  The Schrodinger equation, Klein-Gordon equation (KGE), and Dirac equation are
believed to be the fundamental equations of quantum mechanics. Schrodinger's
equation has a defect in that there are no negative kinetic energy (NKE)
solutions. Dirac's equation has positive kinetic energy (PKE) and NKE branches.
Both branches should have low-momentum, or nonrelativistic, approximations: One
is the Schrodinger equation, and the other is the NKE Schrodinger equation. The
KGE has two problems: It is an equation of the second time derivative so that
the calculated density is not definitely positive, and it is not a Hamiltonian
form. To overcome these problems, the equation should be revised as PKE- and
NKE-decoupled KGEs. The fundamental equations of quantum mechanics after the
modification have at least two merits. They are unitary in that all contain the
first time derivative and are symmetric with respect to PKE and NKE. This
reflects the symmetry of the PKE and NKE matters, as well as, in the author's
opinion, the matter and dark matter of our universe. The problems of
one-dimensional step potentials are resolved by utilizing the modified
fundamental equations for a nonrelativistic particle.
