The Schrodinger versus Dirac equation.

[Alex K]

I haven't dealt much with the Dirac equation as a relativistic wave equation. It is very useful, but kind of awkward in that role. Where the Dirac equation really shines is as an element of quantum field theories such as Quantum Electrodynamics and the Standard Model, where it serves not as a quantum wave function equation for a fixed number of particles, but one level deeper, as a field equation where it describes the creation, annihilation and interaction of arbitrary numbers of particles including virtual ones.

Technically, the Dirac equation as a wave equation can be taken to its non-relativistic limit where it yields the Schrödinger equation (or rather the Pauli equation which is an extension incorporating spin) plus several approximate corrections which can be used to improve atomic physics calculations (see: Foldy-Wouthuysen-Transformation). So in principle, the Dirac equation can do anything the simple Schrödinger equation can because it contains it as a limit, but in practise, as you say, if only approximations to relativity theory are good enough, many calculations are way simpler using the Schrödinger or if necessary, the Pauli equation.