Quantum Week: An Equation That Connects Special Relativity And Quantum Mechanics.

Today is the penultimate day of our Quantum Week and this is the right time to introduce the famous Dirac equation. We started off from 1900 with Max Planck, Albert Einstein and Niels Bohr on day 1. Then on day 2, we saw Rutherford and Bohr split the atom and lay the basic foundation of the atomic model. Day 3 was on the flagship equation of quantum mechanics: the Schrodinger wave equation and day 4 was on an introduction to the mathematics of quantum mechanics along with Heisenberg’s uncertainty principle. Yesterday we learnt Pauli’s exclusion principle and its applications in astrophysics. Today, we will learn about the work of ‘The Strange Man of Physics’ aka Paul Dirac.

Why A New Equation?

Before studying the Dirac equation, let us answer this question. Prior to Dirac, we had the Schrodinger Wave Equation (SWE). This equation was easy to use and provided the total energy and wave function of any quantum state. Wavefunction is the most important parameter in quantum mechanics. Once you know the wavefunction, you can get everything. However, the SWE had a problem: it was too simple. In reality, some corrections need to be added in any system before using it. For example, suppose we apply a potential V to a system and solve the SWE for wavefunction using it. The answer will not be accurate. This is because we have ignored so many other factors that affect the potential. The potential is modified by the electron-electron repulsion, for example. So SWE isn’t a very accurate equation. Moreover, electrons that are orbiting the nucleus are relativistic. SWE doesn’t consider this fact. Hence, in order to synchronize quantum mechanics with special relativity, a new equation was required.

The First Attempt: Klein Gordon Equation

The first attempt to combine relativity and quantum mechanics came in the form of the Klein Gordon Equation. I will try to get into minimum mathematical details in this article. The Schrodinger wave equation is this:

The Schrodinger Wave Equation.

The left side represents the square of the momentum operator divided by twice the mass, which is the non-relativistic kinetic energy. From special relativity, the total energy of a particle is given by the momentum-energy relation: E^2 = p^2 c^2 + m^2 c^4 where p is the momentum. Substituting this in the momentum operator, we get the Klein Gordon equation:

The KG Equation

Don’t worry about the mathematics. It is not very simple. What is more important to understand is the physical significance of this equation. KG equation is the first attempt to combine the two fields of physics but there are two problems in this equation. First is that if we consider the solutions of this equation, we get particles that have negative energy (negative energy solutions). Second is the negative probability density. Schrodinger himself first at this equation but later discarded it.

The Dirac Equation

This is the time Paul Dirac comes into picture. Dirac worked on solving these two problems and combining special relativity and quantum mechanics. With rigorous mathematical efforts, he derived an equation that did solve the problem of negative probability density, but still had negative energy solutions in it. Now you might be thinking that why don’t we just discard the negative energy solutions like classical mechanics. We know that negative energy isn’t possible. The answer is: Algebra doesn’t allow us to do so. To form a complete set of solutions, we need the negative energy solutions too. The Dirac is something like this

The Dirac Equation

Dirac’s Explanation: The Hole Theory

With no other option left, Dirac thought of an explanation to these particles with negative energy. The puzzle that he wanted to solve was that if electrons have positive energy (which they really have) and when a photon interacts with this electron, an electron will decay into negative energy. But this doesn’t happen. Why?

Dirac came up with an idea. He said that the all the negative energy states are already occupied. This description of the vacuum as a “sea” of electrons is called the Dirac sea. Since the Pauli exclusion principle forbids electrons from occupying the same state, any additional electron would be forced to occupy a positive-energy state, and positive-energy electrons would be forbidden from decaying into negative-energy states. Dirac further reasoned that if the negative-energy states are incompletely filled, each unoccupied state – called a hole – would behave like a positively charged particle. 

The Dirac Sea: The black dots are the particles and the white ones are the holes. Notice that the particles have positive energy and anti-particles have negative energy.

Antimatter

Dirac argued that the hole should be a proton, which is positive. He did not realise that fact that the whole should of same mass as that of an electron. In fact, proton is 1,836 times heavier than an electron. It was in 1932, a few years after Dirac’s equation, Carl Anderson discovered the first anti-particle: the positron. An anti-particle has exactly same mass but opposite charge as compared to the ordinary matter. It was the antimatter that actually corresponded to the negative energy solutions. So antimatter was first predicted theoretically.

The comparison between matter and anti-matter.

The Dirac equation was the first successful attempt to combine special relativity and quantum mechanics. However, the fields of general relativity and quantum mechanics are yet to be combined. This is the most important unsolved problem in physics.

Previous In Quantum Week: Pauli’s Exclusion Principle And Its Wonderful Applications In Astronomy.

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