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As we know that the Schrödinger equation presents basis of Quantum Mechanics and analogy with Newton second law in Classical Mechanics, I thought that relativistic interpretation of Schrödinger equation can make general relativity and quantum mechanics closer

$$ \frac {1}{c^2} \frac{\partial^2}{\partial t^2} \psi - \nabla^2 \psi + \frac {m^2 c^2}{\hbar^2} \psi = 0. $$

My question is can we derive Schrödinger equation from this one and what are solutions of this equation representing actually in physical relativistic sense? Which part in equation is associated to relativistic mechanics?

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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. – Qmechanic Apr 13 '13 at 18:03

The equation you give contains no hint about the potential, so you couldn't derive a real Schrödinger equation from it. You could only get Schrödinger equation for a free particle.

If you are interested in the solution, I shall forward you to wikipedia.

The part associated to relativistic mechanics is the last term with mass and $c$.

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