What exactly goes wrong when using the Klein-Gordon equation to calculate the spectrum of hydrogen? In many textbooks and lecture notes, it says that the Klein-Gordon equation was discarded first because when interpreting it as an equation for a single-particle wave function and trying to calculate the spectrum of the hydrogen atom, it gives wrong results.
But what exactly does go wrong when trying to calculate the spectrum of hydrogen from the Klein-Gordon equation instead of using the Schroedinger equation? How doe the wrong results look like? 
Nowhere have I seen a detailled explanation of this issue.
 A: I'm just getting to relativistic QM in my AQM course. The textbook I use as a reference [1] derives the continuity equation for a free particle. The standard procedure to do so is:
$\phi^*  \hat{KG}\phi - (\phi^*  \hat{KG}\phi)^{\dagger}= \dot{\rho}+ div\vec{j}=0$
where $\hat{KG}\phi=0$ is the Klein Gorden equation. As the KG equation has the second derivative with time and this component isn't complex doing this procedure leads to:
$\rho= \frac{i\hbar}{2mc^{2}} (\phi^*\frac{\partial\phi}{\partial t}-\phi\frac{\partial\phi^*}{\partial t})$.
(the usual result for TDSE is $\rho=|\phi^{*}\phi|$)
Now given that under certain I.Cs this does is not always definitely positive it cannot be interpreted as a probability density. However my lecturer, and Dirac himself [2] state that is was perfectly healthy back in the day to treat it as a current density.
In the introduction to the theory of the electron [2] he states: "The Gordon-Klein interpretation can answer such questions [probability of dynamical quantities taking values in certain limits] if they refer to the position of the electron
(by the use of [charge/probability density]), but not if they refer to its momentum, or angular momentum or any other dynamical variable."
I guess this isn't an exact answer but gives an idea as to why it fails fundamentally. An extension is:
Does anyone have a good example of the KG failing for these other variables?
[1]- F.Schwabl Advanced Quantum Mechanic 4th edition
[2]-Dirac, P.A "The quantum theory of the electron" [URL:https://www.rpi.edu/dept/phys/Courses/PHYS6520/DiracElectron.pdf]
