Not much wrong with your English.
I'm assuming the hyphenated words are artifacts of cut and paste. My quantum mechanics and computer programming are a bit rusty, but here goes...
The theories of quantum mechanics resulted in a major revolution in the way we conceive the world and the reality all around us. Nevertheless the Schrödinger equation can be exactly solved only in few situations. The goal of this work was to develop programs (more than one?) in order to numerically determine the evolution of several (several kinds of? multiple?) systems. A C++ code was written to integrate the Schrödinger equation in two and three spatial dimensions and to show the real-time evolution of the wave-packet.
The method used was based on the Lie-Trotter product formula in order to write the time evolution operator as an infinite product of diagonal terms in coordinate and momentum space. This product is written as a finite number of terms by the discretization of the temporal evolution in small intervals ∆t; the error introduced in this way is of the order of (∆t)2) {you have an open parentheses here}. In order to evolve the wave function, first it is multiplied by the factor diagonal on the coordinate basis, then the wave function is written in the basis of the momentum space and is multiplied by the factor diagonal in that basis, and finally the wave function is written back into the coordinate space and multiplied by the third factor of the split evolution operator. The two bases are linked by the Fourier transform.
If a magnetic field is present, it can be shown that its effect can be accomplished with the introduction of an appropriate phase. Any substantial difficulty is {not sure what you want to say here - I guess either Substantial difficulty is introduced (sounds like a problem) or Any substantial difficulty can be introduced (sounds like a useful experimental feature if you want to consider spin interactions, but then "difficulty" sounds odd to me.)} introduced if the spin interaction is also considered. It can be shown that the spin coupling with the magnetic field can be achived with a matrix inserted into the evolution operator and considering the wave function as a two-component spinor.
Fundamental to the development of the code was the validation of the results. To do this, the numerical results have been compared with the analytic solutions for some cases where the exact solution of the Schrödinger equation is available. These include the free particle, the harmonic oscillator, the infinite-height potential barrier and the spin megnetic resonance.
In conclusion, the code can manage any initial state and any potential, showing the real-time evolution of the wave function, and visually displaying the strictly quantum behavior so distant from our daily experience.