Photon polarization

Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. Individual photons are completely polarized. Their polarization state can be elliptical, circular, or linear. The description 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 state vectors, probability amplitudes, unitary operators, and Hermitian operators, 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 energy conservation 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. 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 sunglasses.

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 experiments of Planck and the interpretation of those experiments by Einstein. The correspondence principle then allows the identification of momentum and angular momentum (called spin), as well as energy, with the photon.

Polarization of classical electromagnetic waves

Main article: Sinusoidal plane-wave solutions of the electromagnetic wave equation

A polarization state vector

Main article: Polarization

Jones vector

The polarization of a classical sinusoidal plane wave traveling in the z direction can be characterized by the Jones vector

Failed to parse (Missing texvc executable; please see math/README to configure.): |\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}.

where the angle Failed to parse (Missing texvc executable; please see math/README to configure.): \theta

describes the relation between the amplitudes of the electric fields in the x and y directions. 

Failed to parse (Missing texvc executable; please see math/README to configure.): \theta \ \stackrel{\mathrm{def}}{=}\ \tan^{-1} \left ( { E_y^0 \over E_x^0 } \right )

and

Failed to parse (Missing texvc executable; please see math/README to configure.): \mid \mathbf{E} \mid^2 \ \stackrel{\mathrm{def}}{=}\ \left ( E_x^0 \right )^2 + \left ( E_y^0 \right )^2

This implies

Failed to parse (Missing texvc executable; please see math/README to configure.): E_x^0 = \mid \mathbf{E} \mid \cos \theta

Failed to parse (Missing texvc executable; please see math/README to configure.): E_y^0 = \mid \mathbf{E} \mid \sin \theta

where Failed to parse (Missing texvc executable; please see math/README to configure.): E_x^0

is the amplitude of the wave polarized in the x direction and Failed to parse (Missing texvc executable; please see math/README to configure.):   E_y^0  
is the amplitude of the wave polarized in the y direction.

The angles Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha_x

and Failed to parse (Missing texvc executable; please see math/README to configure.):   \alpha_y  
characterize the phase relationship between the wave polarized in x and the wave polarized in y.

The Jones vector contains all the polarization information for a plane wave. If the Jones vector for a sinusoidal plane wave, the amplitude Failed to parse (Missing texvc executable; please see math/README to configure.): \mid \mathbf{E} \mid , and the dispersion relationship

Failed to parse (Missing texvc executable; please see math/README to configure.): \omega_{ }^{ } = c k\,

are known, then the state of the wave is completely characterized.

Here Failed to parse (Missing texvc executable; please see math/README to configure.): \omega

is the angular frequency of the wave, and Failed to parse (Missing texvc executable; please see math/README to configure.):  c 
is the speed of light.

The notation for the Jones vector is the bra-ket notation of Dirac, which is normally used in a quantum context. The quantum notation is used here in anticipation of the interpretation of the Jones vector as a quantum state vector.

Dual Jones vector

The Jones vector has a dual given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle \psi | \ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix} \psi_x^* & \psi_y^* \end{pmatrix} = \begin{pmatrix} \quad \cos\theta \exp \left ( -i \alpha_x \right ) & \sin\theta \exp \left ( -i \alpha_y \right ) \quad \end{pmatrix}.

Normalization of the Jones vector

The Jones vector is normalized. The inner product of the vector with itself is

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle \psi | \psi\rangle = \begin{pmatrix} \psi_x^* & \psi_y^* \end{pmatrix} \begin{pmatrix} \psi_x \\ \psi_y \end{pmatrix} = 1.

Polarization states

Linear polarization

Main article: Linear polarization

The wave is linearly polarized when the phase angles Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha_x^{ } , \alpha_y

are equal,

Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha_x = \alpha_y \ \stackrel{\mathrm{def}}{=}\ \alpha.

This represents a wave polarized at an angle Failed to parse (Missing texvc executable; please see math/README to configure.): \theta

with respect to the x axis. In that case the Jones vector can be written

Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi\rangle = \begin{pmatrix} \cos\theta \\ \sin\theta \end{pmatrix} \exp \left ( i \alpha \right )

.

The state vectors for linear polarization in x or y are special cases of this state vector.

If unit vectors are defined such that

Failed to parse (Missing texvc executable; please see math/README to configure.): |x\rangle \ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix} 1 \\ 0 \end{pmatrix}

and

Failed to parse (Missing texvc executable; please see math/README to configure.): |y\rangle \ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix} 0 \\ 1 \end{pmatrix}

then the polarization state can written in the "x-y basis" as

Failed to parse (Missing texvc executable; please see math/README to configure.): |\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.

Circular polarization

Main article: Circular polarization

If Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha_y

is rotated by Failed to parse (Missing texvc executable; please see math/README to configure.):    \pi / 2   
radians with respect to Failed to parse (Missing texvc executable; please see math/README to configure.):    \alpha_x    
and the x amplitude equals the y amplitude the wave is circularly polarized. The Jones vector is

Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi\rangle = \begin{pmatrix} \cos\theta \\ \pm i\sin\theta \end{pmatrix} \exp \left ( i \alpha_x \right )

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

Failed to parse (Missing texvc executable; please see math/README to configure.): |R\rangle \ \stackrel{\mathrm{def}}{=}\ {1 \over \sqrt{2}} \begin{pmatrix} 1 \\ i \end{pmatrix}

and

Failed to parse (Missing texvc executable; please see math/README to configure.): |L\rangle \ \stackrel{\mathrm{def}}{=}\ {1 \over \sqrt{2}} \begin{pmatrix} 1 \\ -i \end{pmatrix}

then a circular polarization state can written in the "R-L basis" as

Failed to parse (Missing texvc executable; please see math/README to configure.): |c\rangle = \psi_R |R\rangle + \psi_L |L\rangle

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \psi_R = {1 \over \sqrt{2} } \exp \left ( i \alpha_x -i\theta \right )

and

Failed to parse (Missing texvc executable; please see math/README to configure.): \psi_L = { 1 \over \sqrt{2} } \exp \left ( i \alpha_x +i\theta \right )

.

Any arbitrary state can be written in the R-L basis

Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi\rangle = a_R \exp \left ( i \alpha_x -i \theta \right ) |R\rangle + a_L \exp \left ( i \alpha_x + i \theta \right ) |L\rangle

where

Failed to parse (Missing texvc executable; please see math/README to configure.): 1 = \mid a_R \mid^2 + \mid a_L \mid^2 .

Elliptical polarization

Main article: 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

Failed to parse (Missing texvc executable; please see math/README to configure.): |\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}.

Energy, momentum, and angular momentum of a classical electromagnetic wave

Energy density of classical electromagnetic waves

Energy in a plane wave

Main article: Energy density

The energy per unit volume in classical electromagnetic fields is (cgs units)

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{E}_c = \frac{1}{8\pi} \left [ \mathbf{E}^2( \mathbf{r} , t ) + \mathbf{B}^2( \mathbf{r} , t ) \right ] .

For a plane wave, this becomes

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{E}_c = \frac{\mid \mathbf{E} \mid^2}{8\pi}

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

Failed to parse (Missing texvc executable; please see math/README to configure.): f_x = \frac{ \mid \mathbf{E} \mid^2 \cos^2\theta }{ \mid \mathbf{E} \mid^2 } = \psi_x^*\psi_x

with a similar expression for the y component.

The fraction in both components is

Failed to parse (Missing texvc executable; please see math/README to configure.): \psi_x^*\psi_x + \psi_y^*\psi_y = \langle \psi | \psi\rangle = 1.

Momentum density of classical electromagnetic waves

The momentum density is given by the Poynting vector

Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol { \mathcal{P}} = {1 \over 4\pi c } \mathbf{E}( \mathbf{r}, t ) \times \mathbf{B}( \mathbf{r}, t ).

For a sinusoidal plane wave traveling in the z direction, the momentum is in the z direction and is related to the energy density:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{P} c = \mathcal{E}_c.

The momentum density has been averaged over a wavelength.

Angular momentum density of classical electromagnetic waves

The angular momentum density is

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 ].

For a sinusoidal plane wave the angular momentum is in the z direction and is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 )

where again the density is averaged over a wavelength.

Optical filters and crystals

Passage of a classical wave through a polaroid filter

A polaroid 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

Failed to parse (Missing texvc executable; please see math/README to configure.): f_x = \psi_x^*\psi_x.\,

Example of energy conservation: Passage of a classical wave through a birefringent crystal

An ideal birefringent crystal transforms the polarization state 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 is has for light polarized perpendicular to the axis. Light polarized parallel to the axis are called "extraordinary rays." Light polarized perpendicular to the axis are called "ordinary rays." 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \theta

with respect to the optic axis, the incident state vector can be written

Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi\rangle = \begin{pmatrix} \cos\theta \\ \sin\theta \end{pmatrix}

and the state vector for the emerging wave can be written

Failed to parse (Missing texvc executable; please see math/README to configure.): |\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.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle \psi '| = \langle \psi | \hat{U}^{\dagger}

where Failed to parse (Missing texvc executable; please see math/README to configure.): U^{\dagger}

is the 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle\psi '| \psi '\rangle = \langle\psi |\hat{U}^{\dagger}\hat{U}|\psi\rangle = \langle \psi|\psi\rangle = 1.

In this ideal case, all the energy impinging on the crystal emerges from the crystal. An operator U with the property that

Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{U}^{\dagger}\hat{U} = I,

where I is the 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

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Doubly refracting Calcite from Iceberg claim, Dixon, New Mexico. This 35 pound (16 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{U} \approx I + i\hat{H}

and the adjoint by

Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{U}^{\dagger} \approx I - i\hat{H}^{\dagger}.

Energy conservation then requires

Failed to parse (Missing texvc executable; please see math/README to configure.): 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}.

This requires that

Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{H} = \hat{H}^{\dagger}.

Operators like this that are equal to their adjoints are called Hermitian or self-adjoint.

The infinitesimal transition of the polarization state is

Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi ' \rangle - |\psi\rangle = i\hat{H} |\psi\rangle.

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 article: Photon

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 for electrodynamics that the treatment can be made quantum mechanical with only a reinterpretation of classical quantities. The reinterpretation is based on the experiments of Max Planck and the interpretation of those experiments by Albert Einstein.

The important conclusion from these early experiments is that electromagnetic radiation is composed of irreducible packets of energy, known as photons. The energy of each packet is related to the angular frequency of the wave by the relation

Failed to parse (Missing texvc executable; please see math/README to configure.): \epsilon = \hbar \omega

where Failed to parse (Missing texvc executable; please see math/README to configure.): \hbar

is an experimentally determined quantity known as Planck's constant. If there are Failed to parse (Missing texvc executable; please see math/README to configure.):     N  
photons in a box of volume Failed to parse (Missing texvc executable; please see math/README to configure.):    V 

, the energy in the electromagnetic field is

Failed to parse (Missing texvc executable; please see math/README to configure.): N \hbar \omega

and the energy density is

Failed to parse (Missing texvc executable; please see math/README to configure.): {N \hbar \omega \over V}

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 Failed to parse (Missing texvc executable; please see math/README to configure.): N , the quantum energy density must be the same as the classical energy density

Failed to parse (Missing texvc executable; please see math/README to configure.): {N \hbar \omega \over V} = \mathcal{E}_c = \frac{\mid \mathbf{E} \mid^2}{8\pi}.

The number of photons in the box is then

Failed to parse (Missing texvc executable; please see math/README to configure.): N = \frac{V }{8\pi \hbar \omega}\mid \mathbf{E} \mid^2 .

Momentum

The correspondence principle also determines the momentum and angular momentum of the photon. For momentum

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{P}_c = {N \hbar \omega \over cV} = {N \hbar k \over V}

which implies that the momentum of a photon is

Failed to parse (Missing texvc executable; please see math/README to configure.): \hbar k .\,

Angular momentum and spin

Similarly for the angular momentum

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{L} = { 1 \over \omega } \mathcal{E}_c \left ( \mid \psi_R \mid^2 - \mid \psi_L \mid^2 \right ) = { \hbar \over V } \left ( \mid \psi_R \mid^2 - \mid \psi_L \mid^2 \right )

which implies that the angular momentum of the photon is

Failed to parse (Missing texvc executable; please see math/README to configure.): l_z = \hbar \left ( \mid \psi_R \mid^2 - \mid \psi_L \mid^2 \right ).

the quantum interpretation of this expression is that the photon has a probability of Failed to parse (Missing texvc executable; please see math/README to configure.): \mid \psi_R \mid^2

of having an angular momentum of Failed to parse (Missing texvc executable; please see math/README to configure.):  \hbar  
and a probability of Failed to parse (Missing texvc executable; please see math/README to configure.):   \mid \psi_L \mid^2  
of having an angular momentum of Failed to parse (Missing texvc executable; please see math/README to configure.):  -\hbar  

. We can therefore think of the angular momentum of the photon being quantized as well as the energy. This has indeed been experimentally verified. Photons have only been observed to have angular momenta of Failed to parse (Missing texvc executable; please see math/README to configure.): \pm \hbar .

Spin operator

The spin of the photon is defined as the coefficient of Failed to parse (Missing texvc executable; please see math/README to configure.): \hbar

in the angular momentum calculation. A photon has spin 1 if it is in the Failed to parse (Missing texvc executable; please see math/README to configure.):  | R \rangle  
state and -1 if it is in the Failed to parse (Missing texvc executable; please see math/README to configure.):  | L \rangle  
state. The spin operator is defined as the outer product

Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{S} \ \stackrel{\mathrm{def}}{=}\ |R\rangle \langle R | - |L\rangle \langle L | = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}.

The eigenvectors of the spin operator are Failed to parse (Missing texvc executable; please see math/README to configure.): |R\rangle

and Failed to parse (Missing texvc executable; please see math/README to configure.):  |L\rangle     
with eigenvalues 1 and -1, respectively.

The expected value of a spin measurement on a photon is then

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle \psi |\hat{S} |\psi\rangle = \mid \psi_R \mid^2 - \mid \psi_L \mid^2.

An operator S has been associated with an observable quantity, the angular momentum. The eigenvalues of the operator are the allowed observable values. This has been demonstrated for angular momentum, but it is in general true for any observable quantity.

Spin states

We can write the circularly polarized states as

Failed to parse (Missing texvc executable; please see math/README to configure.): |s\rangle

where s=1 for

Failed to parse (Missing texvc executable; please see math/README to configure.): |R\rangle

and s= -1 for

Failed to parse (Missing texvc executable; please see math/README to configure.): |L\rangle .\,

An arbitrary state can be written

Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi\rangle = \sum_{s=-1,1} a_s \exp \left ( i \alpha_x -i s \theta \right ) |s\rangle

where

Failed to parse (Missing texvc executable; please see math/README to configure.): 1 = \sum_{s=-1,1} \mid a_s \mid^2.

Spin and angular momentum operators in differential form

When the state is written in spin notation, the spin operator can be written

Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{S}_d \rightarrow i { \partial \over \partial \theta}

Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{S}_d^{\dagger} \rightarrow -i { \partial \over \partial \theta}.

The eigenvectors of the differential spin operator are

Failed to parse (Missing texvc executable; please see math/README to configure.): \exp \left ( i \alpha_x -i s \theta \right ) |s\rangle.

To see this note

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 ].

The angular momentum operator is

Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{l}_z = \hbar \hat{S}_d.

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:

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, Fourth Edition, 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 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]

1. The probability amplitude for two successive probabilities is the product of amplitudes for the individual possibilities. For e.g., 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \langle R|x\rangle\langle y|R\rangle,

the product of the individual amplitudes.

1. The amplitude for a process that can take place in 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, Failed to parse (Missing texvc executable; please see math/README to configure.): \langle y|R\rangle\langle R|x\rangle,

plus the amplitude for it to pass as a left circularly polarized photon, Failed to parse (Missing texvc executable; please see math/README to configure.): \langle y|L\rangle\langle L|x\rangle\dots

1. The total probability for the process to occur is the absolute value squared of the total amplitude calculated by 1 and 2.

Uncertainty principle

Main article: Uncertainty principle

This implies Failed to parse (Missing texvc executable; please see math/README to configure.):   \mathbf{V} \cdot \mathbf{W} \le \| \mathbf{V} \|\| \mathbf{W} \| .    

Mathematical preparation

For any legal operators the following inequality, a consequence of the Cauchy-Schwarz inequality, is true.

Failed to parse (Missing texvc executable; please see math/README to configure.): \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.

If B A ψ and A B ψ are defined then

Failed to parse (Missing texvc executable; please see math/README to configure.): \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|

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \left\langle \hat{X} \right\rangle_\psi = \left\langle \psi | \hat{X} \psi \right\rangle

is the operator mean of observable X in the system state ψ and

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta_{\psi} \hat{X} = \sqrt{\langle {\hat{X}}^2\rangle_\psi - \langle {\hat{X}}\rangle_\psi ^2}.

Here

Failed to parse (Missing texvc executable; please see math/README to configure.): \left[{\hat{A}},{\hat{B}}\right] \ \stackrel{\mathrm{def}}{=}\ \hat{A} \hat{B} - \hat{B} \hat{A}

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta_{\psi} \hat{l}_z \, \Delta_{\psi} {\theta} \ge \frac{\hbar}{2},

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 amplitudes 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 experiments. 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
• Photon dynamics in the double-slit experiment
• Theoretical and experimental justification for the Schrödinger equation

References

• Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X. 
• Baym, Gordon (1969). Lectures on Quantum Mechanics. W. A. Benjamin. ISBN 0-8053-0667-6. 
• Dirac, P. A. M. (1958). The Principles of Quantum Mechanics, Fourth Edition. Oxford. ISBN 0-19-851208-2.