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In this paper, we introduce the Schrodinger equation with a general kineticenergy operator. The conservation law is proved and the probability continuityequation is deducted in a general sense. Examples with a Hermitian kineticenergy operator include the standard Schrodinger equation, the relativisticSchrodinger equation, the fractional Schrodinger equation, the Dirac equation,and the deformed Schrodinger equation. We reveal that the Klein-Gordon equationhas a hidden non-Hermitian kinetic energy operator. The probability continuityequation with sources indicates that there exists a different way ofprobability transportation, which is probability teleportation. An averageformula is deducted from the relativistic Schrodinger equation, the Diracequation, and the K-G equation.

The quantum mechanics based on a general kinetic energy