What is the Dirac Equation, and how is it defined?

Paul Adrien Maurice Dirac (1902-1984) proposed at the end of 1928 one of the equations with the greatest importance and implications in the Physics of the current era, and this is because it unifies the principles of quantum mechanics with those of relativity.

Quote (APA)

Evelyn Maitee Marin | Aug. 2022
Industrial Engineer, MSc in Physics, and EdD

This equation can be expressed in various ways, the most compact and simplified being what is considered one of the most aesthetic equations in science:

\(\left( {i\nabla - \frac{{mc}}{h}} \right) = 0\)

Where:
i: imaginary unit
m: rest mass of the electron
ħ: Planck's reduced constant
c: speed of the light
: summation operator of partial derivatives
: mathematical wave function of the electron

The absolute value of the square of the wave function represents the probability to find the particle in a certain position, considering its Energy, speed, among other parameters, as well as its evolution in the time. In other words, the Paul Dirac equation uses matrices acting on vectors and represents an evolution of the Schrödinger equation in relativistic quantum physics.

The Dirac equation was originally used to describe the behavior of an electron devoid of interaction, although its applicability extends to description of subatomic particles when they travel at speeds close to the speed of light. Dirac managed to explain at the subatomic scale the dual behavior of wave and particle that was already known at that time, since he considered the properties of particles such as angular momentum intrinsic or spin.

Another of the significant contributions of the Dirac equation is the prediction of antimatter, whose existence was later demonstrated (in 1932) by Carl D. Anderson using a cloud chamber with which he identified the positron. Likewise, he largely explains the fine structure identified in atomic spectral lines.

The image shows the famous photograph taken during the "Photons and Electrons" conference in 1927 where some of the most outstanding scientists in history are portrayed. In the celestial circumference is Paul Dirac.

Dirac Equation Background

In order to understand the considerations taken by Dirac in the development of his equation, as well as the bases on which his approach was based, it is important to know the theories prior to his model.

First, there is the famous Schrödinger equation of quantum mechanics, published in 1925, which converts quantities into quantum operators. This equation uses the wave function (), taking as its starting point the classical equation of energy E = p2/2m and incorporates the quantization rules for both momentum (p) and energy (AND):

\(ih\frac{\partial }{{\partial t}}\left( {r, t} \right) = \left[ {\frac{{{h^2}}}{{2m}}{\ nabla ^2} + V\left( {r, t} \right)} \right]\left( {r, t} \right)\)

The partial derivative /t expresses the evolution of the system with respect to time. The first term inside the square bracket refers to the Kinetic energy (\({\nabla ^2} = \frac{\partial }{{\partial r}}\left( {r, t} \right)\)), while the second term relates to the potential energy.

Note: in Einstein's theory of relativity, the variables of space and time must enter equally into the equations, which is not the case in the Schrödinger equation, in which time appears as a derivative, and position as a second derivative.

Now, for centuries, scientists have tried to find a model of Physics that unifies the different theories, and in the case of Schrödinger's equation, takes into account the mass (m) and the charge of the electron, but does not consider the relativistic effects that manifest at high speeds. For this reason, in 1926, scientists Oskar Klein and Walter Gordon proposed an equation that does take into account the principles of relativity:

\({\left( {ih\frac{\partial }{{\partial t}}} \right)^2} = \left[ {{m^2}{c^4} + c{{\left( { - ih\bar \nabla } \right)}^2}} \right]\)

The problem with the Klein-Gordon equation is that it is based on Einstein's, in which energy is squared, so this (Klein-Gordon) equation incorporates a squared derivative with respect to time, and this implies that it has two solutions, allowing for negative values ​​of time, and this makes no sense physical. Likewise, it has the inconvenience of generating probability values ​​less than zero as solutions.

Trying to resolve the inconsistencies implied by negative solutions of certain magnitudes that do not support these results, Paul Dirac started from the Klein-Gordon equation to linearize it, and in this procedure, he introduced two parameters in the form of matrices of dimension 4, known as Dirac or also Pauli matrices, and which are a representation of the algebra of the spin. These parameters are denoted as  and ` (in the energy equation, they are represented as E = pc + mc2):

By what is equality is fulfilled, the condition is that ´2 = m2c4

In general, the quantization rules lead to operations with derivatives that apply to scalar wavefunctions, however, as the parameters α and β are 4x4 matrices, the differential operators intervene on a four-dimensional vector (), known as spinor.

The Dirac equation solves the negative energy problem presented by the Klein-Gordon equation, but a negative energy solution still appears; that is, particles with properties similar to those of the other solution but with opposite charge, Dirac called this antiparticles. Furthermore, with the Dirac equation, it is shown that the spin is the result of applying relativistic properties to the quantum world.