# Solutions of relativistic wave equations compared to classical wave functions

In classical quantum mechanics, absolute square of the wave function (i.e. $$|\psi|²$$) means probability density of particle's location, so when we integrate this over certain volume we get the probability for a particle to be in that volume. Does this also apply for solutions of Klein-Gordon equation or Dirac equation or other relativistic wave equations? If not, then how do we get something that can be experimentally measured from the solutions of relativistic wave equations.

We can play around with the interpretation of the wavefunctions in relativistic quantum mechanics but we quickly run into problems with particle creation/destruction. I think it is easier to talk instead about the explicitly measurable quantities in experiments. In both non-relativistic and relativistic QM it is rare to measure the positions of the particles and compare them to your prediction. If instead you view, $$\int \psi^{*}_f\psi_i dx = \text{P}(i\rightarrow f),$$ read as the probability to go from state $$i$$ to state $$f$$. Now you can massage the Schrodinger equation (or Klein-Gordon or Dirac equations) into a probability to go from an initial state to a final state. If you have a single particle transitioning into multiparticle states we would call that a lifetime calculation, if you measure multiple particles going into multiple particles we would call that a scattering calculation.

There are many many experiments measuring lifetimes and particle scatterings. These can be predicted in both relativistic and non-relativistic QM, often to very impressive accuracy.

The algebra is complicated and is covered in sections called "Time-dependent purturbation theory" in most quantum books, and would be covered in a second semester undergrad quantum class (and a second semester graduate course).