The KG equation describes the behavior of a spin zero particle and hence would be the correct equation to use for an approximate description of pionic atoms (atomic states with a substituted for an electron). Unfortunately, the pion is very short lived, and these "atomic" states have a very short lifetime. In addition, because the pion is so much more massive than the electron, it is bound in a very small orbit. The orbit is so small that there is a significant probability that the pion will overlap with the nucleus, where it will interact strongly, further broadening the states. These effects make it difficult to study pionic atoms, and direct tests of the applicability of the KG equation to such states is a topic of current research.
In any case, the study of the structure of (perhaps hypothetical) atomic states with spin zero constituents is an interesting intellectual question. Comparing results obtained from the KG equation with those we will obtain later from the Dirac equation will tell us how much of the observed fine structure is due to relativity alone and how much is due to the spin of the electron. Similarly, comparison of the Zeeman effect predicted by each equation helps us separate effects due to the orbital motion of the bound particle (all that we have in the KG theory) from additional effects present in the Dirac theory.
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