Dirac equation

The Dirac equation is a relativistic quantum mechanical wave equation formulated by Paul Dirac in 1928. It provides a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. It also accounts in a natural way for the nature of particle spin. The equation demands the existence of antiparticles and actually predated their experimental discovery, making the discovery of the positron, the antiparticle of the electron, one of the greatest triumphs of modern theoretical physics.

Related Topics:
Relativistic - Quantum mechanical - Paul Dirac - 1928 - Elementary - Spin-½ - Electron - Special relativity - Antiparticle - Positron

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Since the Dirac equation was originally invented to describe the electron, we will generally speak of "electrons" in this article. Actually, the equation also applies to quarks, which are also elementary spin-½ particles. A modified Dirac equation can be used to approximately describe protons and neutrons, which are not elementary particles (they are made up of quarks). Another modification of the Dirac equation, called the Majorana equation), is used to describe neutrinos.

Related Topics:
Quark - Proton - Neutron - Majorana equation - Neutrino

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The Dirac equation is

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: left(lpha_0 mc^2 + sum_{j = 1}^3 lpha_j p_j , c ight) psi (mathbf{x},t) = i hbar rac{partialpsi}{partial t} (mathbf{x},t)

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where m is the rest mass of the electron, c is the speed of light, p is the momentum operator, hbar is the reduced Planck's constant, x and t are the space and time coordinates respectively, and ψ(x, t) is a four-component wavefunction. (The wavefunction has to be formulated as a four-component spinor, rather than a simple scalar, due to the demands of special relativity. The physical meanings of the components are discussed below.)

Related Topics:
Rest mass - Speed of light - Momentum - Planck's constant - Space - Time - Wavefunction - Spinor - Scalar

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The αs are linear operators that act on the wavefunction. Their most fundamental property is that they must anticommute with each other. In other words,

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:lpha_ilpha_j = -lpha_jlpha_i,

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where i e j, and i and j range from zero to three. The simplest way to obtain such properties is with 4×4 matrices. There is no set of matrices of smaller dimension fulfilling the anticommutation requirements. That four dimensional matrices are necessary turns out to have physical significance.

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A convenient (but not unique) choice of lphas is

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:lpha_0 = egin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & -1 & 0 \ 0 & 0 & 0 & -1 end{bmatrix} quad lpha_1 = egin{bmatrix} 0 & 0 & 0 & 1 \ 0 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \ 1 & 0 & 0 & 0 end{bmatrix} ,

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:lpha_2 = egin{bmatrix} 0 & 0 & 0 & -i \ 0 & 0 & i & 0 \ 0 & -i& 0 & 0 \ i & 0 & 0 & 0 end{bmatrix} quad lpha_3 = egin{bmatrix} 0 & 0 & 1 & 0 \ 0 & 0 & 0 & -1 \ 1 & 0 & 0 & 0 \ 0 & -1 & 0 & 0 end{bmatrix} ,

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known as Dirac matrices. All possible choices are related by similarity transformations because Dirac spinors are unique representation theoretically.

Related Topics:
Similarity transformation - Representation theoretically

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The Dirac equation describes the probability amplitudes for a single electron. This is a single-particle theory, in other words it does not account for the creation and destruction of the particles. It gives a good prediction of the magnetic moment of the electron and explains much of the fine structure observed in atomic spectral lines. It also explains the spin of the electron. Two of the four solutions of the equation correspond to the two spin states of the electron. The other two solutions make the peculiar prediction that there exist an infinite set of quantum states in which the electron possesses negative energy. This strange result led Dirac to predict, via a remarkable hypothesis known as "hole theory", the existence of particles behaving like positively-charged electrons. This prediction was verified by the discovery of the positron in 1932.

Related Topics:
Probability amplitude - Fine structure - Atom - Spectral line - Energy - Positron - 1932

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Despite these successes, the theory is flawed by its neglect of the possibility of creating and destroying particles, one of the basic consequences of relativity. This difficulty is resolved by reformulating it as a quantum field theory. Adding a quantized electromagnetic field to this theory leads to the theory of quantum electrodynamics (QED).

Related Topics:
Quantum field theory - Electromagnetic field - Quantum electrodynamics

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A similar equation for spin 3/2 particles is called the Rarita-Schwinger equation.

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