Ciao! Non so se questo è il forum adatto.. cmq, devo tradurre la sintesi della relazione per la laurea triennale in inglese. Io ho provato a scriverla, potete per favore darle un'occhiata? Grazie
Quantum mechanics has 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 for few situations; it is so interesting to develop programs 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 real- time the evolution of the wave-packet.
The method it was used is based on Lie-Trotter formula in order to write the time evolution operator as an infinitive product of terms diagonal in coordinate and momentum space. These product is written as a finite number of terms by the discretization of the temporal evolution in small intervals ∆t. The error introduced this way is of order of $(∆t)^2$. 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 difference is introduced if the also the spin interaction is considered.
A fundamental part in the development of the code has been 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.
In conclusion, the code can manage any initial state and any potential, showing real-time the evolution of the wave function and visually displaying the strictly quantum behavior so distant from our daily experience.