I have been studying quantum mechanics. I use many rectangular potential barrier to approximate real potential barrier to know transmission coefficient. I read some journal (link : http://aip.scitation.org/doi/abs/10.1063/1.338082 ; http://aip.scitation.org/doi/abs/10.1063/1.330428), them tell to solved Schrodinger equation use numerical method. Why Schrodinger equation solved by numerical methods? And why the paper use approximation rectangular function for real potential barrier ? What special from rectangular function ?
Solving the Shrodinger Equation (either time dependent or time-independent) is equivalent to solving a linear partial-differential equation. There are few potentials for which nice closed form solutions exist. As a consequence we often have to use numerical methods to find the solutions (and in the case of time independent bound states, to find the energy eigenvalues).
As an aside, much of what is taught in intro level QM courses consists of looking at the few cases where nice closed form solutions exist (and later on, looking at a systematic method of dealing with the cases with no nice closed form solutions, this method is called perturbation theory).