I know there are significant differences in the nature of the Schrödinger, Pauli, and Dirac equations. Although I know a bit about how each works, I don't understand the relative limitations of each equation relative to the others in terms of their usefulness in describing physical phenomena.
For instance, the Schrödinger equation is not relativistically invariant, so the Dirac equation would be more suitable if you are working with a relativistic problem/situation.
Can someone summarize some of the crucial applications each equation relatively more useful than the others (if any)?
Note: if someone sees what I'm driving at and can edit this to make it better articulated, that would be great.
Best Answer
The Schrodinger equation describes the time evolution of a non-relativistic quantum state:
$$ i{\frac {\partial }{\partial t}}|\psi \rangle = H | \psi \rangle $$
The Klein-Gordon equation describes relativistic spinless particles, but being a 2nd order differential equation, it gives rise to unphysical negative probability densities:
$$ (\partial^{{\mu }}\partial_{{\mu }} + m^{2})\psi =0 $$
The Dirac equation comes from linearising the K-G equation, is relativistic and describes fermions, i.e spin $1/2$ particles. It also naturally describes anti-particles. It is worth to note that solutions to the Dirac equation are automatically solutions to the K-G equation:
$$ (i \gamma ^{\mu }\partial _{\mu } -m)\psi =0 $$
Finally, the Pauli equation is just a modification to the Schrodinger equation to incoropate spin interactions of fermions with an external electromagnetic field described by a vector potential $\mathbf{A}$, therefore it is still non-relativistic:
$$ \left[{\frac {1}{2m}}({\boldsymbol {\sigma }}\cdot ({\mathbf {p}}-q{\mathbf {A}}))^{2}+q\phi \right]|\psi \rangle =i\hbar {\frac {\partial }{\partial t}}|\psi \rangle $$