How the works of Schrodinger, Pauli, and Dirac threw light on the Existence of Antimatter

Getting to the science that anti-matters.

Antimatter is usually found during high-speed collisions and sometimes as a convenient plot twist in science fiction stories. It’s just like normal matter, but opposite in terms of properties like electrical charge and baryon numbers. But the fact that scientists were able to realize the structure of a particle not found on Earth is almost as awe-inspiring as antimatter itself.

Our understanding of physics is an ever-changing sea, constantly evolving with the rivers of insights brought about by the geniuses of our time and teeming with discoveries yet to be made.

At the beginning of the 20th century, traditional physics had gone through an upheaval. Albert Einstein’s theory of special and general relativity had altered our perception of space, motion and time. The blossoming field of Quantum Mechanics defied all previous notions of the fundamental building blocks of nature.

By the late 1920s, the works of Einstein and Planck told us that light is not only a wave but also a particle. de Broglie further showed us that in fact, all matter has this dual nature.

The brilliance of Erwin Schrodinger

In 1926, Erwin Schrodinger’s famous equation described how to find allowed energy levels of quantum mechanical systems. This equation cleared up previous inconsistencies in physics, like the strange interference pattern in the famous double-slit experiment.

The Schrodinger equation

But there was a problem — or two.

This new theory didn’t align with Einstein’s relativity. Whereas Einstein showed us that the dimensions of time and space are two sides of the same coin depending on your frame of reference, Schrodinger could track a particle’s wave function depending on only one clock — the clock in the observer’s reference frame.

This meant that the Schrodinger equation is valid only for slow-moving objects!

Hold up. Don’t subatomic particles move at velocities near the speed of light? This signals to us that the Schrodinger equation is not complete.

Another problem arose in the description of particles as simple wave functions. This implies the particle’s distribution of momenta and positions should have no internal properties. But they do! These particles have the property of spin.

Wolfgang Pauli’s revelation

What is spin? Although it sounds like rotation, it really doesn’t mean the particles rotate but is an intrinsic form of angular momentum. This property was discovered by Wolfgang Pauli.

Pauli told us electrons must be following a “rule.” This rule, the Pauli exclusion principle, helped explain the electron energy levels in atoms.

So what does this mean? There should only be one electron per atomic orbital.

Source: xkcd

But what do we find when we take a closer look at the orbital? 2 big fat electrons!

Pauli quickly modified his theory, introducing a ‘degree of freedom’. He said electrons could have an up property and down property. So every orbital could have one up and one down electron and all is well in Pauli’s Land of Exclusion. This up and down property he discovered is actually just spin.

Alright, so this was discovered a little while after Schrodinger’s Equation. Can’t we just ignore this pesky little property and get approximately the same answer? Absolutely not!

Enter: A magnetic field.

With the presence of a magnetic field, spin direction takes center stage. This causes the Schrodinger equation to break down for speedy electrons and electrons in an electromagnetic field.

Paul Dirac’s Discovery

But not for long. in 1928, Paul Dirac stepped in to save the day and started working on this problem. Dirac worked toward a fully relativistic version of Schrodinger’s equation that applies for electrons too.

Let’s retrace his steps to see how he came to the groundbreaking conclusion of antimatter.

Step 1: Write down E = mc² {In its full form with momemtum}

Step 2: Introduce the quantum mechanical expressions for energy and momentum.

Step 3: The result: An unintelligible mishmash of math.

Luckily for us, Dirac stumbled upon a simple idea that untangled the mess of mathematics into an ingenious equation. Instead of just 2 components, up and down electrons, he realized there must be 4 degrees of freedom.

The Dirac Equation

This solution was elegant in the sense that it combined both quantum mechanics (Plank’s constant) and relativity (the speed of light). And it predicted the motion of an electron at any speed even in an electromagnetic field!

By this point, you should have realized that the insights in physics are bitter-sweet. Every new idea ties up old quandaries but is certain to bring up new questions. Dirac’s revelation was no different.

Dirac still did not know what the other 2 degrees of freedom were.

When calculating the energy of the electron using Dirac’s equation, it seemed that electrons can exist in states of negative energy. But negative energy can’t be right — there must a bottom to the energy well!

The Dirac Sea

Dirac came up with the solution to this by imagining an endless sea of electrons. These electrons occupy all possible negative energy states.

Now when we take away an electron we get an absence of the electron, a hole. The hole can actually be considered as a particle in its own right. It moves and acts like it has the mass of the missing electron with a positive charge.

Now let’s say a positive energy electron stumbled upon a hole. It would combine with this hole, simultaneously annihilate and release an explosion of energy equivalent to their masses.

Antimatter behaves similarly the holes in the Dirac Sea; the supposed hole is actually the equivalent of a positron, the electron’s antimatter counterpart.

The Dirac sea might be imaginary, but Dirac’s negative energy solutions describe something that is very real — antimatter.

In fact, the two missing components or degrees of freedom are the two spin directions of the antimatter counterpart.

This idea was confirmed when the positron was spotted by Carl Anderson in 1932.

Antimatter

Antimatter has been created at facilities in CERN as a byproduct of the super fast collisions at the Large Hadron Collider. During the Big Bang, the original large collision, a huge amount of antimatter and matter must have been created. If they were emitted in equal amounts, annihilation must have occurred, leaving no matter behind. But our universe, teeming with matter, is living proof that there must have been asymmetry in the start of the universe.

Dirac’s insight flipped a new chapter in our understanding of our universe with its prediction of antimatter. The careful re-examination of some fundamental theories of physics led to a pretty amazing discovery. Even more astounding discoveries just might be waiting right under our noses.