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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.

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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).

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