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Quantum mechanics
${\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }$ 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 Davisson–Germer Double-slit Elitzur–Vaidman Franck–Hertz Leggett–Garg inequality Mach–Zehnder Popper Quantum eraser Delayed-choice Schrödinger's cat Stern–Gerlach Wheeler's delayed-choice
Formulations Overview Heisenberg Interaction Matrix Phase-space Schrödinger Sum-over-histories (path integral)
Equations Dirac Klein–Gordon Pauli Rydberg Schrödinger
Interpretations Bayesian Consistent histories Copenhagen de Broglie–Bohm Ensemble Hidden-variable Local Superdeterminism Many-worlds Objective collapse Quantum logic Relational Transactional Von Neumann–Wigner
Advanced topics Relativistic quantum mechanics Quantum field theory Quantum information science Quantum computing Quantum chaos EPR paradox Density matrix Scattering theory Quantum statistical mechanics Quantum machine learning
Scientists Aharonov Bell Bethe Blackett Bloch Bohm Bohr Born Bose de Broglie Compton Dirac Davisson Debye Ehrenfest Einstein Everett Fock Fermi Feynman Glauber Gutzwiller Heisenberg Hilbert Jordan Kramers Lamb Landau Laue Moseley Millikan Onnes Pauli Planck Rabi Raman Rydberg Schrödinger Simmons Sommerfeld von Neumann Weyl Wien Wigner Zeeman Zeilinger
v t e

In physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light c, and can accommodate massless particles. The theory has application in high energy physics,^[1] particle physics and accelerator physics,^[2] as well as atomic physics, chemistry^[3] and condensed matter physics.^[4][5] Non-relativistic quantum mechanics refers to the mathematical formulation of quantum mechanics applied in the context of Galilean relativity, more specifically quantizing the equations of classical mechanics by replacing dynamical variables by operators. Relativistic quantum mechanics (RQM) is quantum mechanics applied with special relativity. Although the earlier formulations, like the Schrödinger picture and Heisenberg 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 of antimatter, spin magnetic moments of elementary spin 1⁄2 fermions, fine structure, and quantum dynamics of charged particles in electromagnetic fields.^[6] The key result is the Dirac equation, from which these predictions emerge automatically. By contrast, in non-relativistic quantum mechanics, terms have to be introduced artificially into the Hamiltonian operator to achieve agreement with experimental observations.

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

Paul Dirac's work between 1927 to 1933 shaped the synthesis of special relativity and quantum mechanics.^[8] His work was instrumental, as he formulated the Dirac equation and also originated quantum electrodynamics, both of which were successful in combining the two theories.^[9]

In this article, the equations are written in familiar 3D vector calculus notation and use hats for operators (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 the Einstein summation convention is used. SI units are used here; Gaussian units and natural units are common alternatives. All equations are in the position representation; for the momentum representation the equations have to be Fourier transformed – see position and momentum space.

Combining special relativity and quantum mechanics

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

A postulate of quantum mechanics is that the time evolution of any quantum system is given by the Schrödinger equation:

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

using a suitable Hamiltonian operator Ĥ corresponding to the system. The solution is a complex-valued wavefunction ψ(r, t), a function of the 3D position vector r of the particle at time t, describing the behavior of the system.

Every particle has a non-negative spin quantum number s. The number 2s is an integer, odd for fermions and even for bosons. Each s has 2s + 1 z-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 1920s Pauli, Kronig, Uhlenbeck and Goudsmit were the first to propose the concept of spin. The inclusion of spin in the wavefunction incorporates the Pauli exclusion principle (1925) and the more general spin–statistics theorem (1939) due to Fierz, rederived by Pauli a year later. This is the explanation for a diverse range of subatomic particle behavior and phenomena: from the electronic configurations of atoms, nuclei (and therefore all elements on the periodic table and their chemistry), to the quark configurations and colour charge (hence the properties of baryons and mesons).

A fundamental prediction of special relativity is the relativistic energy–momentum relation; for a particle of rest mass m, and in a particular frame of reference with energy E and 3-momentum p with magnitude in terms of the dot product ${\displaystyle p={\sqrt {\mathbf {p} \cdot \mathbf {p} }}}$, it is:^[10]

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

These equations are used together with the energy and momentum operators, which are respectively:

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

to construct a relativistic wave equation (RWE): a partial differential equation consistent with the energy–momentum relation, and is solved for ψ 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 time partial 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).

The Heisenberg picture is another formulation of QM, in which case the wavefunction ψ is time-independent, and the operators A(t) contain the time dependence, governed by the equation of motion:

${\displaystyle {\frac {d}{dt}}A={\frac {1}{i\hbar }}+{\frac {\partial }{\partial t}}A\,,}$

This equation is also true in RQM, provided the Heisenberg operators are modified to be consistent with SR.^[11][12]

Historically, around 1926, Schrödinger and Heisenberg show that wave mechanics and matrix mechanics are equivalent, later furthered by Dirac using transformation theory.

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

Space and time

In classical 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 a many particle system ψ(r₁, r₂, r₃, ..., t, σ₁, σ₂, σ₃...).

In relativistic mechanics, the spatial coordinates and coordinate time are not absolute; any two observers moving relative to each other can measure different locations and times of events. The position and time coordinates combine naturally into a four-dimensional spacetime position X = (ct, r) corresponding to events, and the energy and 3-momentum combine naturally into the four-momentum P = (E/c, p) of a dynamic particle, as measured in some reference frame, change according to a Lorentz 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 proper orthochronous Lorentz transformation (r, t) → Λ(r, t) in Minkowski space, all one-particle quantum states ψ_σ locally transform under some representation D of the Lorentz group:^{[13] [14]}

${\displaystyle {\mathrm{\psi <!--\; \psi \; -->}}_{\mathrm{\sigma <!--\; \sigma \; -->}}(\mathbf{r},t)\to <!--\; \to \; -->D(\mathrm{\Lambda <!--\; \Lambda \; -->}){\mathrm{\psi <!--\; \psi \; -->}}_{\mathrm{\sigma <!--\; \sigma \; -->}}}$Zdroj:https://en.wikipedia.org?pojem=Relativistic_quantum_mechanics
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