Significance of mc/h constant in Klein-Gordon equaiton The are several ways, in which one can write the Klein-Gordon equation, the most straightforward being probably the following:
$$
\hbar^2 \partial_t^2 \psi(x) = (\hbar^2 c^2 \Delta + m^2c^4) \psi(x)
$$
However it is possible to use the d'Alembert operator $\square$ and write KG equation like this:
$$
(\square+\mu^2)\psi = 0
$$
where $\mu := \frac{mc}{\hbar}$. the $\mu$ in natural units ($c = \hbar = 1$) is obviously just the mass of the particle, however I wonder if is it possible to interpret it somehow in non-natural units (e.g.: $[\mu] = m^{-1}$). I mean something along the lines of $mc^2$, which we can interpret as energy and therefore easily analyze particle collisions in terms of possible masses that appear in such events. Just a soft question, because I don't have a lot of experience with relativistic quantum mechanics.
 A: OK, so I realized that $\mu$ is the inverse of the Compton's wavelength for a particle.
$$
\lambda_C := \frac{\hbar}{mc}
$$
From F. Schwabl's "Quantum Mechanics":
"Compton wavelength (...) can be interpreted as the de Broglie wavelength of highly relativistic electrons" (p. 135). That is the sort of thing I was looking for, but I'm not sure if my question was well understood.
A: First, in units $c = \hbar = 1$, energy, momenta, and mass have the same units : $[E]=[\vec P]=[\mu]= [m]= M= L^{-1}$, and there is no possibility of "non-natural units".
Secondly, the energy is no $mc^2$, if you make a energy/momentum Fourier transformation of the KG equation, you obtain : $(p_0^2 - \vec p^2 - m^2) \psi(p)=0$, where $p_0$ represents the energy. This means that the only authorized momenta Fourier components $\psi(p)$ correspond to $p_0^2 - \vec p^2 - m^2=0$, which is the correct equation between energy, momentum, and mass, for relativistic fields/particles.
Third, the KG equation represents the equation for a free field, without interactions. If you want to study interactions, you have to add some interactions terms in the Lagrangian (which define the possible particles and possible masses), and, within theories like Quantum Field Theory, analyse the probability of possible outcomes.
