 # 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 the 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 the Dirac equation, new spin properties and magnetic moment were assigned, and the magnetic moment was given as:

 $$\begin{array}{l}\mu _{D}=\frac{qS}{m}\end{array}$$

where,

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

## What is Dirac equation?

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

### Dirac Equation Formula

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

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
• $$\begin{array}{l}\hbar \, is \, the \, Planck \, constant\end{array}$$

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, the Dirac field is used.
• The 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 spinors 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 space-time: 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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## Frequently Asked Questions – FAQs

Q1

### Who suggested the concept of a matter-wave?

De Broglie suggested the concept of a matter-wave.

Q2

### What is the value of spin g-factor?

The value of spin g-factor is 1.
Q3

### What is g-factor?

g-factor is a dimensionless quantity used to characterise the atom’s magnetic moment and angular momentum. It is also known as the g value or dimensionless magnetic moment.

Q4

### What is the total probability of finding the particle in space?

The total probability of finding the particle in space is unity.
Q5

### Define group velocity

Group velocity is defined as the velocity of the wave through which the shape of the wave amplitudes results in modulation.

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