Dirac Equation

Before introducing the Dirac equation, it was difficult to explain the behaviour of the particles as the particles with higher velocities were not studied. But Dirac equation introduced four new components to the wave. These four components were divided into two energy states: positive and negative. Both energy states have a spin of 1/2 up and down. Using Dirac equation, new spin properties and magnetic moment were assigned and the magnetic moment was given as:

\(\mu _{D}=\frac{qS}{m}\)

where,

  • S is the spin vector
  • q is the charge
  • m is the mass

What is Dirac equation?

Dirac equation is a relativistic wave equation which explained that for all half spin electrons and quarks are parity inversion (sign inversion of spatial coordinates) is symmetrical. The equation was first explained in the year 1928 by P. A. M. Dirac. The equation is used to predict the existence of antiparticles. The equation also supports solution for free moving electrons.

Dirac equation formula

\(\left ( \beta mc^{2}+c\sum_{n=1}^{3}\alpha _{n}p_{n} \right )\psi (x,t)=i\hbar\frac{\partial \psi (x,t)}{\partial t}\)

Where,

  • 𝜓=𝜓(x,t) is the electron wave function
  • M is the electron mass at rest
  • X, t is the spacetime coordinates
  • p1, p2, p3 are the momentum components
  • c is the speed of light
  • \(\hbar\) is the Planck constant

All these physical constants are the reflection of special relativity and quantum mechanics. The purpose of formulating this equation was to study the relative motion of the electron and to treat the atom as consistent with relativity.

Applications of Dirac Equation

  • In quantum mechanics, to resolve paradoxical features, Dirac field is used.
  • Dirac sea was studied with the help of “hole theory” according to which there are many negative charged electrons occupied in the vacuum and they are at eigenstate.

Other Formulations of the Equation

  • Polar form: With the help of Lorentz transformation, Dirac spinor can be represented using two degrees of freedom ie; as the derivatives of scalar and pseudoscalar bi-linear quantities.
  • As a differential equation: Spinor function of the Dirac equation for three out of four components can be represented as a partial differential equation for one component.
  • Curved spacetime: The equation can also be represented in curved space-time.

What is Dirac Field?

Dirac field is an example of the fermion field in which the canonical time equal communication relations are replaced with the canonical time equal anti-communication relations.

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Questions Related To Dirac Equations

Q1. Who suggested the concept of a matter wave?

Ans: de Broglie suggested the concept of a matter wave.

Q2. What is the value of spin g-factor?

Ans:  The value of spin g-factor is 1.

Q3. What is g-factor?

Ans: g-factor is a dimensionless quantity that is used to characterize the magnetic moment and angular momentum of the atom. It is also known as g value or dimensionless magnetic moment.

Q4. What is the total probability of finding the particle in space?

Ans: The total probability of finding the particle in the space is unity.

Q5. Define group velocity.

Ans: Group velocity is defined as the velocity of the wave through which the shape of the wave amplitudes resulting in modulation.

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