ \stackrel{\mathrm{def}}{=}\  \hat{U} |\psi\rangle. </math>
 
While the initial state was linearly polarized, the final state is elliptically polarized. The birefringent crystal alters the character of the polarization.
 
====Dual of the final state====
 
[[Image:Calcite.jpg|right|thumb|400px|A calcite crystal laid upon a paper with some letters showing the double refraction]]
 
The initial polarization state is transformed into the final state with the [[Operator (physics)|operator]] U. The dual of the final state is given by
 
:<math>  \langle \psi '|  =  \langle \psi | \hat{U}^{\dagger} </math>
 
where <math>  U^{\dagger} </math> is the [[Hermitian adjoint|adjoint]] of U, the complex conjugate transpose of the matrix.
 
====Unitary operators and energy conservation====
 
The fraction of energy that emerges from the crystal is
 
:<math>\langle\psi '| \psi '\rangle = \langle\psi |\hat{U}^{\dagger}\hat{U}|\psi\rangle = \langle \psi|\psi\rangle = 1.</math>
 
In this ideal case, all the energy impinging on the crystal emerges from the crystal. An operator U with the property that
 
:<math>\hat{U}^{\dagger}\hat{U} = I,</math>
 
where I is the [[identity function|identity operator]] and U is called a [[unitary operator]]. The unitary property is necessary to ensure [[energy conservation]] in state transformations.
 
====Hermitian operators and energy conservation====
 
[[Image:Calcite-HUGE.jpg|thumb|right|Doubly refracting Calcite from Iceberg claim, Dixon, New Mexico. This 35 pound (16&nbsp;kg) crystal, on display at the [[National Museum of Natural History]], is one of the largest single crystals in the United States.]]
 
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
 
:<math>    \hat{U}  \approx I + i\hat{H} </math>
 
and the adjoint by
 
:<math>    \hat{U}^{\dagger}  \approx I - i\hat{H}^{\dagger}. </math>
 
Energy conservation then requires
 
:<math>  I =  \hat{U}^{\dagger} \hat{U} \approx \left ( I - i\hat{H}^{\dagger} \right ) \left ( I + i\hat{H} \right ) \approx I - i\hat{H}^{\dagger} + i\hat{H}. </math>
 
This requires that
 
:<math>    \hat{H} = \hat{H}^{\dagger}. </math>
 
Operators like this that are equal to their adjoints are called [[Self-adjoint operator|Hermitian]] or self-adjoint.
 
The infinitesimal transition of the polarization state is
 
:<math> |\psi ' \rangle  - |\psi\rangle  =  i\hat{H} |\psi\rangle.   </math>
 
Thus, energy conservation requires that infinitesimal transformations of a polarization state occur through the action of a Hermitian operator.
 
==Photons: The connection to quantum mechanics==
 
{{Main|Photon}}
 
===Energy, momentum, and angular momentum of photons===
====Energy====
 
<!--  Commented out: [[Image:Max-Planck-und-Albert-Einstein.jpg|thumb||Max Planck presents [[Albert Einstein]] with the Max Planck medal, Berlin June 28, 1929]] -->
 
The treatment to this point has been [[Classical physics|classical]]. It is a testament, however, to the generality of [[Maxwell's equations]] for electrodynamics that the treatment can be made [[Quantum mechanics|quantum mechanical]] with only a reinterpretation of classical quantities. The reinterpretation is based on the theories of [[Max Planck]] and the interpretation by [[Albert Einstein]] of those theories and of other experiments.
 
Einsteins's conclusion from early experiments on the [[photoelectric effect]] is that electromagnetic radiation is composed of irreducible packets of energy, known as [[photon]]s. The energy of each packet is related to the angular frequency of the wave by the relation
 
:<math>    \epsilon = \hbar \omega </math>
 
where <math>  \hbar  </math> is an experimentally determined quantity known as [[Planck's constant]]. If there are <math>    N  </math> photons in a box of volume <math>  V </math>, the energy in the electromagnetic field is
 
:<math>    N \hbar \omega </math>
 
and the energy density is
 
:<math>    {N \hbar \omega \over V} </math>
 
The energy of a photon can be related to classical fields through the [[correspondence principle]] which states that for a large number of photons, the quantum and classical treatments must agree. Thus, for very large <math>    N  </math>, the quantum energy density must be the same as the classical energy density
 
:<math>    {N \hbar \omega \over V} = \mathcal{E}_c = \frac{\mid \mathbf{E} \mid^2}{8\pi}. </math>
 
The number of photons in the box is then
 
:<math>    N  = \frac{V }{8\pi \hbar \omega}\mid  \mathbf{E} \mid^2 . </math>
 
====Momentum====
 
The correspondence principle also determines the momentum and angular momentum of the photon. For momentum
 
:<math> \mathcal{P}_z =  {N \hbar \omega \over cV} = {N \hbar k_z \over V}  </math>
 
where kz is the wave number. This implies that the momentum of a photon is
 
:<math>  p_z=\hbar k_z .\,  </math>
 
====Angular momentum and spin====
 
Similarly for the spin angular momentum
 
:<math> \mathcal{L}  = { 1  \over \omega } \mathcal{E}_c \left ( \mid \psi_R \mid^2 - \mid \psi_L \mid^2 \right ) = { N\hbar  \over V }  \left ( \mid \psi_R \mid^2 - \mid \psi_L \mid^2 \right )</math>
 
where Ec is field strength. This implies that the spin angular momentum of the photon is
 
:<math> l_z = \hbar  \left ( \mid \psi_R \mid^2 - \mid \psi_L \mid^2 \right ). </math>
 
the quantum interpretation of this expression is that the photon has a probability of <math>  \mid \psi_R \mid^2  </math> of having a spin angular momentum of <math> \hbar  </math> and a probability of <math>  \mid \psi_L \mid^2  </math> of having a spin angular momentum of <math> -\hbar  </math>. We can therefore think of the spin angular momentum of the photon being quantized as well as the energy. The angular momentum of classical light has been verified.<ref>{{cite journal |author=Beth, R.A. |title=Direct detection of the angular momentum of light |journal=Phys. Rev. |volume=48 |issue=5 |page=471 |year=1935 |doi=10.1103/PhysRev.48.471 |url=http://prola.aps.org/abstract/PR/v48/i5/p471_1|bibcode = 1935PhRv...48..471B }}</ref> Photons have only been observed to have spin angular momenta of <math> \pm \hbar </math>.{{Citation needed|reason=there is no reference to an actual expermiment which measures this, and not just the classical angular momentum.|date=January 2013}}
 
=====Spin operator=====
 
The [[Spin (physics)|spin]] of the photon is defined as the coefficient of <math> \hbar  </math> in the spin angular momentum calculation. A photon has spin 1 if it is in the <math> | R \rangle  </math> state and -1 if it is in the <math> | L \rangle  </math> state. The spin operator is defined as the [[outer product]]
 
:<math>  \hat{S} \ \stackrel{\mathrm{def}}{=}\  |R\rangle \langle R | - |L\rangle \langle L |  =  \begin{pmatrix} 0 & -i    \\ i & 0  \end{pmatrix}.   </math>
 
The [[eigenvector]]s of the spin operator are <math>  |R\rangle    </math> and <math> |L\rangle    </math> with [[eigenvalue]]s 1 and -1, respectively.
 
The expected value of a spin measurement on a photon is then
 
:<math>    \langle \psi |\hat{S} |\psi\rangle  = \mid \psi_R \mid^2 - \mid \psi_L \mid^2.  </math>
 
An operator S has been associated with an observable quantity, the spin angular momentum. The eigenvalues of the operator are the allowed observable values. This has been demonstrated for spin angular momentum, but it is in general true for any observable quantity.
 
=====Spin states=====
 
We can write the circularly polarized states as
 
:<math>    |s\rangle  </math>
 
where s=1 for
 
:<math>    |R\rangle    </math>
 
and s= -1 for
 
:<math>    |L\rangle .\,  </math>
 
An arbitrary state can be written
 
:<math>  |\psi\rangle  = \sum_{s=-1,1} a_s  \exp \left ( i \alpha_x -i s \theta \right ) |s\rangle </math>
 
where
 
:<math>    \sum_{s=-1,1} \mid a_s \mid^2=1.   </math>
 
=====Spin and angular momentum operators in differential form=====
 
When the state is written in spin notation, the spin operator can be written
 
:<math>  \hat{S}_d  \rightarrow i { \partial \over \partial \theta}    </math>
 
:<math>  \hat{S}_d^{\dagger}  \rightarrow -i { \partial \over \partial \theta}.    </math>
 
The eigenvectors of the differential spin operator are
 
:<math>    \exp \left ( i \alpha_x -i s \theta \right ) |s\rangle. </math>
 
To see this note
 
:<math>  \hat{S}_d      \exp \left ( i \alpha_x -i s \theta \right ) |s\rangle  \rightarrow i { \partial \over \partial \theta}  \exp \left ( i \alpha_x -i s \theta \right ) |s\rangle = s \left [ \exp \left ( i \alpha_x -i s \theta \right ) |s\rangle \right ]. </math>
 
The spin angular momentum operator is
 
:<math>  \hat{l}_z = \hbar \hat{S}_d.  </math>
 
===The nature of probability in quantum mechanics===
====Probability for a single photon====
There are two ways in which probability can be applied to the behavior of photons; probability can be used to calculate the probable number of photons in a particular state, or probability can be used to calculate the likelihood of a single photon to be in a particular state. The former interpretation violates energy conservation. The latter interpretation is the viable, if nonintuitive, option. Dirac explains this in the context of the [[double-slit experiment]]:
<blockquote>Some time before the discovery of quantum mechanics people realized that the connection between light waves and photons must be of a statistical character. What they did not clearly realize, however, was that the wave function gives information about the probability of '''one''' photon being in a particular place and not the probable number of photons in that place. The importance of the distinction can be made clear in the following way. Suppose we have a beam of light consisting of a large number of photons split up into two components of equal intensity. On the assumption that the beam is connected with the probable number of photons in it, we should have half the total number going into each component. If the two components are now made to interfere, we should require a photon in one component to be able to interfere with one in the other. Sometimes these two photons would have to annihilate one another and other times they would have to produce four photons. This would contradict the conservation of energy. The new theory, which connects the wave function with probabilities for one photon gets over the difficulty by making each photon go partly into each of the two components. Each photon then interferes only with itself. Interference between two different photons never occurs.<br>&mdash;Paul Dirac, '''The Principles of Quantum Mechanics,''' Fourth Edition, Chapter 1</blockquote>
 
====Probability amplitudes====
The probability for a photon to be in a particular polarization state depends on the fields as calculated by the classical Maxwell's equations. The polarization state of the photon is proportional to the field. The probability itself is quadratic in the fields and consequently is also quadratic in the quantum state of polarization. In quantum mechanics, therefore, the state or [[probability amplitude]] contains the basic probability information. 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]
 
:<blockquote>
# The probability amplitude for two successive probabilities is the product of amplitudes for the individual possibilities. For example, the amplitude for the x polarized photon to be right circularly polarized '''and''' for the right circularly polarized photon to pass through the y-polaroid is <math>\langle R|x\rangle\langle y|R\rangle,</math> the product of the individual amplitudes.
# The amplitude for a process that can take place in one of several '''indistinguishable''' ways is the sum of amplitudes for each of the individual ways. For example, the total amplitude for the x polarized photon to pass through the y-polaroid is the sum of the amplitudes for it to pass as a right circularly polarized photon, <math>\langle y|R\rangle\langle R|x\rangle,</math> plus the amplitude for it to pass as a left circularly polarized photon, <math>\langle y|L\rangle\langle L|x\rangle\dots</math>
# The total probability for the process to occur is the absolute value squared of the total amplitude calculated by 1 and 2.
</blockquote>
 
===Uncertainty principle===
 
{{Main|Uncertainty principle}}
[[Image:Cauchy schwarz inequality.svg|right|thumb|400px|Cauchy-Schwarz inequality in Euclidean space. <math>  \mathbf{V} \cdot \mathbf{W} = \| \mathbf{V} \|\| \mathbf{W} \| \cos a .  </math> This implies
<math>  \mathbf{V} \cdot \mathbf{W} \le \| \mathbf{V} \|\| \mathbf{W} \| .    </math>
]]
 
====Mathematical preparation====
For any legal operators the following inequality, a consequence of the [[Cauchy-Schwarz inequality]], is true.
 
: <math> \frac{1}{4} |\langle (\hat{A} \hat{B} - \hat{B} \hat{A} )x | x \rangle|^2\leq \| \hat{A} x \|^2 \| \hat{B} x \|^2.</math>
 
If ''B A'' ψ and ''A B'' ψ are defined then
:<math>
\Delta_{\psi} \hat{A} \, \Delta_{\psi} \hat{B} \ge \frac{1}{2} \left|\left\langle\left[{\hat{A}},{\hat{B}}\right]\right\rangle_\psi\right|
</math>
where
 
:<math>\left\langle \hat{X} \right\rangle_\psi = \left\langle \psi | \hat{X} \psi \right\rangle</math>
 
is the operator [[mean]] of observable ''X'' in the system state ψ and
 
:<math>\Delta_{\psi} \hat{X} = \sqrt{\langle {\hat{X}}^2\rangle_\psi - \langle {\hat{X}}\rangle_\psi ^2}.</math>
 
Here
 
:<math>
\left[{\hat{A}},{\hat{B}}\right] \ \stackrel{\mathrm{def}}{=}\  \hat{A} \hat{B} - \hat{B} \hat{A}
</math>
 
is called the [[commutator]] of A and B.
 
This is a purely mathematical result. No reference has been made to any physical quantity or principle. It simply states that the uncertainty of an operator acting on a state times the uncertainty of another operator acting on the state is not necessarily zero.
 
====Application to angular momentum====
The connection to physics can be made if we identify the operators with physical operators such as the angular momentum and the polarization angle. We have then
 
:<math>
\Delta_{\psi} \hat{l}_z \, \Delta_{\psi} {\theta} \ge \frac{\hbar}{2}, </math>
 
which simply states that angular momentum '''and''' the polarization angle cannot be measured simultaneously with infinite accuracy.
 
===States, probability amplitudes, unitary and Hermitian operators, and eigenvectors===
 
Much of the mathematical apparatus of quantum mechanics appears in the classical description of a polarized sinusoidal electromagnetic wave. The Jones vector for a classical wave, for instance, is identical with the quantum polarization state vector for a photon. The right and left circular components of the Jones vector can be interpreted as [[probability amplitude]]s of spin states of the photon. Energy conservation requires that the states be transformed with a unitary operation. This implies that infinitesimal transformations are transformed with a Hermitian operator. 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. The allowed observable values are determined by the eigenvalues of the operators associated with the observable. 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. They have been found to be true for many other physical systems. In fact, the typical program is to assume the concepts of this section and then to infer the unknown dynamics of a physical system. This was done, for instance, with the dynamics of electrons. 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]]. The solution of this equation for atoms led to the explanation of the [[Balmer series]] for atomic spectra and consequently formed a basis for all of atomic physics and chemistry.
 
This is not the only occasion in which Maxwell's equations have forced a restructuring of Newtonian mechanics. Maxwell's equations are relativistically consistent. [[Special relativity]] resulted from attempts to make classical mechanics consistent with Maxwell's equations (see, for example, [[Moving magnet and conductor problem]]).
 
==See also==
 
*[[Stern–Gerlach experiment]]
*[[Wave–particle duality]]
*[[Double-slit experiment]]
*[[Theoretical and experimental justification for the Schrödinger equation]]
 
==References==
{{Reflist}}
 
*{{cite book |author=Jackson, John D.|title=Classical Electrodynamics (3rd ed.)|publisher=Wiley|year=1998|isbn=0-471-30932-X}}
*{{cite book |author=Baym, Gordon |title=Lectures on Quantum Mechanics|publisher=W. A. Benjamin|year=1969|isbn=0-8053-0667-6}}
*{{cite book |author=Dirac, P. A. M. |title=The Principles of Quantum Mechanics, Fourth Edition|publisher=Oxford|year=1958|isbn=0-19-851208-2}}
 
{{Physics-footer}}
 
{{DEFAULTSORT:Photon Polarization}}
[[Category:Quantum mechanics]]
[[Category:Concepts in physics]]
[[Category:Polarization (waves)]]

Latest revision as of 19:09, 18 July 2014

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