Photon polarization is the
quantum mechanical
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
description of the
classical polarized sinusoidal
A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in m ...
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
electromagnetic wave
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
. An individual
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
can be described as having right or left
circular polarization
In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to t ...
, or a
superposition of the two. Equivalently, a photon can be described as having horizontal or vertical
linear polarization
In electrodynamics, linear polarization or plane polarization of electromagnetic radiation is a confinement of the electric field vector or magnetic field vector to a given plane along the direction of propagation. The term ''linear polarizati ...
, 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
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
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 amplitude
In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The modulus squared of this quantity represents a probability density.
Probability amplitudes provide a relationship between the quan ...
s,
unitary operator
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the con ...
s, and
Hermitian operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to it ...
s, emerge naturally from the classical
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
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
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (res ...
. Unitary operators emerge from the classical requirement of the
conservation of energy
In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...
of a classical wave propagating through lossless media that alter the polarization state of the wave. Hermitian operators then follow for infinitesimal transformations of a classical polarization state.
Many of the implications of the mathematical machinery are easily verified experimentally. In fact, many of the experiments can be performed with
polaroid
Polaroid may refer to:
* Polaroid Corporation, an American company known for its instant film and cameras
* Polaroid camera, a brand of instant camera formerly produced by Polaroid Corporation
* Polaroid film, instant film, and photographs
* Polar ...
sunglass lenses.
The connection with quantum mechanics is made through the identification of a minimum packet size, called a
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
, for energy in the electromagnetic field. The identification is based on the theories of
Planck
Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918.
Planck made many substantial contributions to theoretical p ...
and the interpretation of those theories by
Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
. The
correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says t ...
then allows the identification of momentum and angular momentum (called
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
), as well as energy, with the photon.
Polarization of classical electromagnetic waves
Polarization states
Linear polarization
The wave is linearly polarized (or plane polarized) when the phase angles
are
equal,
:
This represents a wave with
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
polarized at an angle
with respect to the x axis.
In this case the Jones vector
:
can be written with a single phase:
:
The state vectors for linear polarization in x or y are special cases of this state vector.
If unit vectors are defined such that
:
and
:
then the linearly polarized polarization state can be written in the "x-y basis" as
:
Circular polarization
If the phase angles
and
differ by exactly
and the x amplitude equals the y amplitude the wave is
circularly polarized
In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to t ...
. The Jones vector then becomes
:
where the plus sign indicates left circular polarization and the minus sign indicates right 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
:
and
:
then an arbitrary polarization state can be written in the "R-L basis" as
:
where
:
and
:
We can see that
:
Elliptical polarization
The general case in which the electric field rotates in the x-y plane and has variable magnitude is called
elliptical polarization
In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. An elli ...
. The state vector is given by
:
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
and then having the real parts of its components interpreted as x and y coordinates respectively. That is:
:
If only the traced out shape and the direction of the rotation of is considered when interpreting the polarization state, i.e. only
:
(where and are defined as above) and whether it is overall more right circularly or left circularly polarized (i.e. whether 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
:
and the direction of rotation will remain the same. In other words, there is no physical difference between two polarization states
and
, between which only a phase factor differs.
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 . For a circularly polarized state,
M will be a circle with radius 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
The
energy per unit volume in classical electromagnetic fields is (cgs units) and also Planck unit
:
For a plane wave, this becomes
:
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
:
with a similar expression for the y component resulting in
.
The fraction in both components is
:
Momentum density of classical electromagnetic waves
The momentum density is given by the
Poynting vector
In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area per unit time) or '' power flow'' of an electromagnetic field. The SI unit of the Poynting vector is the watt ...
:
For a sinusoidal plane wave traveling in the z direction, the momentum is in the z direction and is related to the energy density:
:
The momentum density has been averaged over a wavelength.
Angular momentum density of classical electromagnetic waves
Electromagnetic waves can have both
orbital and
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
angular momentum.
The total angular momentum density is
:
For a sinusoidal plane wave propagating along
axis the orbital angular momentum density vanishes. The spin angular momentum density is in the
direction and is given by
:
where again the density is averaged over a wavelength.
Optical filters and crystals
Passage of a classical wave through a polaroid filter
A
linear filter
Linear filters process time-varying input signals to produce output signals, subject to the constraint of linearity. In most cases these linear filters are also time invariant (or shift invariant) in which case they can be analyzed exactly using ...
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
:
Example of energy conservation: Passage of a classical wave through a birefringent crystal
An ideal
birefringent
Birefringence is the optics, optical property of a material having a refractive index that depends on the Polarization (waves), polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or ...
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
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
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
with respect to the optic axis, the incident state vector can be written
:
and the state vector for the emerging wave can be written
:
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
The initial polarization state is transformed into the final state with the
operator U. The dual of the final state is given by
:
where
is the
adjoint
In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type
:(''Ax'', ''y'') = (''x'', ''By'').
Specifically, adjoin ...
of U, the complex conjugate transpose of the matrix.
Unitary operators and energy conservation
The fraction of energy that emerges from the crystal is
:
In this ideal case, all the energy impinging on the crystal emerges from the crystal. An operator U with the property that
:
where I is the
identity operator
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), a ...
and U is called a
unitary operator
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the con ...
. The unitary property is necessary to ensure
energy conservation
Energy conservation is the effort to reduce wasteful energy consumption by using fewer energy services. This can be done by using energy more effectively (using less energy for continuous service) or changing one's behavior to use less service (f ...
in state transformations.
Hermitian operators and energy conservation
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 {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
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.
Photons: The connection to quantum mechanics
Energy, momentum, and angular momentum of photons
Energy
The treatment to this point has been
classical. It is a testament, however, to the generality of
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
for electrodynamics that the treatment can be made
quantum mechanical
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
with only a reinterpretation of classical quantities. The reinterpretation is based on the theories of
Max Planck
Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918.
Planck made many substantial contributions to theoretical p ...
and the interpretation by
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
of those theories and of other experiments.
Einstein's conclusion from early experiments on the
photoelectric effect
The photoelectric effect is the emission of electrons when electromagnetic radiation, such as light, hits a material. Electrons emitted in this manner are called photoelectrons. The phenomenon is studied in condensed matter physics, and solid st ...
is that electromagnetic radiation is composed of irreducible packets of energy, known as
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
s. The energy of each packet is related to the angular frequency of the wave by the relation
:
where
is an experimentally determined quantity known as
Planck's constant. If there are
photons in a box of volume
, the energy in the electromagnetic field is
:
and the energy density is
:
The
photon energy
Photon energy is the energy carried by a single photon. The amount of energy is directly proportional to the photon's electromagnetic frequency and thus, equivalently, is inversely proportional to the wavelength. The higher the photon's frequency, ...
can be related to classical fields through the
correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says t ...
which states that for a large number of photons, the quantum and classical treatments must agree. Thus, for very large
, the quantum energy density must be the same as the classical energy density
:
The number of photons in the box is then
:
Momentum
The correspondence principle also determines the momentum and angular momentum of the photon. For momentum
:
where
is the wave number. This implies that the momentum of a photon is
:
Angular momentum and spin
Similarly for the spin angular momentum
:
where
is field strength. This implies that the spin angular momentum of the photon is
:
the quantum interpretation of this expression is that the photon has a probability of
of having a spin angular momentum of
and a probability of
of having a spin angular momentum of
. 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.
A photon that is linearly polarized (plane polarized) is in a superposition of equal amounts of the left-handed and right-handed states.
=Spin operator
=
The
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
of the photon is defined as the coefficient of
in the spin angular momentum calculation. A photon has spin 1 if it is in the
state and -1 if it is in the
state. The spin operator is defined as the
outer product
In linear algebra, the outer product of two coordinate vector
In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An ea ...
:
The
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of the spin operator are
and
with
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s 1 and -1, respectively.
The expected value of a spin measurement on a photon is then
:
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
:
where ''s''=1 for
and ''s''= -1 for
. An arbitrary state can be written
:
where
and
are phase angles, ''θ'' is the angle by which the frame of reference is rotated, and
:
=Spin and angular momentum operators in differential form
=
When the state is written in spin notation, the spin operator can be written
:
:
The eigenvectors of the differential spin operator are
:
To see this note
:
The spin angular momentum operator is
:
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
In modern physics, the double-slit experiment is a demonstration that light and matter can display characteristics of both classically defined waves and particles; moreover, it displays the fundamentally probabilistic nature of quantum mechanics ...
:
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.
—Paul Dirac, The Principles of Quantum Mechanics, 1930, Chapter 1
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
In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The modulus squared of this quantity represents a probability density.
Probability amplitudes provide a relationship between the quan ...
contains the basic probability information. In general, the rules for combining probability amplitudes look very much like the classical rules for composition of probabilities:
he following quote is from Baym, Chapter 1
# 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 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, plus the amplitude for it to pass as a left circularly polarized photon,
# The total probability for the process to occur is the absolute value squared of the total amplitude calculated by 1 and 2.
Uncertainty principle
Mathematical preparation
For any legal operators the following inequality, a consequence of the
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics.
The inequality for sums was published by . The corresponding inequality fo ...
, is true.
:
If ''B A'' ψ and ''A B'' ψ are defined, then by subtracting the means and re-inserting in the above formula, we deduce
:
where
:
is the operator
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
of observable ''X'' in the system state ψ and
:
Here
:
is called the
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...
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 one operator times the uncertainty of another operator has a lower bound.
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
:
which means that angular momentum and the polarization angle cannot be measured simultaneously with infinite accuracy. (The polarization angle can be measured by checking whether the photon can pass through a polarizing filter oriented at a particular angle, or a
polarizing beam splitter. This results in a yes/no answer which, if the photon was plane-polarized at some other angle, depends on the difference between the two angles.)
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
In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The modulus squared of this quantity represents a probability density.
Probability amplitudes provide a relationship between the quan ...
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
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
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
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motion ...
. The solution of this equation for atoms led to the explanation of the
Balmer series
The Balmer series, or Balmer lines in atomic physics, is one of a set of six named series describing the spectral line emissions of the hydrogen atom. The Balmer series is calculated using the Balmer formula, an empirical equation discovered b ...
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
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws o ...
resulted from attempts to make classical mechanics consistent with Maxwell's equations (see, for example,
Moving magnet and conductor problem
The moving magnet and conductor problem is a famous thought experiment, originating in the 19th century, concerning the intersection of classical electromagnetism and special relativity. In it, the current in a conductor moving with constant vel ...
).
See also
*
Angular momentum of light
The angular momentum of light is a Euclidean vector, vector quantity that expresses the amount of dynamical rotation present in the electromagnetic field of the light. While traveling approximately in a straight line, a beam of light can also be r ...
**
Spin angular momentum of light
The spin angular momentum of light (SAM) is the component of angular momentum of light that is associated with the spin (physics), quantum spin and the rotation between the polarization (waves), polarization degrees of freedom of the photon.
Int ...
**
Orbital angular momentum of light The orbital angular momentum of light (OAM) is the component of angular momentum of a light beam that is dependent on the field spatial distribution, and not on the polarization. It can be further split into an internal and an external OAM. The in ...
*
Quantum decoherence
Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave ...
*
Stern–Gerlach experiment
The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. Thus an atomic-scale system was shown to have intrinsically quantum properties. In the original experiment, silver atoms were sent throug ...
*
Wave–particle duality
Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave. It expresses the inability of the classical concepts "particle" or "wave" to fully describe the ...
*
Double-slit experiment
In modern physics, the double-slit experiment is a demonstration that light and matter can display characteristics of both classically defined waves and particles; moreover, it displays the fundamentally probabilistic nature of quantum mechanics ...
*
*
Spin polarization
Spin polarization is the degree to which the spin, i.e., the intrinsic angular momentum of elementary particles, is aligned with a given direction. This property may pertain to the spin, hence to the magnetic moment, of conduction electrons in fe ...
References
Further reading
*
*
*
{{DEFAULTSORT:Photon Polarization
Quantum mechanics
Physical phenomena
Polarization (waves)