By contrast, in non-relativistic quantum mechanics, terms have to be introduced artificially into the Hamiltonian operator to achieve agreement with experimental observations.
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.
[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.
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.
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.
Under a proper orthochronous Lorentz transformation (r, t) → Λ(r, t) in Minkowski space, all one-particle quantum states ψσ locally transform under some representation D of the Lorentz group:[13] [14] where D(Λ) is a finite-dimensional representation, in other words a (2s + 1)×(2s + 1) square matrix .
The classical Hamiltonian for a particle in a potential is the kinetic energy p·p/2m plus the potential energy V(r, t), with the corresponding quantum operator in the Schrödinger picture: 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.
Another problem, less obvious and more severe, is that it can be shown to be nonlocal and can even violate causality: if the particle is initially localized at a point r0 so that ψ(r0, t = 0) is finite and zero elsewhere, then at any later time the equation predicts delocalization ψ(r, t) ≠ 0 everywhere, even for |r| > ct which means the particle could arrive at a point before a pulse of light could.
For a particle in an externally applied magnetic field B, the interaction term[18] has to be added to the above non-relativistic Hamiltonian.
This equation can be factored into the form:[22][23] where α = (α1, α2, α3) and β are not simply numbers or vectors, but 4 × 4 Hermitian matrices that are required to anticommute for i ≠ j: and square to the identity matrix: so that terms with mixed second-order derivatives cancel while the second-order derivatives purely in space and time remain.
The matrices multiplying ψ suggest it isn't a scalar wavefunction as permitted in the KG equation, but must instead be a four-component entity.
In non-relativistic quantum mechanics, the square modulus of the wavefunction ψ gives the probability density function ρ = |ψ|2.
For one charged particle of electric charge q in an electromagnetic field, given by the magnetic vector potential A(r, t) defined by the magnetic field B = ∇ × A, and electric scalar potential ϕ(r, t), this is:[27] where Pμ is the four-momentum that has a corresponding 4-momentum operator, and Aμ the four-potential.
They have applications to quaternions and to the SO(2) and SO(3) Lie groups, because they satisfy the important commutator [ , ] and anticommutator [ , ]+ relations respectively: where εabc is the three-dimensional Levi-Civita symbol.
The gamma matrices form bases in Clifford algebra, and have a connection to the components of the flat spacetime Minkowski metric ηαβ in the anticommutation relation: (This can be extended to curved spacetime by introducing vierbeins, but is not the subject of special relativity).
[31][32] Considering the factorization of the KG equation above, and more rigorously by Lorentz group theory, it becomes apparent to introduce spin in the form of matrices.
The wavefunctions are multicomponent spinor fields, which can be represented as column vectors of functions of space and time: where the expression on the right is the Hermitian conjugate.
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.
For a particle of rest mass m, the total angular momentum tensor is: where the star denotes the Hodge dual, and is the Pauli–Lubanski pseudovector.
[41] In 1926, the Thomas precession is discovered: relativistic corrections to the spin of elementary particles with application in the spin–orbit interaction of atoms and rotation of macroscopic objects.
In classical electromagnetism and special relativity, an electron moving with a velocity v through an electric field E but not a magnetic field B, will in its own frame of reference experience a Lorentz-transformed magnetic field B′: In the non-relativistic limit v << c: so the non-relativistic spin interaction Hamiltonian becomes:[44] 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 factor of 1⁄2.
It can be shown[45] 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: In the case of RQM, the factor of 1⁄2 is predicted by the Dirac equation.
[44] The events which led to and established RQM, and the continuation beyond into quantum electrodynamics (QED), are summarized below [see, for example, R. Resnick and R. Eisberg (1985),[46] and P.W Atkins (1974)[47]].
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.
In 1916, Sommerfeld explains fine structure; the splitting of the spectral lines of atoms due to first order relativistic corrections.
The Compton effect of 1923 provided more evidence that special relativity does apply; in this case to a particle description of photon–electron scattering.
By 1927, Davisson and Germer and separately G. Thomson successfully diffract electrons, providing experimental evidence of wave-particle duality.
In 1947, the Lamb shift was discovered: a small difference in the 2S1⁄2 and 2P1⁄2 levels of hydrogen, due to the interaction between the electron and vacuum.
Lamb and Retherford experimentally measure stimulated radio-frequency transitions the 2S1⁄2 and 2P1⁄2 hydrogen levels by microwave radiation.