Dirac Equation


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Dirac equation

[di′rak i′kwā·zhən]
(quantum mechanics)
A relativistic wave equation for an electron in an electromagnetic field, in which the wave function has four components corresponding to four internal states specified by a two-valued spin coordinate and an energy coordinate which can have a positive or negative value.

Dirac Equation

 

a quantum equation for the motion of an electron, meeting the requirements of the theory of relativity; established by Dirac in 1928. It follows from the Dirac equation that an electron has a characteristic mechanical moment of angular momentum—spin—equal to ħ/2 and a characteristic magnetic moment equal to the Bohr magneton /2mc, which were previously (1925) discovered experimentally (e and m are the charge and mass of the electron, c is the velocity of light, and ħ is Planck’s constant). The Dirac equation has made it possible to obtain a more accurate formula of the energy levels of hydrogen and hydrogen-like atoms, which includes the fine structure of the levels; it has also helped explain the Zeeman effect. With the Dirac equation as the basis, formulas have been obtained for the probabilities of scattering photons by free electrons (Compton effect) and radiation emitted by a decelerating electron (bremstrahlung); these formulas have been experimentally confirmed. However, a systematic relativistic description of the motion of an electron is provided by quantum electro-dynamics.

A characteristic feature of the Dirac equation is that its solutions include those that correspond to negative values of energy for the free motion of a particle (corresponding to the negative mass of a particle). This presented a difficulty for the theory, since all the mechanical laws for a particle in such states would be incorrect, although transitions in such states are possible in quantum theory. The real physical sense of transitions to a negative energy level were elucidated later, when the possibility of particle interconversion was proved. It followed from the Dirac equation that a new particle must exist (an antiparticle with respect to the electron) with the mass of an electron and a positive charge: in 1932 such a particle was actually discovered by C. D. Anderson and called the positron. This was a great success for the Dirac theory of the electron. The passage of an electron from a state of negative energy to one of positive energy and the reverse are interpreted as the process of the formation of an electron-positron pair and the annihilation of such a pair.

The Dirac equation is also valid for particles with spin ½ (in ħ units)—mu-mesons and the neutrino. With the proton and neutron, which also have spin ½, it leads to incorrect values of the magnetic moments. The magnetic moment for the Dirac proton should be equal to the nuclear magneton /2Mc (m is the mass of the proton) and that of the neutron to zero since it is uncharged. Experiments show that the magnetic moment of the proton is about 2.8 mangetons and the magnetic moment of the neutron is negative, with an absolute value equal to about two-thirds of the magnetic moment of a proton. The anomalous magnetic moments of these particles are due to their strong interactions.

REFERENCE

Broglie, L. de. Magnitnyi elektron. Kharkov, 1936. (Translated from French.)
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Groups are usually introduced relatively late in physics education, he admits, but he begins with them in order to arrive at a semblance of the Dirac equation.
Nishina succeeded to derive the famous Klein--Nishina formula, (8,9) calculating the intensity distribution of the scattered wave in the Compton scattering based on the Dirac equation.
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The Close Relation Between the Maxwell System and the Dirac Equation when the Electric Field is Parallel to the Magnetic Field, Ingeniare, Revista chilena de ingenieria: 16(1), 43-47, (2008).
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