  We will derive a rigorous real time propagator for the Non-relativistic
Quantum Mechanic $L^2$ transition probability amplitude and for the
Non-relativistic wave function. The propagator will be explicitly given in
terms of the time evolution operator. The derivation will be for all
self-adjoint nonvector potential Hamiltonians. For systems with potential that
carries at most a finite number of singularity and discontinuities, we will
show that our propagator can be written in the form of a rigorous real time,
time sliced Feynman path integral via improper Riemann integrals. We will also
derive the Feynman path integral in Nonstandard Analysis Formulation. Finally,
we will compute the propagator for the harmonic oscillator using the
Nonstandard Analysis Feynman path integral formuluation; we will compute the
propagator without using any knowledge of classical properties of the harmonic
oscillator.
