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## Antimatter and Dirac’s Equation

Yesterday I introduced Paul Dirac, number 10 in “The Guardian’s” list of the 10 best physicists. I mentioned that his main contributions to physics were (i) predicting antimatter, which he did in 1928, and (ii) producing an equation (now called the Dirac equation) which describes the behaviour of a sub-atomic particle such as an electron travelling at close to the speed of light (a so-called relativistic theory). This equation was also published in 1928.

## The Dirac Equation

In 1928 Dirac wrote a paper in which he published what we now call the Dirac Equation.

This is a relativistic form of Schrödinger’s wave equation for an electron. The wave equation was published by Erwin Schrödinger two years earlier in 1926, and describes how the quantum state of a physical system changes with time.

### The Schrödinger eqation

The various terms in this equation need some explaining. Starting with the terms to the left of the equality, and going from left to right, we have $i$ is the imaginary number, remember $i = \sqrt{-1}$. The next term $\hbar$ is just Planck’s constant divided by two times pi, i.e. $\hbar = h/2\pi$. The next term $\partial/\partial t \text{ } \psi(\vec{r},t)$ is the partial derivative with respect to time of the wave function $\psi(\vec{r},t)$.

Now, moving to the right hand side of the equality, we have
$m$ which is the mass of the particle, $V$ is its potential energy, $\nabla^{2}$ is the Laplacian. The Laplacian, $\nabla^{2} \psi(\vec{r},t)$ is simply the divergence of the gradient of the wave function, $\nabla \cdot \nabla \psi(\vec{r},t)$.

In plain language, what the Schrödinger equation means “total energy equals kinetic energy plus potential energy”, but the terms take unfamiliar forms for reasons explained below.

## The 10 best physicists – no. 10 – Paul Dirac

At number 10 in “The Guardian’s” 10 best physicists is English theoretical physicist Paul Dirac.

## Dirac’s brief biography

Dirac was born in Bristol in the south-west of England in 1902. He died in 1984. He was brought up in Bristol. His father was Swiss-French, his mother was English. He did his undergraduate degree at Bristol University studying engineering. However, he was unable to find work as an engineer, and so instead undertook a second degree, this time in mathematics, at the same institute. He then went to Cambridge to do his PhD, working on General Relativity and Quantum Mechanics, under the supervision of Ralph Fowler. The title of his PhD thesis was simply “Quantum Mechanics”.

The front cover of Paul Dirac’s PhD Thesis, submitted in 1927 to Saint John’s College, Cambridge.

## Dirac’s main achievements

Dirac’s place in this top 10 list is due to two main things, his prediction of the existence of antimatter, and for the equation which describes the motion of a fundamental particle such as an electron when it is travelling near the speed of light. Both of these will be described in more detail in future blogs. Dirac won the Nobel prize for Physics in 1933, he shared it with Erwin Shrödinger “for the discovery of new productive forms of atomic theory”.

### Antimatter

The theoretical prediction for which Dirac is most famous to people outside of physics is his idea of antimatter, which of course has become a firm favourite of science fiction. His basic idea was that every fundamental particle has an anti-particle. So, for example, an electron has an anti-particle which would have the same mass and the opposite electric charge. We call this anti-electron a positron. A proton would have an anti-proton and so on. Anti-matter was predicted by Dirac in 1928 and was experimentally verified in 1932 with the discovery of the positron.

### The Dirac equation

Dirac is most famous amongst physicists for what is now known as “Dirac’s equation”. This is an equation which describes the relativistic behaviour of an electron, and therefore unified quantum mechanics with special relativity. Relativistic means travelling near the speed of light.

The terms in this equation need a little explaining. Rather than explaining them in this blog, I will do so in a series of future blogs, as I will need to give some background. Not only do I need to explain the terms in this equation, but this equation cannot be understood in isolation, one has to also understand Schrödinger’s equation.

For example, the term $\psi(x,t)$ is the so-called “wave-function” of the particle, and $\nabla^{2}$ is the so-called Laplacian. $i \text{ is the imaginary number, that is } \sqrt{-1}$. Now you see why I need to give some background!!