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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 ?

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  • $\begingroup$ Well, you should show us the particular journal article if you're interested in a particular answer. But in general, why do you expect that we can solve an arbitrary ODE? $\endgroup$ – zeldredge May 2 '17 at 0:32
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    $\begingroup$ There are very few potentials for which you can find analytic solutions. $\endgroup$ – OON May 2 '17 at 0:56
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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).

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  • $\begingroup$ Numerical methods are generally not covered, even though they are essential. Physics is a stool with three legs: theory, experiment, and computation. I suppose it is thought that there is not enough time in intro courses for numerical methods. Too bad. $\endgroup$ – garyp Jun 3 '17 at 2:16

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