Dirac equation

Dirac Equation

The Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. It provides a description of elementary spin-½ particles, such as electrons, that incorporates both quantum mechanics and the theory of special relativity. The equation is significant for predicting the existence of antimatter and for its role in the development of quantum field theory.

Historical Context

Before the Dirac equation, the Schrödinger equation was used to describe quantum systems. However, the Schrödinger equation is not consistent with the principles of special relativity. Dirac sought to find an equation that would be first-order in both space and time derivatives, unlike the second-order Klein-Gordon equation, which also attempted to incorporate relativity but failed to account for the intrinsic spin of particles.

Mathematical Formulation

The Dirac equation is given by:

<math>i \hbar \frac{\partial \psi}{\partial t} = \left( c \boldsymbol{\alpha} \cdot \mathbf{p} + \beta mc^2 \right) \psi</math>

where:

• <math>\psi</math> is the wave function of the particle, a four-component spinor.
• <math>\mathbf{p}</math> is the momentum operator.
• <math>m</math> is the rest mass of the particle.
• <math>c</math> is the speed of light.
• <math>\hbar</math> is the reduced Planck's constant.
• <math>\boldsymbol{\alpha}</math> and <math>\beta</math> are matrices that satisfy the anticommutation relations:

<math>\{ \alpha_i, \alpha_j \} = 2 \delta_{ij} I, \quad \{ \alpha_i, \beta \} = 0, \quad \beta^2 = I</math>

These matrices are typically represented using the Pauli matrices and the identity matrix.

Physical Implications

The Dirac equation successfully predicts the existence of antiparticles. For example, the equation's solutions for electrons imply the existence of positrons, which were later discovered experimentally by Carl Anderson in 1932. The equation also naturally incorporates the concept of spin, a fundamental property of particles.

Solutions and Applications

The solutions to the Dirac equation are four-component spinors, which account for both the particle and antiparticle states, as well as the two possible spin states. The equation is fundamental in the field of quantum electrodynamics (QED) and has applications in particle physics, condensed matter physics, and quantum computing.

See Also

• Quantum mechanics
• Special relativity
• Antimatter
• Quantum field theory

References

• Dirac, P. A. M. (1928). "The Quantum Theory of the Electron". Proceedings of the Royal Society A. 117 (778): 610–624.
• Anderson, C. D. (1932). "The Positive Electron". Physical Review. 43 (6): 491–494.

External Links

• Dirac Equation on Wikipedia