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Motivation

It's rough to study PHYS 125 here without actually systematically studies Q.M. first. But I will give a shot and preview Dirac Equation (essentially a relativistic Schrödinger equation) here and write a summary for its key features.


In general, Schrodinger's equation refers to the following, and we most will only deal with it's 1-D form.
$$
i \hbar \frac{\partial}{\partial t}\vert\Psi(\mathbf{r},t)\rangle = \hat H\vert\Psi(\mathbf{r},t)\rangle
$$

Following Griffiths Chapter 7, we shall see what happens is the mean speed of the particle is too fast that momentum requires relativistic correction.


Operator in Classical Equations

After familierizing oneself with the concept of operators and how we can obtain observables by combining (H, x, p) operators according to classical formulas, we can use this machinenary to 're-derive' Schrodinger's equation. First, recall a classical energy-momentum relation:
\begin{equation}
\frac{\mathbf{p}^2}{2m} + V = E
\end{equation}
Apply what Q.M. taught us about observable (p) being a operator:
$$
\begin{cases}
\mathbf{p}\rightarrow-i \hbar\nabla \\
E\rightarrow-i \hbar \frac{\partial}{\partial t}
\end{cases}
$$
And apply the whole equation (as an operator), on a wave equation $\Psi$, we get Schrodinger's equation:
$$
-\frac{\hbar ^2}{2m}\nabla^2\Psi + V\Psi = i\hbar\frac{\partial\Psi}{\partial t}
$$


Klein-Gordon Equation

KG equation can be obtains by the very same logic --- using Q.M. operators to replace some variable in classical formulas and see where it leads us to. From relativity, we know that $E^2 - p^2c^2=m^2c^4$ (I want to let $c=1$ so badly). Of course we are neglecting potential energy here because we are dealing with particle physics. We can also write this in more informative Einstein notation :
$$
p^\mu p_\mu - m^2c^2 = 0
$$
If now we replace each $p_\mu \rightarrow i\hbar \partial_\mu$, with $\partial_\mu=\frac{\partial}{\partial x^\mu}$ (you can never be too familiar with co/contravariant). We can write out a bit just to be clear:
$$
\partial_0=\frac{1}{c}\frac{\partial}{\partial t},\partial_1=\frac{\partial}{\partial x},\partial_2=\frac{\partial}{\partial y},\partial_3=\frac{\partial}{\partial z}
$$
We can drag all of these into our original relativistic energy-momentum equation:
$$
-\hbar^2\partial^\mu\partial_\mu\psi - m^2c^2\psi = 0
$$
or it's equivalent to write as:
$$
-\frac{1}{c^2}\frac{\partial^2\psi}{\partial t^2}+\nabla^2\psi = (\frac{mc}{\hbar})^2\psi
$$
Which is the KG equation. Schrodinger actually discovered this before the non-relativistic one and abandoned it because it failed to reduce to Bohr's model (because electron is fermion and KG equation is in fact for Bose-Einstein statistical particles).