Equations which accidentally uncovered mysteries of nature
Today, we will talk about 3 equations, one which led to inception of Quantum Mechanics(QM), other gave the foundations of General Relativity and finally, quite aptly, an equation which was the first successful attempt to combine QM and relativity.
The pursuit of the theoretical physicists is to explain different phenomenons by a common underlying theory. A theory would consist of mathematical rules which has to explain the observations. The approach generally, is to fit the experimental data to a mathematical model. However, there have been many instances, where Mathematics and the beauty of it, forced the hand (or should I say, the mind) of these physicists to look at the equations and try to “fit the nature” based on it. We will see what, initially-though-to-be-bizarre implications it led to, and how it unfolded to discover something fundamental about out nature itself.
The Origin Of Quantum Mechanics
Normal matter, due to the internal motion of particles within it, emits electromagnetic radiation (let’s call it heat glow). Matter like the sun (surface temp ~6000K) shines yellow, our human body(surface temp~300K) emits low frequency infrared light and so on.
Above is the experimentally derived graph with the distribution of brightness w.r.t. frequency for a constant temperature black-body.
Till the dawn of the 20th century, this curve was a mathematical mystery. Rayleigh and Jeans set out to explain this with a new concept that an object’s heat energy would be equally shared among all possible energy states in which the particles could vibrate. They assumed that particles could vibrate with any amount of energy, as low as possible.
The resulting equation(shown above and as black curve in the graph), predicted that the intensity should eventually approach infinity as the frequency increases. (This is popularly known as Ultraviolet Catastrophe)
Max Planck resolved this catastrophe as an accident. In what he described as a moment of desperation, he tried to introduce a maths trick where he assumed that a particle could only vibrate in multiples of some minimum energy. He assigned a number to this minimum energy (hν) where h is a very small number and ν is frequency. He didn’t believe in the quantised energy states, rather he believed that the term ‘h’ would get cancelled out in the final equation as there should be no minimum energy state. But it didn’t.
Rather, this equation, which he came up with, perfectly predicted the black body curve.What had initially been introduced as a trick, now hinted that h is a real, fundamental entity of nature. It led to the idea that energy is actually quantised. h is famously known as Planck Constant. This was the origin of Quantum Mechanics which would later lead to quantum theories by likes Bohr, Heisenberg and Schrodinger.
General Relativity
Einstein originally believed in the concept of a static universe, a notion which was believed to be true at that time. In 1915, he had proposed the General Theory of Relativity. He was unsatisfied with his equations as they predicted that whole of cosmos would eventually contract under the effects of gravity predicting the end of the universe with a collapse onto itself.
To counteract this possibility, in 1917, Einstein added a cosmological constant(Λ). He assigned it a value which would just counteract the gravity so that universe stays static (in an unstable equilibrium, though). He would later go on to call this idea, “the greatest stupidity of his life”. In 1931, the Hubble telescope disrupted the physics community by discovering that the universe is actually expanding and not static.
So, the universe is not static. What did Einstein do? He abandoned the concept of (Λ) and proposed a model of the universe which is continuously expanding where Λ=0.
Note: This is very well possible, all forces are attractive doesn’t mean the universe can’t expand now. Just like a ball can be observed to be is in upward motion inspite of a downward force (due to an initial velocity), the universe could also be expanding currently with only gravity existing.
This Einstein’s model of an expanding universe with Λ=0 was thought to be true, until 1998, when our understanding was gonna change forever.
Two teams of astronomers, carried out observations on distant supernovae to show that the galaxies were not just moving away from us but the galaxies were actually accelerating away from us. That means there is an underlying energy which is forcing them away from us. This implied that Λ, the cosmological constant does exist and Λ > 0. This constant Λ which exists, yet is never witnessed directly, is known as the dark energy of the vacuum.
A mathematical tweak to the initial GR equation to produce a static universe, only to be later abandoned due to the observation of an expanding universe, not to be resurfaced before a century nearly, to predict the most bizarre properties of vacuum, namely the dark energy is what a single ‘popped’ term did for our understanding of the cosmos.
Dirac Equations
By early 20th century, we understood the particle-wave nature of matter. Schrodinger Equation predicted the evolution of the wave function for any quantum system. However, it came with its own issues.
- Schrodinger Equation could only predict the wave function of slow moving particles.
As we know from Special Relativity, values of distance covered and time experienced become observer-dependent. This was a big problem since sub-atomic particles do travel at very high velocities. - Schrodinger Equations predicted wrong results for electrons present in magnetic field
Physicists had identified a hidden quantum state of an electron (called electron spin, which can take on 2 values) which was not considered in the Schrodinger Wave Function and in a magnetic field, the spin directions are very important.
It took the genius of Paul Dirac in 1928 to resolve these issues in the Schrodinger Equation. Trying to model an electron with 2 possible spin states, he started with Einstein’s equations and tried to replace the energy and momentum with their respective terms in the quantum form. The resulting equations were pretty complex and could be called mathematically ‘ugly’.
In trying to resolve the mess, he stumbled upon a beautiful idea. Rather than solving for a 2-component electron system, he realised that a 4-component system collapsed the complexity in the equations to reduce it into a beautiful form.
He had just introduced 2 new degrees of freedom in the 4-component electron, without realising what they even represented. The resulting equation being so elegant made him believe that he was on the right path.
His equations were widely successful in predicting the evolution of a fast moving electron in a magnetic field (solving problem 1 and 2 above).
- But what are these 2 extra components ?
Computing the energy for these 2 new states, Dirac found that these particles could exhibit negative energy and a charge, exactly opposite that of an electron. Let’s call them anti-electron. Theories to explain these implied that if an electron would interact with an anti-electron, they both would annihilate and release all their energy.
The extra components of the Dirac Equation had predicted the existence of anti-matter. Not long after 1928, experimental physicist successfully observed the anti-electron (called a positron) in cosmic rays. Anti matter is real and Dirac was able to predict and calculate their properties even before we had ever realised of its existence.
The pursuit of beauty in mathematics bore its fruit for Dirac and the rest of us. There is no pre-defined approach and pursuing some form of beauty or elegance in the theory over experimentation implications has been a long, philosophical debate among the best minds in physics.
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