|MadSci Network: Physics|
The reason the author has made this approach is so they are able to use integral calculus. The parameter that L<<D allows for an integration to be performed over the whole dumbbell of length D. This is a powerful tool in physics, which comes from mathematics and allows a summation over a very small distance delta l to be performed over an infinitesimally small distance dl. Various tricks can be used by making this approach in order to form the general solution which is much easier than computing the full summation. Basically, splitting up the problem into smaller parts makes it easier and, although I do not have a copy of the text, I assume this is the author's reasoning.
With respect to the solution of a free electron's wavefunction, Schrodinger's equation solves the Hamiltonian given a potential in which the electron lies. If the electron is completely unbound then there is no potential to put into the equation and it cannot be solved as an infinite number of solutions exist for the eigenvalues which satisfy the boundary conditions of the problem. If we look at the solution of the finite width, infinite potential well solution (which is analytically solvable) to a 1D problem, we can sub in a zero depth, infinite width well to the answer and this will approximate the free electron solution (but this will have no realistic solution).
I hope this isn't too obscure an answer,
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