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Prove that there are a limited number of bound solutions for the semi-infi nite well.

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An electron is in a finite square well that is 0.6 eV deep, and 2.1 nm wide. Calculate the value of the dimensionless potential (to 3 s.f).
Consider a three-dimensional infinite-well modeled as a cube of dimensions L x L x L. The length L is such that the ground state energy of one electron confined to this box is 0.50eV. (a) Write down the four lowest energy states and evaluate their corresponding degeneracy. (b) If 15 (total) electrons are placed in the box, find the Fermi energy of the system. (c) What is the total energy of the 15-electron system? (d) How much energy would be required to lift an electron from Fermi energy of part (b) to the first excited state? Need full detailed answers and explanations to understand the concept.