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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.

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    $\begingroup$ Although I am not aware of the historical circumstances, I would guess that the spin-orbital interaction is not reproduced by the Klein-Gordon equation since it describes a particle with zero spin. Another point is that if electrons were Klein-Gordon particles they wouldn't obey the exclusion principle which would lead to the complete meltdown of chemistry. There are also several conceptual issues arising from the single-particle interpretation of the KG equation: the conserved probability is not positive-definite and causality is violated (although this violation is exponentially small). $\endgroup$ – Prof. Legolasov Jan 17 '15 at 10:36
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    $\begingroup$ Some Googling suggests the solution was published in V. A. Fock, Z. Phys. 40, 632 (1926). However I can't find a copy of this paper so I can't confirm that it does describe the application of the KG equation to the hydrogen atom. $\endgroup$ – John Rennie Jan 17 '15 at 11:49
  • $\begingroup$ I've learned KG is a minimum must for each component of wave function of any spins. (Dirac electron has 4 components, each satisfies KG) I think the main flaw of KG is negative probability as Hindsight pointed out. But hydrogen spectrum maybe another problem of KG. $\endgroup$ – noel_lapin Jan 17 '15 at 13:10
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    $\begingroup$ I found two documents but not yet well understand. (1)physics.drexel.edu/~bob/PHYS516_11/Frobenius.pdf (2)arxiv.org/abs/1006.3971 $\endgroup$ – noel_lapin Jan 18 '15 at 6:51
  • $\begingroup$ Not exactly what OP is asking, but for a connection between Schr. eq. and Klein-Gordon eq, see e.g. A. Zee, QFT in a Nutshell, Chap. III.5, and this Phys.SE post plus links therein. $\endgroup$ – Qmechanic Feb 1 '15 at 21:38

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