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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.

When an electron is stationary it has an electric field around it. The strength of that electric field (called the electrostatic field) is given by the equation

$E_{r} = \frac{ 1 }{ 4 \pi \varepsilon_{0} } \cdot \frac{ e }{ r^{2} } \text{ (Equ. 1) }$

where $E_{r}$ is the radial component of the electric field (which, in this case, is the only component), $e$ is the charge on the electron, $r$ is the radial distance from the electron, and $\varepsilon_{0}$ is the permittivity of free space $(8.854 \times 10^{-12} \text{ F/m or C}^{2} \text{/N/m}^{2})$.

An electron is surrounded by a radial electric field

If the electron is moving in a wire we have a current. A moving electron produces a magnetic field, but this magnetic field is constant because the electron’s motion has a constant velocity. In order for an electron to radiate Electromagnetic (EM) radiation, it needs to accelerate. Remember, acceleration means its velocity needs to change, which can be achieved either by changing the electron’s speed or its direction. Both types of change will produce an acceleration, and hence produce EM radiation.

In 1907 J.J. Thomson, who discovered the electron in 1897, presented an argument which helps us understand why an accelerated electron produces EM radiation.

## Thomson’s argument of why an accelerated electron radiates

Let us suppose we have an electron which is initially at the origin of a coordinate system at time $t=0$. Thus the electric field lines radiate from the origin as shown in this diagram.

At time $t=0$ the electron is at the origin of the x-y coordinate system. We consider a sphere which is a distance $r(=ct)$ away from the centre.

We are now going to accelerate it in a time $\Delta t$ to the right, along the x-axis, and we shall see how this affects the electric field surrounding this electron. We are going to accelerate it to a small velocity $\Delta v$ which is much less than the speed of light $c$ (so that we do not have to consider relativistic effects). The acceleration is given by $a = \Delta v / \Delta t$.

We are going to consider the electric field lines at some distance $r$ from the electron. This distance is shown by the black dashed circle in the diagram above. Because of the finite propagation speed of EM radiation, the sphere can also be said to have a radius of $r=ct$. The radius of the sphere $r$ is chosen to be large enough so that the distance moved by the electron in the time $\Delta t$ is much less than the radius of this sphere. That is, $r \gg \Delta v \times \Delta t$.

At a time $t$ later, the electron has moved a distance $\Delta v \, t$ to the right. Because the black dashed sphere is large, with $r \gg ct$, the electric field lines outside of this sphere are indisturbed by the motion of the electron, as its change of position has not had time to propogate out to such distances. However, close to the electron the field lines are centred on the new position of the electron. These new field lines are represented in red in the figure below, with the black dashed lines representing the original field lines from when the electron was at the origin. The diagram also shows the original black dashed circle centred on the origin, but also a new solid red circle centred on the new position of the electron.

The electron is accelerated to the right along the x-axis. The black dot represents its position at time $t=0$, the red dot its position at time $t=t$. The black dashed circle is large enough so that its radius $r \gg ct$, where $c$ is the speed of light. Therefore the field lines outside of this black dashed circle are unaffected by the movement of the electron. The red solid lines represent the new field lines due to the electron’s new position, with the red solid circle being centred on the electron’s new position.

There is a transition region where the new field lines (the red ones) connect up with the old field lines (the black ones), in a shell. As the electron moves for a time $\Delta t$, the thickness of this transition region (a shell) is $\Delta t \, c$. We are going to consider the radial component and the tangential component (the tangential component is at right angles to the radial component) of the electric field in this shell. That is, the radial and the tangential components of the electric field line from point $A \text{ to point } C$ in the diagram below.

In the transition region the electric field line goes from point $A \text{ to point }C$. We are going to split this field line into its radial and tangential components $E_{r} \text{ and } E_{\theta}$.

As the electron has moved a distance $\Delta v \, t$, and we are considering a field line which is at an angle $\theta$ to the x-axis, the ratio of the tangential component of the electric field to the radial component is just given by the tangential and radial components of the triangle $ADC$, that is

$\frac{ E_{\theta} }{ E_{r} }= \frac{ \Delta v \, t \sin \theta }{ c \, \Delta t }$

But, the radial component $E_{r}$ is given in Equ. 1 above, so we can write

$E_{\theta} = E_{r} \cdot \frac{ \Delta v \, t \sin \theta }{ c \, \Delta t } = \frac{ 1 }{ 4 \pi \varepsilon_{0} } \cdot \frac{ e }{ r^{2} } \cdot \frac{ \Delta v \, t \sin \theta }{ c \, \Delta t }$

Re-ordering out terms we can write

$E_{\theta} = e \cdot \frac{ 1 }{ 4 \pi \varepsilon_{0} } \cdot \frac{ \Delta v }{ \Delta t } \cdot \sin \theta \cdot \frac{ 1 }{ cr^{2} } \cdot t =$

Remembering that $r=ct$ and the acceleration of the electron $a = \Delta v / \Delta t = \ddot{r}$ we can write

$\boxed{ E_{\theta} = e \cdot \frac{ 1 }{ 4 \pi \varepsilon_{0} } \cdot \frac{ \ddot{r} \sin \theta }{ c^{2} r } }$

So, this is the pulse of EM radiation which is produced by an accelerating electron. Notice that

1. it depends on the acceleration $\ddot{r}$ of the electron, the larger the acceleration the greater the EM pulse.
2. it is dependent on $1/r \text{ and not } 1/r^{2}$, and
3. it is tangential (at right angles) to the direction of motion of the electron.

## The discovery of the electron

I was in Cambridge in early February with my son, and on our 3rd day there we went out to the Cavendish Laboratory and the Institute of Astronomy. In the road that runs from the main road to the Cavendish is JJ Thomson Avenue. As this picture shows, the avenue is named after the famous English physicist J.J. Thomson.

The sign on JJ Thomson Avenue, the road which leads from the main road to the Cavendish laboratories in Cambridge

In 1897 J.J. Thomson discovered the first sub-atomic particle, the electron. He was doing experiments with cathode ray tubes, which had been discovered in the mid 1880s, and decided to see how they might be affected by magnetic and electric fields. He found that the cathode rays were deflected by the magnetic and electric fields, showing that they must be made up of charged particles.

A cathode ray tube. Electrons travel from the negative end (the cathode) towards the positive terminal (the anode). The green glow is produced by the charged electrons interacting with phosphorous in the glass, which then fluoresces. The shadow of the Maltese cross appears on the glass.

More than this, be was able to measure the shape of curve produced by the fields, and using the known strength of the fields he used, was able to calculate the mass and charge of the particles. By this time, chemists had fairly accurately determined the masses of atoms, and had shown that e.g. Carbon had a mass some 12 times the mass of Hydrogen.

Thomson found that the mass of the particles he was deflecting in his magnetic and electric fields werethousands of times less massive than the mass of the lightest known element, Hydrogen. This of course indicated that what he had discovered was sub-atomic, a constituent of atoms.

Thomson thus showed that cathode rays are a stream of electrons. The very high voltage (thousands of volts) between the positive (anode) and negative (cathode) terminals causes electrons in the cathode to be accelerated to a high velocity as they are attracted towards the anode. Cathode ray tubes were the basis for most TVs and computer monitors until this last 10 years, when more efficient Liquid Crystal Displays have largely replaced them.

As I will describe in a future blog in more details, some 12-13 years after Thomson’s work, one of his ex-students Ernest Rutherford showed that most of the mass of an atom resides in the centre, in its nucleus. The electrons orbit the nucleus, but it is the electrons which are important in e.g. giving elements their chemical properties and forming bonds with other atoms.

One anecdote about Thomson is that he used to get annoyed when people would write to him and spell his name with the more traditional “p” in it, Thompson. In replying to any such correspondence, he would insert random letter “p”s into the person’s name.