The Schrodinger Equation regularly fails. In this video we look at two upgraded equations (including the famous Dirac Equation) that work in both quantum and relativistic environments.

The Schrodinger Equation is famous, and rightly so. It's the governing equation of a theory called quantum mechanics. It can very accurately predict how quantum systems (i.e. very small systems) will behave through space and over time. The basic premise of it is that it adds together a system's kinetic and potential energies and equates this to the system's total energy. This is seemingly pretty common sense, but the Schrodinger Equation is "quantized", meaning measurements on the system only give very specific results. We can also never predict exactly which measurement outcome we will get, but only the probabilities of each possible outcome. The Schrodinger equation also has "measurement operators", which are the math equivalent of making a measurement on the system.

Importantly, the Schrodinger Equation is not relativistic. In other words, it does not account for the strange effects we see when relativity is accounted for. We know that when objects move at high speeds relative to each other, that they noticeably measure distances and times differently to each other. Because these effects are not accounted for, the Schrodinger Equation does not always accurately predict the behaviour of small systems that may be moving at high speeds. It also treats time as a universal variable (i.e. everybody measures time in the same way), which is not how relativity deals with what it calls "the fourth dimension".

To save quantum mechanics in these high-speed scenarios, we need to look at some other equations that are both quantised and relativistic. The first equation of this sort that we'll look at is known as the Klein-Gordon Equation. To get this equation we start with Einstein's famous mass-energy relation (E = mc^2). But in reality, we start with the full version of this equation which also involves momentum. Taking this full mass-energy equivalence relation, we can then quantise it and derive the Klein-Gordon Equation.

The Klein-Gordon Equation accurately predicts the behaviour of spin-0 particles. In other words, it does not account for spin. But it is quantum and relativistic. It also has a "psi" quantity in it just like the Schrodinger equation, but here "psi" is charge density, not probability density. This is because the Klein-Gordon Equation allows negative solutions for the square modulus of psi, which previously we interpreted as probability. It makes no sense to have negative probabilities, and instead this equation deals with the behaviours of particles with positive, negative, and zero charge.

To account for spin, then, we need to look at yet another equation. Remember, spin is angular momentum that is inherent to a particle (without it moving along a curved path or rotating). The equation that starts to account for spin is the very famous Dirac Equation. It's highly complicated, but can be essentially thought of as the square root of the Klein-Gordon Equation. It has four complex degrees of freedom in its "psi" quantity. The first two of these look like the quantum wave function "psi", but the remaining two encode details for systems that are quantum and also relativistic.

When Dirac came up with his equation, he realized that some potential solutions allowed for particles similar to the ones we know, but with the exact opposite charge. For example, electron-like particles with +1 unit of charge rather than -1 were allowed. Dirac thought this was initially a mistake, but we eventually found particles like this to exist! We now call them antiparticles, which make up antimatter. In other words, what was initially thought to be an accident of under-constrained mathematics, actually provided a wonderful prediction for phenomena never seen before!

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Timestamps:
0:00 - Understanding the Schrodinger Equation
3:50 - Relativistic Quantum Mechanics
5:05 - The Klein-Gordon Equation
7:42 - The Dirac Equation

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