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photon polarization
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{{Short description|Quantum explanation of electromagnetic polarization}}Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, 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 degree of freedom, which forms a fundamental basis for an understanding of more complicated quantum phenomena. Much of the mathematical machinery of quantum mechanics, such as state vectors, probability amplitudes, unitary operators, and Hermitian operators, emerge naturally from the classical Maxwell’s equations in the description. 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 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 sunglass lenses.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 Planck and the interpretation of those theories by Einstein. The correspondence principle then allows the identification of momentum and angular momentum (called spin), as well as energy, with the photon.- the content below is remote from Wikipedia
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Polarization of classical electromagnetic waves
{{duplication|section=yes|dupe=Sinusoidal plane-wave solutions of the electromagnetic wave equation|date=July 2014}}Polarization states
Linear polarization
missing image!
- Mudflats-polariser.jpg -
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 alpha_x , ,; alpha_y are equal,
- Mudflats-polariser.jpg -
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.
alpha_x = alpha_y stackrel{mathrm{def}}{=} alpha.
This represents a wave with phase alpha polarized at an angle theta with respect to the x axis. In this case the Jones vector
|psirangle = begin{pmatrix} costheta exp left ( i alpha_x right ) sintheta exp left ( i alpha_y right ) end{pmatrix}
can be written with a single phase:
|psirangle = begin{pmatrix} costheta sintheta 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
|xrangle stackrel{mathrm{def}}{=} begin{pmatrix} 1 0 end{pmatrix}
and
|yrangle stackrel{mathrm{def}}{=} begin{pmatrix} 0 1 end{pmatrix}
then the linearly polarized polarization state can be written in the “x-y basis” as
|psirangle = costheta exp left ( i alpha right ) |xrangle + sintheta exp left ( i alpha right ) |yrangle = psi_x |xrangle + psi_y |yrangle.
Circular polarization
If the phase angles alpha_x and alpha_y differ by exactly pi / 2 and the x amplitude equals the y amplitude the wave is circularly polarized. The Jones vector then becomes
|psirangle = frac{1}{sqrt{2}}begin{pmatrix} 1 pm i end{pmatrix} exp left ( i alpha_x right )
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
|mathrm{R}rangle stackrel{mathrm{def}}{=} {1 over sqrt{2}} begin{pmatrix} 1 i end{pmatrix}
and
|mathrm{L}rangle stackrel{mathrm{def}}{=} {1 over sqrt{2}} begin{pmatrix} 1 -i end{pmatrix}
then an arbitrary polarization state can be written in the “R-L basis” as
|psirangle = psi_{rm R} |mathrm{R}rangle + psi_{rm L} |mathrm{L}rangle
wherepsi_{rm R} = langle mathrm{R}|psirangle = frac{1}{sqrt{2}} left(costheta exp(ialpha_x) - isintheta exp(ialpha_y)right)andpsi_{rm L} = langle mathrm{L}|psirangle = frac{1}{sqrt{2}} left(costhetaexp(ialpha_x) + isinthetaexp(ialpha_y)right).We can see that
1 = |psi_{rm R}|^2 + |psi_{rm L}|^2 .
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
|psirangle stackrel{mathrm{def}}{=} begin{pmatrix} psi_x psi_y end{pmatrix} = begin{pmatrix} costheta exp left ( i alpha_x right ) sintheta exp left ( i alpha_y right ) end{pmatrix}.
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 e^{iomega t} and then having the real parts of its components interpreted as x and y coordinates respectively. That is:begin{pmatrix}x(t)y(t)end{pmatrix} = begin{pmatrix}Re(e^{iomega t}psi_x) Re(e^{iomega t}psi_y)end{pmatrix} = Releft[e^{iomega t}begin{pmatrix}psi_x psi_yend{pmatrix}right] = Releft(e^{iomega t}|psirangleright).If only the traced out shape and the direction of the rotation of {{math|(x(t), y(t))}} is considered when interpreting the polarization state, i.e. onlyM(|psirangle) = left.left{Big( x(t),,y(t) Big),right|,forall,t right}(where {{math|x(t)}} and {{math|y(t)}} are defined as above) and whether it is overall more right circularly or left circularly polarized (i.e. whether {{math|{{!}}ÏR{{!}} > {{!}}ÏL{{!}}}} 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, sinceM(e^{ialpha}|psirangle) = M(|psirangle), alphainmathbb{R}and the direction of rotation will remain the same. In other words, there is no physical difference between two polarization states |psirangle and e^{ialpha}|psirangle, 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 {{math|tan(θ)}}. For a circularly polarized state, M 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
The energy per unit volume in classical electromagnetic fields is (cgs units) and also Planck unit
mathcal{E}_c = frac{1}{8pi} left [ mathbf{E}^2( mathbf{r} , t ) + mathbf{B}^2( mathbf{r} , t ) right ] .
For a plane wave, this becomes
mathcal{E}_c = frac{mid mathbf{E} mid^2}{8pi}
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
f_x = frac{ | mathbf{E} |^2 cos^2theta }{ vert mathbf{E} vert^2 } = psi_x^*psi_x = cos^2 theta
with a similar expression for the y component resulting in f_y=sin^2theta.The fraction in both components is
psi_x^*psi_x + psi_y^*psi_y = langle psi | psirangle = 1.
Momentum density of classical electromagnetic waves
The momentum density is given by the Poynting vector
boldsymbol { mathcal{P}} = {1 over 4pi 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:
mathcal{P}_z c = mathcal{E}_c.
The momentum density has been averaged over a wavelength.Angular momentum density of classical electromagnetic waves
Electromagnetic waves can have both orbital and spin angular momentum.JOURNAL, Allen, L., Beijersbergen, M.W., Spreeuw, R.J.C., Woerdman, J.P., Han Woerdman, Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Physical Review A, 45, 11, 8186â9, June 1992, 10.1103/PhysRevA.45.8185, 9906912, 1992PhRvA..45.8185A, The total angular momentum density is
boldsymbol { mathcal{L} } = mathbf{r} times boldsymbol { mathcal{P} } = {1 over 4pi c } mathbf{r} times left [ mathbf{E}( mathbf{r}, t ) times mathbf{B}( mathbf{r}, t ) right ].
For a sinusoidal plane wave propagating along z axis the orbital angular momentum density vanishes. The spin angular momentum density is in the z direction and is given by
mathcal{L} = { {vert mathbf{E} vert^2} over {8piomega} } left ( leftvert langle mathrm{R} | psirangle rightvert^2 - leftvert langle mathrm{L} | psirangle rightvert^2 right ) = frac{ 1 }{ omega } mathcal{E}_c left ( vert psi_{rm R} vert^2 - vert psi_{rm L} vert^2 right )
where again the density is averaged over a wavelength.Optical filters and crystals
Passage of a classical wave through a polaroid filter
250px|thumb|right|Linear polarizationA 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
f_x = psi_x^*psi_x = cos^2theta.,
Example of energy conservation: Passage of a classical wave through a birefringent crystal
An ideal 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 theta with respect to the optic axis, the incident state vector can be written
|psirangle = begin{pmatrix} costheta sintheta end{pmatrix}
and the state vector for the emerging wave can be written
|psi ‘rangle = begin{pmatrix} costheta exp left ( i alpha_x right ) sintheta 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} costheta sintheta end{pmatrix}
stackrel{mathrm{def}}{=} hat{U} |psirangle.
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
missing image!
- Calcite.jpg -
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 U. The dual of the final state is given by
- Calcite.jpg -
A calcite crystal laid upon a paper with some letters showing the double refraction
langle psi ‘| = langle psi | hat{U}^{dagger}
where 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 islanglepsi ‘| psi ‘rangle = langlepsi |hat{U}^{dagger}hat{U}|psirangle = langle psi|psirangle = 1.In this ideal case, all the energy impinging on the crystal emerges from the crystal. An operator U with the property thathat{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
missing image!
- Calcite-HUGE.jpg -
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
- Calcite-HUGE.jpg -
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.
hat{U} approx I + ihat{H}
and the adjoint by
hat{U}^{dagger} approx I - ihat{H}^{dagger}.
Energy conservation then requires
I = hat{U}^{dagger} hat{U} approx left ( I - ihat{H}^{dagger} right ) left ( I + ihat{H} right ) approx I - ihat{H}^{dagger} + ihat{H}.
This requires that
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
|psi ‘ rangle - |psirangle = ihat{H} |psirangle.
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 for electrodynamics that the treatment can be made 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.{{Citation needed|date=August 2020}}Einstein’s conclusion from early experiments on the photoelectric effect 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
epsilon = hbar omega
where hbar is an experimentally determined quantity known as Planck’s constant. If there are N photons in a box of volume V , the energy in the electromagnetic field is
N hbar omega
and the energy density is
{N hbar omega over V}
The photon energy 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 N , the quantum energy density must be the same as the classical energy density
{N hbar omega over V} = mathcal{E}_c = frac{vert mathbf{E} vert^2}{8pi}.
The number of photons in the box is then
N = frac{V }{8pi hbar omega}vert mathbf{E} vert^2 .
Momentum
The correspondence principle also determines the momentum and angular momentum of the photon. For momentum
mathcal{P}_z = {N hbar omega over cV} = {N hbar k_z over V}
where k_z is the wave number. This implies that the momentum of a photon is
p_z = hbar k_z .,
Angular momentum and spin
Similarly for the spin angular momentum
mathcal{L} = frac{ 1 }{ omega } mathcal{E}_c left ( vert psi_{rm R} vert^2 - vert psi_{rm L} vert^2 right )
frac{ Nhbar }{ V } left ( vert psi_{rm R} vert^2 - vert psi_{rm L} vert^2 right )
where mathcal{E}_c is field strength. This implies that the spin angular momentum of the photon is
l_z = hbar left ( vert psi_{rm R} vert^2 - vert psi_{rm L} vert^2 right ).
the quantum interpretation of this expression is that the photon has a probability of mid psi_{rm R} mid^2 of having a spin angular momentum of hbar and a probability of mid psi_{rm L} mid^2 of having a spin angular momentum of -hbar . 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.JOURNAL, Beth, R.A., Direct detection of the angular momentum of light, Phys. Rev., 48, 5, 471, 1935, 10.1103/PhysRev.48.471, 1935PhRv...48..471B, 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 of the photon is defined as the coefficient of hbar in the spin angular momentum calculation. A photon has spin 1 if it is in the | R rangle state and â1 if it is in the | L rangle state. The spin operator is defined as the outer product
hat{S} stackrel{mathrm{def}}{=} |mathrm{R}rangle langle mathrm{R} | - |mathrm{L}rangle langle mathrm{L} | = begin{pmatrix} 0 & -i i & 0 end{pmatrix}.
The eigenvectors of the spin operator are |mathrm{R}rangle and |mathrm{L}rangle with eigenvalues 1 and â1, respectively.The expected value of a spin measurement on a photon is then
langle psi |hat{S} |psirangle = vert psi_{rm R} vert^2 - vert psi_{rm L} vert^2.
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
{{see also|Circular polarization#Quantum mechanics}}We can write the circularly polarized states as
|srangle
where s=1 for |mathrm{R}rangle and s= â1 for |mathrm{L}rangle. An arbitrary state can be written
|psirangle = sum_{s=-1,1} a_s exp left ( i alpha_x -i s theta right ) |srangle
where alpha_1 and alpha_{-1} are phase angles, θ is the angle by which the frame of reference is rotated, and
sum_{s=-1,1} mid a_s mid^2=1.
Spin and angular momentum operators in differential form
When the state is written in spin notation, the spin operator can be written
hat{S}_d rightarrow i { partial over partial theta}
hat{S}_d^{dagger} rightarrow -i { partial over partial theta}.
The eigenvectors of the differential spin operator are
hat{S}_d^{dagger} rightarrow -i { partial over partial theta}.
exp left ( i alpha_x -i s theta right ) |srangle.
To see this note
hat{S}_d exp left ( i alpha_x -i s theta right ) |srangle rightarrow i { partial over partial theta} exp left ( i alpha_x -i s theta right ) |srangle = s left [ exp left ( i alpha_x -i s theta right ) |srangle right ].
The spin angular momentum operator is
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: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]{{clarify|reason=These rules are expressed much too loosely. What are “two successive probabilities” in scientific terms? What sort of statements, states, or sets of inequalities qualify as “possibilities“? What is a “way“? There seem to be a lot of unstated heuristics at work here, and the reader is left with the impression that QM is a collection of rules-of-thumb that have not been formalized.|date=June 2015}}- 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 langle R|xranglelangle y|Rrangle, 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, langle y|Rranglelangle R|xrangle, plus the amplitude for it to pass as a left circularly polarized photon, langle y|Lranglelangle L|xrangledots
- The total probability for the process to occur is the absolute value squared of the total amplitude calculated by 1 and 2.
Uncertainty principle
Image:Cauchy schwarz inequality.svg|right|thumb|400px|CauchyâSchwarz inequality in Euclidean space. mathbf{V} cdot mathbf{W} = left| mathbf{V} right| left| mathbf{W} right| cos a . This implies
mathbf{V} cdot mathbf{W} le left| mathbf{V} right| left| mathbf{W} right| .
Mathematical preparation
For any legal{{clarify|reason=The word legal is not generally used in formal scientific writing.|date=June 2015}} operators the following inequality, a consequence of the CauchyâSchwarz inequality, is true.
frac{1}{4} left|langle (hat{A} hat{B} - hat{B} hat{A} )x | x rangleright|^2leq left| hat{A} x right|^2 left| hat{B} x right|^2.
If B A Ï and A B Ï are defined, then by subtracting the means and re-inserting in the above formula, we deduceDelta_{psi} hat{A} , Delta_{psi} hat{B} ge frac{1}{2} left|leftlangleleft[{hat{A}},{hat{B}}right]rightrangle_psiright|whereleftlangle hat{X} rightrangle_psi = leftlangle psi right| hat{X} left| psi rightrangleis the operator mean of observable X in the system state Ï andDelta_{psi} hat{X} = sqrt{langle {hat{X}}^2rangle_psi - langle {hat{X}}rangle_psi ^2}.Hereleft[{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 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 thenDelta_{psi} hat{L}_z , Delta_{psi} {theta} ge frac{hbar}{2}, 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 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 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{{dubious|reason=I didn’t see where Maxwell’s equations “forced a restructuring of Newtonian mechanics” in the previous paragraphs. In the following, yes, but that has nothing to do with quantum mechanics or the purpose of the article.|date=August 2016}} 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
- Angular momentum of light
- Quantum decoherence
- SternâGerlach experiment
- Waveâparticle duality
- Double-slit experiment
- Theoretical and experimental justification for the Schrödinger equation
- Spin polarization
References
{{Reflist}}Further reading
- BOOK, Jackson, John D., Classical Electrodynamics, 3rd, Wiley, 1998, 0-471-30932-X,
- BOOK, Baym, Gordon, Lectures on Quantum Mechanics, W. A. Benjamin, 1969, 0-8053-0667-6,
- BOOK, Dirac, P. A. M., The Principles of Quantum Mechanics, Fourth, Oxford, 1958, 0-19-851208-2,
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