- #1

- 202

- 2

I don't know if this is the right place to post this topic, if there is a better place, please let me know and I'll move my post.

I have to write a resume of my project and I have to write it in English. My mother language is Italian and I'm not so good in writing in English, so can you please check if my translation is correct? Thank you very much.

Quantum mechanics represented a big revolution in the way we conceive the world and the reality all around us. Nevertheless the Schr ̈odinger equation can be exactly solved only in few situations; it is thus interesting to develop pro- grams in order to numerically determine the evolution of several systems. That was, indeed, the goal of this work, where a C++ code was written to integrate the Schr ̈odinger equation in two and three spatial dimensions and to show the real-time evolution of the wave-packet.

The method used is based on Lie-Trotter formula in order to write the time evolution operator as an infinite product of terms diagonal 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 intro- duced in this way is of the order of (∆t)2). 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 it’s multiplied by the factor diagonal in that basis and finally the wave function is written back in the coordinate space and multiplied by the third factor of the split evolution operator. The two basis 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 in- troduced 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 in the evolution operator and considering the wave function as a two-component spinor.

A fundamental part in 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 Schr ̈odinger 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.