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Quantum mechanics
${\displaystyle i\hslash \frac{d}{dt}|\mathrm{\Psi }⟩=\hat{H}|\mathrm{\Psi }⟩}$ Schrödinger equation
Introduction Glossary History
Background Classical mechanics Old quantum theory Bra–ket notation Hamiltonian Interference
Fundamentals Complementarity Decoherence Entanglement Energy level Measurement Nonlocality Quantum number State Superposition Symmetry Tunnelling Uncertainty Wave function Collapse
Experiments Bell's inequality CHSH inequality Davisson–Germer Double-slit Elitzur–Vaidman Franck–Hertz Leggett inequality Leggett–Garg inequality Mach–Zehnder Popper Quantum eraser Delayed-choice Schrödinger's cat Stern–Gerlach Wheeler's delayed-choice
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Equations Dirac Klein–Gordon Pauli Rydberg Schrödinger
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Inphysics,relativistic quantum mechanics (RQM) is anyPoincaré-covariant formulation ofquantum mechanics (QM). This theory is applicable tomassive particles propagating at allvelocities up to those comparable to thespeed of light c, and can accommodatemassless particles. The theory has application inhigh-energy physics,^[1]particle physics andaccelerator physics,^[2] as well asatomic physics,chemistry^[3] andcondensed matter physics.^[4][5]Non-relativistic quantum mechanics refers to themathematical formulation of quantum mechanics applied in the context ofGalilean relativity, more specifically quantizing the equations ofclassical mechanics by replacing dynamical variables byoperators.Relativistic quantum mechanics (RQM) is quantum mechanics applied withspecial relativity. Although the earlier formulations, like theSchrödinger picture andHeisenberg picture were originally formulated in a non-relativistic background, a few of them (e.g. the Dirac or path-integral formalism) also work with special relativity.

Key features common to all RQMs include: the prediction ofantimatter,spin magnetic moments ofelementaryspin-1/2fermions,fine structure, and quantum dynamics ofcharged particles inelectromagnetic fields.^[6] The key result is theDirac equation, from which these predictions emerge automatically. By contrast, in non-relativistic quantum mechanics, terms have to be introduced artificially into theHamiltonian operator to achieve agreement with experimental observations.

The most successful (and most widely used) RQM isrelativisticquantum field theory (QFT), in which elementary particles are interpreted asfield quanta. A unique consequence of QFT that has been tested against other RQMs is the failure of conservation of particle number, for example, inmatter creation andannihilation.^[7]

Nevertheless, QFT wrongly predicts the existence of matter-antimatter symmetry. A recently developed RQM theory, without the flows of previous single-particle RQM, offers a possible explanation for the observed matter-antimatter asymmetry.^[8]

Paul Dirac's work between 1927 and 1933 shaped the synthesis of special relativity and quantum mechanics.^[9] His work was instrumental, as he formulated the Dirac equation and also originatedquantum electrodynamics, both of which were successful in combining the two theories.^[10]

In this article, the equations are written in familiar 3Dvector calculus notation and use hats foroperators (not necessarily in the literature), and where space and time components can be collected,tensor index notation is shown also (frequently used in the literature), in addition theEinstein summation convention is used.SI units are used here;Gaussian units andnatural units are common alternatives. All equations are in the position representation; for the momentum representation the equations have to beFourier-transformed – seeposition and momentum space.

One approach is to modify the Schrödinger picture to be consistent with special relativity.^[2]

Apostulate of quantum mechanics is that thetime evolution of any quantum system is given by theSchrödinger equation:

${\displaystyle i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}\psi =\hat{H}\psi }$

using a suitableHamiltonian operatorĤ corresponding to the system. The solution is acomplex-valuedwavefunctionψ(r,t), afunction of the3Dposition vectorr of the particle at timet, describing the behavior of the system.

Every particle has a non-negativespin quantum numbers. The number2s is an integer, odd forfermions and even forbosons. Eachs has2s + 1z-projection quantum numbers;σ = s,s − 1, ... , −s + 1, −s.^[a] This is an additional discrete variable the wavefunction requires;ψ(r, t, σ).

Historically, in the early 1920sPauli,Kronig,Uhlenbeck andGoudsmit were the first to propose the concept of spin. The inclusion of spin in the wavefunction incorporates thePauli exclusion principle (1925) and the more generalspin–statistics theorem (1939) due toFierz, rederived by Pauli a year later. This is the explanation for a diverse range ofsubatomic particle behavior and phenomena: from theelectronic configurations of atoms, nuclei (and therefore allelements on theperiodic table and theirchemistry), to the quark configurations andcolour charge (hence the properties ofbaryons andmesons).

A fundamental prediction of special relativity is the relativisticenergy–momentum relation; for a particle ofrest massm, and in a particularframe of reference withenergyE and 3-momentump withmagnitude in terms of thedot product${\displaystyle p=\sqrt{\mathbf{p}\cdot \mathbf{p}}}$, it is:^[11]

${\displaystyle E^2=c^2\mathbf{p}\cdot \mathbf{p}+(m{c}^2{)}^2.}$

These equations are used together with theenergy andmomentumoperators, which are respectively:

${\displaystyle \hat{E}=i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t},\hat{\mathbf{p}}=-i\hslash \mathrm{\nabla },}$

to construct arelativistic wave equation (RWE): apartial differential equation consistent with the energy–momentum relation, and is solved forψ to predict the quantum dynamics of the particle. For space and time to be placed on equal footing, as in relativity, the orders of space and timepartial derivatives should be equal, and ideally as low as possible, so that no initial values of the derivatives need to be specified. This is important for probability interpretations, exemplified below. The lowest possible order of any differential equation is the first (zeroth order derivatives would not form a differential equation).

TheHeisenberg picture is another formulation of QM, in which case the wavefunctionψ istime-independent, and the operatorsA(t) contain the time dependence, governed by the equation of motion:

${\displaystyle \frac{d}{dt}A=\frac{1}{i\hslash }[A,\hat{H}]+\frac{\mathrm{\partial }}{\mathrm{\partial }t}A,}$

This equation is also true in RQM, provided the Heisenberg operators are modified to be consistent with SR.^[12][13]

Historically, around 1926,Schrödinger andHeisenberg show that wave mechanics andmatrix mechanics are equivalent, later furthered by Dirac usingtransformation theory.

A more modern approach to RWEs, first introduced during the time RWEs were developing for particles of any spin, is to applyrepresentations of the Lorentz group.

Inclassical mechanics and non-relativistic QM, time is an absolute quantity all observers and particles can always agree on, "ticking away" in the background independent of space. Thus in non-relativistic QM one has for amany particle systemψ(r₁,r₂,r₃, ...,t,σ₁,σ₂,σ₃...).

Inrelativistic mechanics, thespatial coordinates andcoordinate time arenot absolute; any two observers moving relative to each other can measure different locations and times ofevents. The position and time coordinates combine naturally into afour-dimensional spacetime positionX = (ct,r) corresponding to events, and the energy and 3-momentum combine naturally into thefour-momentumP = (E/c,p) of a dynamic particle, as measured insomereference frame, change according to aLorentz transformation as one measures in a different frame boosted and/or rotated relative the original frame in consideration. The derivative operators, and hence the energy and 3-momentum operators, are also non-invariant and change under Lorentz transformations.

Under a properorthochronous Lorentz transformation(r,t) → Λ(r,t) inMinkowski space, all one-particle quantum statesψ_σ locally transform under somerepresentationD of theLorentz group:^[14][15]

${\displaystyle {\psi }_{\sigma }(\mathbf{r},t)\to D(\mathrm{\Lambda }){\psi }_{\sigma }({\mathrm{\Lambda }}^{-1}(\mathbf{r},t))}$

whereD(Λ) is a finite-dimensional representation, in other words a(2s + 1) × (2s + 1)square matrix . Again,ψ is thought of as acolumn vector containing components with the(2s + 1) allowed values ofσ. Thequantum numberss andσ as well as other labels, continuous or discrete, representing other quantum numbers are suppressed. One value ofσ may occur more than once depending on the representation.

Theclassical Hamiltonian for a particle in apotential is thekinetic energyp·p/2m plus thepotential energyV(r,t), with the corresponding quantum operator in theSchrödinger picture:

${\displaystyle \hat{H}=\frac{\hat{\mathbf{p}}\cdot \hat{\mathbf{p}}}{2m}+V(\mathbf{r},t)}$

and substituting this into the above Schrödinger equation gives a non-relativistic QM equation for the wavefunction: the procedure is a straightforward substitution of a simple expression. By contrast this is not as easy in RQM; the energy–momentum equation is quadratic in energyand momentum leading to difficulties. Naively setting:

${\displaystyle \hat{H}=\hat{E}=\sqrt{c^2\hat{\mathbf{p}}\cdot \hat{\mathbf{p}}+(m{c}^2{)}^2}\Rightarrow i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}\psi =\sqrt{c^2\hat{\mathbf{p}}\cdot \hat{\mathbf{p}}+(m{c}^2{)}^2}\psi }$

is not helpful for several reasons. The square root of the operators cannot be used as it stands; it would have to be expanded in apower series before the momentum operator, raised to a power in each term, could act onψ. As a result of the power series, the space and timederivatives arecompletely asymmetric: infinite-order in space derivatives but only first order in the time derivative, which is inelegant and unwieldy. Again, there is the problem of the non-invariance of the energy operator, equated to the square root which is also not invariant. Another problem, less obvious and more severe, is that it can be shown to benonlocal and can evenviolatecausality: if the particle is initially localized at a pointr₀ so thatψ(r₀,t = 0) is finite and zero elsewhere, then at any later time the equation predicts delocalizationψ(r,t) ≠ 0 everywhere, even for|r| >ct which means the particle could arrive at a point before a pulse of light could. This would have to be remedied by the additional constraintψ(|r| >ct,t) = 0.^[16]

There is also the problem of incorporating spin in the Hamiltonian, which isn't a prediction of the non-relativistic Schrödinger theory. Particles with spin have a corresponding spin magnetic moment quantized in units ofμ_B, theBohr magneton:^[17][18]

${\displaystyle {\hat{\mathit{\mu }}}_S=-\frac{g{\mu }_B}{\hslash }\hat{\mathbf{S}},\left|{\mathit{\mu }}_S\right|=-g{\mu }_B\sigma ,}$

whereg is the (spin)g-factor for the particle, andS thespin operator, so they interact with electromagnetic fields. For a particle in an externally appliedmagnetic fieldB, the interaction term^[19]

${\displaystyle {\hat{H}}_B=-\mathbf{B}\cdot {\hat{\mathit{\mu }}}_S}$

has to be added to the above non-relativistic Hamiltonian. On the contrary; a relativistic Hamiltonian introduces spinautomatically as a requirement of enforcing the relativistic energy-momentum relation.^[20]

Relativistic Hamiltonians are analogous to those of non-relativistic QM in the following respect; there are terms includingrest mass and interaction terms with externally applied fields, similar to the classical potential energy term, as well as momentum terms like the classical kinetic energy term. A key difference is that relativistic Hamiltonians contain spin operators in the form ofmatrices, in which thematrix multiplication runs over the spin indexσ, so in general a relativistic Hamiltonian:

${\displaystyle \hat{H}=\hat{H}(\mathbf{r},t,\hat{\mathbf{p}},\hat{\mathbf{S}})}$

is a function of space, time, and the momentum and spin operators.

Substituting the energy and momentum operators directly into the energy–momentum relation may at first sight seem appealing, to obtain theKlein–Gordon equation:^[21]

${\displaystyle {\hat{E}}^2\psi =c^2\hat{\mathbf{p}}\cdot \hat{\mathbf{p}}\psi +(m{c}^2{)}^2\psi ,}$

and was discovered by many people because of the straightforward way of obtaining it, notably by Schrödinger in 1925 before he found the non-relativistic equation named after him, and by Klein and Gordon in 1927, who included electromagnetic interactions in the equation. Thisisrelativistically invariant, yet this equation alone isn't a sufficient foundation for RQM for at least two reasons: one is that negative-energy states are solutions,^[2][22] another is the density (given below), and this equation as it stands is only applicable to spinless particles. This equation can be factored into the form:^[23][24]

${\displaystyle \left(\hat{E}-c\mathit{\alpha }\cdot \hat{\mathbf{p}}-\beta m{c}^2\right)\left(\hat{E}+c\mathit{\alpha }\cdot \hat{\mathbf{p}}+\beta m{c}^2\right)\psi =0,}$

whereα = (α₁,α₂,α₃) andβ are not simply numbers or vectors, but 4 × 4Hermitian matrices that are required toanticommute fori ≠j:

${\displaystyle {\alpha }_i\beta =-\beta {\alpha }_i,{\alpha }_i{\alpha }_j=-{\alpha }_j{\alpha }_i,}$

and square to theidentity matrix:

${\displaystyle {\alpha }_i^2={\beta }^2=I,}$

so that terms with mixed second-order derivatives cancel while the second-order derivatives purely in space and time remain. The first factor:

${\displaystyle \left(\hat{E}-c\mathit{\alpha }\cdot \hat{\mathbf{p}}-\beta m{c}^2\right)\psi =0\iff \hat{H}=c\mathit{\alpha }\cdot \hat{\mathbf{p}}+\beta m{c}^2}$

is the Dirac equation. The other factor is also the Dirac equation, but for a particle ofnegative mass.^[23] Each factor is relativistically invariant. The reasoning can be done the other way round: propose the Hamiltonian in the above form, as Dirac did in 1928, then pre-multiply the equation by the other factor of operatorsE +cα ·p +βmc², and comparison with the KG equation determines the constraints onα andβ. The positive mass equation can continue to be used without loss of continuity. The matrices multiplyingψ suggest it isn't a scalar wavefunction as permitted in the KG equation, but must instead be a four-component entity. The Dirac equation still predicts negative energy solutions,^[6][25] so Dirac postulated that negative energy states are always occupied, because according to thePauli principle,electronic transitions from positive to negative energy levels inatoms would be forbidden. SeeDirac sea for details.

In non-relativistic quantum mechanics, thesquare modulus of thewavefunctionψ gives theprobability density functionρ = |ψ|². This is theCopenhagen interpretation, circa 1927. In RQM, whileψ(r,t) is a wavefunction, the probability interpretation is not the same as in non-relativistic QM. Some RWEs do not predict a probability densityρ orprobability currentj (really meaningprobability current density) because they arenotpositive-definite functions of space and time. TheDirac equation does:^[26]

${\displaystyle \rho ={\psi }^{†}\psi ,\mathbf{j}={\psi }^{†}{\gamma }^0\mathit{\gamma }\psi \rightleftharpoons J^{\mu }={\psi }^{†}{\gamma }^0{\gamma }^{\mu }\psi }$

where the dagger denotes theHermitian adjoint (authors usually writeψ =ψ^†γ⁰ for theDirac adjoint) andJ^μ is theprobability four-current, while theKlein–Gordon equation does not:^[27]

${\displaystyle \rho =\frac{i\hslash }{2m{c}^2}\left({\psi }^{\ast }\frac{\mathrm{\partial }\psi }{\mathrm{\partial }t}-\psi \frac{\mathrm{\partial }{\psi }^{\ast }}{\mathrm{\partial }t}\right),\mathbf{j}=-\frac{i\hslash }{2m}\left({\psi }^{\ast }\mathrm{\nabla }\psi -\psi \mathrm{\nabla }{\psi }^{\ast }\right)\rightleftharpoons J^{\mu }=\frac{i\hslash }{2m}({\psi }^{\ast }{\mathrm{\partial }}^{\mu }\psi -\psi {\mathrm{\partial }}^{\mu }{\psi }^{\ast })}$

where∂^μ is thefour-gradient. Since the initial values of bothψ and∂ψ/∂t may be freely chosen, the density can be negative.

Instead, what appears look at first sight a "probability density" and "probability current" has to be reinterpreted ascharge density andcurrent density when multiplied byelectric charge. Then, the wavefunctionψ is not a wavefunction at all, but reinterpreted as afield.^[16] The density and current of electric charge always satisfy acontinuity equation:

${\displaystyle \frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \mathbf{J}=0\rightleftharpoons {\mathrm{\partial }}_{\mu }{J}^{\mu }=0,}$

as charge is aconserved quantity. Probability density and current also satisfy a continuity equation because probability is conserved, however this is only possible in the absence of interactions.

Including interactions in RWEs is generally difficult.Minimal coupling is a simple way to include the electromagnetic interaction. For one charged particle ofelectric chargeq in an electromagnetic field, given by themagnetic vector potentialA(r,t) defined by the magnetic fieldB = ∇ ×A, andelectric scalar potentialϕ(r,t), this is:^[28]

${\displaystyle \hat{E}\to \hat{E}-q\varphi ,\hat{\mathbf{p}}\to \hat{\mathbf{p}}-q\mathbf{A}\rightleftharpoons {\hat{P}}_{\mu }\to {\hat{P}}_{\mu }-q{A}_{\mu }}$

whereP_μ is thefour-momentum that has a corresponding4-momentum operator, andA_μ thefour-potential. In the following, the non-relativistic limit refers to the limiting cases:

${\displaystyle E-e\varphi \approx m{c}^2,\mathbf{p}\approx m\mathbf{v},}$

that is, the total energy of the particle is approximately the rest energy for small electric potentials, and the momentum is approximately the classical momentum.

In RQM, the KG equation admits the minimal coupling prescription;

${\displaystyle {(\hat{E}-q\varphi )}^2\psi =c^2{(\hat{\mathbf{p}}-q\mathbf{A})}^2\psi +(m{c}^2{)}^2\psi \rightleftharpoons \left[({\hat{P}}_{\mu }-q{A}_{\mu })({\hat{P}}^{\mu }-q{A}^{\mu })-{(mc)}^2\right]\psi =0.}$

In the case where the charge is zero, the equation reduces trivially to the free KG equation so nonzero charge is assumed below. This is a scalar equation that is invariant under theirreducible one-dimensional scalar(0,0) representation of the Lorentz group. This means that all of its solutions will belong to a direct sum of(0,0) representations. Solutions that do not belong to the irreducible(0,0) representation will have two or moreindependent components. Such solutions cannot in general describe particles with nonzero spin since spin components are not independent. Other constraint will have to be imposed for that, e.g. the Dirac equation for spin ⁠1/2⁠, see below. Thus if a system satisfies the KG equationonly, it can only be interpreted as a system with zero spin.

The electromagnetic field is treated classically according toMaxwell's equations and the particle is described by a wavefunction, the solution to the KG equation. The equation is, as it stands, not always very useful, because massive spinless particles, such as theπ-mesons, experience the much stronger strong interaction in addition to the electromagnetic interaction. It does, however, correctly describe charged spinless bosons in the absence of other interactions.

The KG equation is applicable to spinless chargedbosons in an external electromagnetic potential.^[2] As such, the equation cannot be applied to the description of atoms, since the electron is a spin ⁠1/2⁠ particle. In the non-relativistic limit the equation reduces to the Schrödinger equation for a spinless charged particle in an electromagnetic field:^[19]

${\displaystyle \left(i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}-q\varphi \right)\psi =\frac{1}{2m}{(\hat{\mathbf{p}}-q\mathbf{A})}^2\psi \iff \hat{H}=\frac{1}{2m}{(\hat{\mathbf{p}}-q\mathbf{A})}^2+q\varphi .}$

Non relativistically, spin wasphenomenologically introduced in thePauli equation byPauli in 1927 for particles in anelectromagnetic field:

${\displaystyle \left(i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}-q\varphi \right)\psi =\left[\frac{1}{2m}{(\mathit{\sigma }\cdot (\mathbf{p}-q\mathbf{A}))}^2\right]\psi \iff \hat{H}=\frac{1}{2m}{(\mathit{\sigma }\cdot (\mathbf{p}-q\mathbf{A}))}^2+q\varphi }$

by means of the 2 × 2Pauli matrices, andψ is not just a scalar wavefunction as in the non-relativistic Schrödinger equation, but a two-componentspinor field:

${\displaystyle \psi =\left(\begin{array}{c}{\psi }_{\uparrow }\\ {\psi }_{\downarrow }\end{array}\right)}$

where the subscripts ↑ and ↓ refer to the "spin up" (σ = +⁠1/2⁠) and "spin down" (σ = −⁠1/2⁠) states.^[b]

In RQM, the Dirac equation can also incorporate minimal coupling, rewritten from above;

${\displaystyle \left(i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}-q\varphi \right)\psi ={\gamma }^0\left[c\mathit{\gamma }\cdot (\hat{\mathbf{p}}-q\mathbf{A})-m{c}^2\right]\psi \rightleftharpoons \left[{\gamma }^{\mu }({\hat{P}}_{\mu }-q{A}_{\mu })-m{c}^2\right]\psi =0}$

and was the first equation to accuratelypredict spin, a consequence of the 4 × 4gamma matricesγ⁰ =β,γ = (γ₁,γ₂,γ₃) =βα = (βα₁,βα₂,βα₃). There is a 4 × 4 identity matrix pre-multiplying the energy operator (including the potential energy term), conventionally not written for simplicity and clarity (i.e. treated like the number 1). Hereψ is a four-component spinor field, which is conventionally split into two two-component spinors in the form:^[c]

${\displaystyle \psi =\left(\begin{array}{c}{\psi }_{+}\\ {\psi }_{-}\end{array}\right)=\left(\begin{array}{c}{\psi }_{+\uparrow }\\ {\psi }_{+\downarrow }\\ {\psi }_{-\uparrow }\\ {\psi }_{-\downarrow }\end{array}\right)}$

The 2-spinorψ₊ corresponds to a particle with 4-momentum(E,p) and chargeq and two spin states (σ = ±⁠1/2⁠, as before). The other 2-spinorψ₋ corresponds to a similar particle with the same mass and spin states, butnegative 4-momentum−(E,p) andnegative charge−q, that is, negative energy states,time-reversed momentum, andnegated charge. This was the first interpretation and prediction of a particle andcorrespondingantiparticle. SeeDirac spinor andbispinor for further description of these spinors. In the non-relativistic limit the Dirac equation reduces to the Pauli equation (seeDirac equation for how). When applied a one-electron atom or ion, settingA =0 andϕ to the appropriate electrostatic potential, additional relativistic terms include thespin–orbit interaction, electrongyromagnetic ratio, andDarwin term. In ordinary QM these terms have to be put in by hand and treated usingperturbation theory. The positive energies do account accurately for the fine structure.

Within RQM, for massless particles the Dirac equation reduces to:

${\displaystyle \left(\frac{\hat{E}}{c}+\mathit{\sigma }\cdot \hat{\mathbf{p}}\right){\psi }_{+}=0,\left(\frac{\hat{E}}{c}-\mathit{\sigma }\cdot \hat{\mathbf{p}}\right){\psi }_{-}=0\rightleftharpoons {\sigma }^{\mu }{\hat{P}}_{\mu }{\psi }_{+}=0,{\sigma }_{\mu }{\hat{P}}^{\mu }{\psi }_{-}=0,}$

the first of which is theWeyl equation, a considerable simplification applicable for masslessneutrinos.^[29] This time there is a 2 × 2identity matrix pre-multiplying the energy operator conventionally not written. In RQM it is useful to take this as the zeroth Pauli matrixσ₀ which couples to the energy operator (time derivative), just as the other three matrices couple to the momentum operator (spatial derivatives).

The Pauli and gamma matrices were introduced here, in theoretical physics, rather thanpure mathematics itself. They have applications toquaternions and to theSO(2) andSO(3)Lie groups, because they satisfy the importantcommutator [ , ] andanticommutator [ , ]₊ relations respectively:

${\displaystyle \left[{\sigma }_a,{\sigma }_b\right]=2i{\epsilon }_{abc}{\sigma }_c,{\left[{\sigma }_a,{\sigma }_b\right]}_{+}=2{\delta }_{ab}{\sigma }_0}$

whereε_abc is thethree-dimensionalLevi-Civita symbol. The gamma matrices formbases inClifford algebra, and have a connection to the components of the flat spacetimeMinkowski metricη^αβ in the anticommutation relation:

${\displaystyle {\left[{\gamma }^{\alpha },{\gamma }^{\beta }\right]}_{+}={\gamma }^{\alpha }{\gamma }^{\beta }+{\gamma }^{\beta }{\gamma }^{\alpha }=2{\eta }^{\alpha \beta },}$

(This can be extended tocurved spacetime by introducingvierbeins, but is not the subject of special relativity).

In 1929, theBreit equation was found to describe two or more electromagnetically interacting massive spin ⁠1/2⁠ fermions to first-order relativistic corrections; one of the first attempts to describe such a relativistic quantummany-particle system. This is, however, still only an approximation, and the Hamiltonian includes numerous long and complicated sums.

Thehelicity operator is defined by;

${\displaystyle \hat{h}=\hat{\mathbf{S}}\cdot \frac{\hat{\mathbf{p}}}{|\mathbf{p}|}=\hat{\mathbf{S}}\cdot \frac{c\hat{\mathbf{p}}}{\sqrt{E^2-(m_0{c}^2{)}^2}}}$

wherep is the momentum operator,S the spin operator for a particle of spins,E is the total energy of the particle, andm₀ its rest mass. Helicity indicates the orientations of the spin and translational momentum vectors.^[30] Helicity is frame-dependent because of the 3-momentum in the definition, and is quantized due to spin quantization, which has discrete positive values for parallel alignment, and negative values for antiparallel alignment.

An automatic occurrence in the Dirac equation (and the Weyl equation) is the projection of the spin ⁠1/2⁠ operator on the 3-momentum (timesc),σ ·cp, which is the helicity (for the spin ⁠1/2⁠ case) times${\displaystyle \sqrt{E^2-(m_0{c}^2{)}^2}}$.

For massless particles the helicity simplifies to:

${\displaystyle \hat{h}=\hat{\mathbf{S}}\cdot \frac{c\hat{\mathbf{p}}}{E}}$

The Dirac equation can only describe particles of spin ⁠1/2⁠. Beyond the Dirac equation, RWEs have been applied tofree particles of various spins. In 1936, Dirac extended his equation to all fermions, three years laterFierz and Pauli rederived the same equation.^[31] TheBargmann–Wigner equations were found in 1948 using Lorentz group theory, applicable for all free particles with any spin.^[32][33] Considering the factorization of the KG equation above, and more rigorously byLorentz group theory, it becomes apparent to introduce spin in the form of matrices.

The wavefunctions are multicomponentspinor fields, which can be represented ascolumn vectors offunctions of space and time:

where the expression on the right is theHermitian conjugate. For amassive particle of spins, there are2s + 1 components for the particle, and another2s + 1 for the correspondingantiparticle (there are2s + 1 possibleσ values in each case), altogether forming a2(2s + 1)-component spinor field:

with the + subscript indicating the particle and − subscript for the antiparticle. However, formassless particles of spins, there are only ever two-component spinor fields; one is for the particle in one helicity state corresponding to +s and the other for the antiparticle in the opposite helicity state corresponding to −s:

${\displaystyle \psi (\mathbf{r},t)=\left(\begin{array}{c}{\psi }_{+}(\mathbf{r},t)\\ {\psi }_{-}(\mathbf{r},t)\end{array}\right)}$

According to the relativistic energy-momentum relation, all massless particles travel at the speed of light, so particles traveling at the speed of light are also described by two-component spinors. Historically,Élie Cartan found the most general form ofspinors in 1913, prior to the spinors revealed in the RWEs following the year 1927.

For equations describing higher-spin particles, the inclusion of interactions is nowhere near as simple minimal coupling, they lead to incorrect predictions and self-inconsistencies.^[34] For spin greater than⁠ħ/2⁠, the RWE is not fixed by the particle's mass, spin, and electric charge; the electromagnetic moments (electric dipole moments andmagnetic dipole moments) allowed by thespin quantum number are arbitrary. (Theoretically,magnetic charge would contribute also). For example, the spin ⁠1/2⁠ case only allows a magnetic dipole, but for spin 1 particles magnetic quadrupoles and electric dipoles are also possible.^[29] For more on this topic, seemultipole expansion and (for example) Cédric Lorcé (2009).^[35][36]

The Schrödinger/Pauli velocity operator can be defined for a massive particle using the classical definitionp =mv, and substituting quantum operators in the usual way:^[37]

${\displaystyle \hat{\mathbf{v}}=\frac{1}{m}\hat{\mathbf{p}}}$

which has eigenvalues that takeany value. In RQM, the Dirac theory, it is:

${\displaystyle \hat{\mathbf{v}}=\frac{i}{\hslash }\left[\hat{H},\hat{\mathbf{r}}\right]}$

which must have eigenvalues between ±c. SeeFoldy–Wouthuysen transformation for more theoretical background.

The Hamiltonian operators in the Schrödinger picture are one approach to forming the differential equations forψ. An equivalent alternative is to determine aLagrangian (really meaningLagrangian density), then generate the differential equation by thefield-theoretic Euler–Lagrange equation:

${\displaystyle {\mathrm{\partial }}_{\mu }\left(\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }({\mathrm{\partial }}_{\mu }\psi )}\right)-\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }\psi }=0}$

For some RWEs, a Lagrangian can be found by inspection. For example, the Dirac Lagrangian is:^[38]

${\displaystyle \mathcal{L}=\overline{\psi }({\gamma }^{\mu }{P}_{\mu }-mc)\psi }$

and Klein–Gordon Lagrangian is:

${\displaystyle \mathcal{L}=-\frac{{\hslash }^2}{m}{\eta }^{\mu \nu }{\mathrm{\partial }}_{\mu }{\psi }^{\ast }{\mathrm{\partial }}_{\nu }\psi -m{c}^2{\psi }^{\ast }\psi .}$

This is not possible for all RWEs; and is one reason the Lorentz group theoretic approach is important and appealing: fundamental invariance and symmetries in space and time can be used to derive RWEs using appropriate group representations. The Lagrangian approach with field interpretation ofψ is the subject of QFT rather than RQM: Feynman'spath integral formulation uses invariant Lagrangians rather than Hamiltonian operators, since the latter can become extremely complicated, see (for example) Weinberg (1995).^[39]

In non-relativistic QM, theangular momentum operator is formed from the classicalpseudovector definitionL =r ×p. In RQM, the position and momentum operators are inserted directly where they appear in the orbitalrelativistic angular momentum tensor defined from the four-dimensional position and momentum of the particle, equivalently abivector in theexterior algebra formalism:^[40][d]

${\displaystyle M^{\alpha \beta }=X^{\alpha }{P}^{\beta }-X^{\beta }{P}^{\alpha }=2X^{[\alpha }{P}^{\beta ]}\rightleftharpoons \mathbf{M}=\mathbf{X}\wedge \mathbf{P},}$

which are six components altogether: three are the non-relativistic 3-orbital angular momenta;M¹² =L³,M²³ =L¹,M³¹ =L², and the other threeM⁰¹,M⁰²,M⁰³ are boosts of thecentre of mass of the rotating object. An additional relativistic-quantum term has to be added for particles with spin. For a particle of rest massm, thetotal angular momentum tensor is:

${\displaystyle J^{\alpha \beta }=2X^{[\alpha }{P}^{\beta ]}+\frac{1}{m^2}{\epsilon }^{\alpha \beta \gamma \delta }{W}_{\gamma }{p}_{\delta }\rightleftharpoons \mathbf{J}=\mathbf{X}\wedge \mathbf{P}+\frac{1}{m^2}\star (\mathbf{W}\wedge \mathbf{P})}$

where the star denotes theHodge star operator, and

${\displaystyle W_{\alpha }=\frac{1}{2}{\epsilon }_{\alpha \beta \gamma \delta }{M}^{\beta \gamma }{p}^{\delta }\rightleftharpoons \mathbf{W}=\star (\mathbf{M}\wedge \mathbf{P})}$

is thePauli–Lubanski pseudovector.^[41] For more on relativistic spin, see (for example) Troshin & Tyurin (1994).^[42]

In 1926, theThomas precession is discovered: relativistic corrections to the spin of elementary particles with application in thespin–orbit interaction of atoms and rotation of macroscopic objects.^[43][44] In 1939 Wigner derived the Thomas precession.

Inclassical electromagnetism and special relativity, an electron moving with a velocityv through an electric fieldE but not a magnetic fieldB, will in its own frame of reference experience aLorentz-transformed magnetic fieldB′:

${\displaystyle {\mathbf{B}}^{\prime }=\frac{\mathbf{E}\times \mathbf{v}}{c^2\sqrt{1-{\left(v/c\right)}^2}}.}$

In the non-relativistic limitv <<c:

${\displaystyle {\mathbf{B}}^{\prime }=\frac{\mathbf{E}\times \mathbf{v}}{c^2},}$

so the non-relativistic spin interaction Hamiltonian becomes:^[45]

${\displaystyle \hat{H}=-{\mathbf{B}}^{\prime }\cdot {\hat{\mathit{\mu }}}_S=-\left(\mathbf{B}+\frac{\mathbf{E}\times \mathbf{v}}{c^2}\right)\cdot {\hat{\mathit{\mu }}}_S,}$

where the first term is already the non-relativistic magnetic moment interaction, and the second term the relativistic correction of order(v/c)², but this disagrees with experimental atomic spectra by a factor⁠1/2⁠. It was pointed out by L. Thomas that there is a second relativistic effect: An electric field component perpendicular to the electron velocity causes an additional acceleration of the electron perpendicular to its instantaneous velocity, so the electron moves in a curved path. The electron moves in arotating frame of reference, and this additional precession of the electron is called theThomas precession. It can be shown^[46] that the net result of this effect is that the spin–orbit interaction is reduced by half, as if the magnetic field experienced by the electron has only one-half the value, and the relativistic correction in the Hamiltonian is:

${\displaystyle \hat{H}=-{\mathbf{B}}^{\prime }\cdot {\hat{\mathit{\mu }}}_S=-\left(\mathbf{B}+\frac{\mathbf{E}\times \mathbf{v}}{2c^2}\right)\cdot {\hat{\mathit{\mu }}}_S.}$

In the case of RQM, the factor of⁠1/2⁠ is predicted by the Dirac equation.^[45]

The events which led to and established RQM, and the continuation beyond intoquantum electrodynamics (QED), are summarized below [see, for example, R. Resnick and R. Eisberg (1985),^[47] andP.W Atkins (1974)^[48]]. More than half a century of experimental and theoretical research from the 1890s through to the 1950s in the new and mysterious quantum theory as it was up and coming revealed that a number of phenomena cannot be explained by QM alone. SR, found at the turn of the 20th century, was found to be anecessary component, leading to unification: RQM. Theoretical predictions and experiments mainly focused on the newly foundatomic physics,nuclear physics, andparticle physics; by consideringspectroscopy,diffraction andscattering of particles, and the electrons and nuclei within atoms and molecules. Numerous results are attributed to the effects of spin.

Albert Einstein in 1905 explained of thephotoelectric effect; a particle description of light asphotons. In 1916,Sommerfeld explainsfine structure; the splitting of thespectral lines ofatoms due to first order relativistic corrections. TheCompton effect of 1923 provided more evidence that special relativity does apply; in this case to a particle description of photon–electron scattering.de Broglie extendswave–particle duality tomatter: thede Broglie relations, which are consistent with special relativity and quantum mechanics. By 1927,Davisson andGermer and separatelyG. Thomson successfully diffract electrons, providing experimental evidence of wave-particle duality.

• 1897J. J. Thomson discovers theelectron and measures itsmass-to-charge ratio. Discovery of theZeeman effect: the splitting aspectral line into several components in the presence of a static magnetic field.
• 1908Millikan measures the charge on the electron and finds experimental evidence of its quantization, in theoil drop experiment.
• 1911Alpha particle scattering in theGeiger–Marsden experiment, led byRutherford, showed that atoms possess an internal structure: theatomic nucleus.^[49]
• 1913 TheStark effect is discovered: splitting of spectral lines due to a staticelectric field (compare with the Zeeman effect).
• 1922Stern–Gerlach experiment: experimental evidence of spin and its quantization.
• 1924Stoner studies splitting ofenergy levels inmagnetic fields.
• 1932 Experimental discovery of theneutron byChadwick, andpositrons byAnderson, confirming the theoretical prediction of positrons.
• 1958 Discovery of theMössbauer effect: resonant and recoil-free emission and absorption ofgamma radiation by atomic nuclei bound in a solid, useful for accurate measurements ofgravitational redshift andtime dilation, and in the analysis of nuclear electromagnetic moments inhyperfine interactions.^[50]

In 1935, Einstein,Rosen,Podolsky published a paper^[51] concerningquantum entanglement of particles, questioningquantum nonlocality and the apparent violation of causality upheld in SR: particles can appear to interact instantaneously at arbitrary distances. This was a misconception since information is not and cannot be transferred in the entangled states; rather the information transmission is in the process of measurement by two observers (one observer has to send a signal to the other, which cannot exceedc). QM doesnot violate SR.^[52][53] In 1959,Bohm andAharonov publish a paper^[54] on theAharonov–Bohm effect, questioning the status of electromagnetic potentials in QM. TheEM field tensor andEM 4-potential formulations are both applicable in SR, but in QM the potentials enter the Hamiltonian (see above) and influence the motion of charged particles even in regions where the fields are zero. In 1964,Bell's theorem was published in a paper on the EPR paradox,^[55] showing that QM cannot be derived fromlocal hidden-variable theories if locality is to be maintained.

In 1947, the Lamb shift was discovered: a small difference in the²S_1⁄2 and²P_1⁄2 levels of hydrogen, due to the interaction between the electron and vacuum.Lamb andRetherford experimentally measure stimulated radio-frequency transitions the²S_1⁄2 and²P_1⁄2 hydrogen levels bymicrowave radiation.^[56] An explanation of the Lamb shift is presented byBethe. Papers on the effect were published in the early 1950s.^[57]

• 1927Dirac establishes the field of QED, also coining the term "quantum electrodynamics".^[58]
• 1943Tomonaga begins work onrenormalization, influential in QED.
• 1947Schwinger calculates theanomalous magnetic moment of the electron.Kusch measures of the anomalous magnetic electron moment, confirming one of QED's great predictions.

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