Consider the elementary example of a particle in a box, with V(x) = co everywhere, except...

 

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Consider the elementary example of a particle in a box, with V(x) = co everywhere, except 0<x<L, where V = 0. (a) Since all the particle's energy is kinetic, use the approximate density functional in one dimension for the kinetic energy that you derived in problem 2.2, namely, 7lo [n] = 1.645dx n³ (x) to estimate the energy for single, double, and triple occupation of the box, where each energy level is singly occupied. Use the box eigenfunctions to determine the density. Compare the energies you obtain with the exact energies. (b) For a large occupation number >> 1, obtain an expression for the exact total kinetic energy of the particles in the box. Consider the elementary example of a particle in a box, with V(x) = co everywhere, except 0<x<L, where V = 0. (a) Since all the particle's energy is kinetic, use the approximate density functional in one dimension for the kinetic energy that you derived in problem 2.2, namely, 7lo [n] = 1.645dx n³ (x) to estimate the energy for single, double, and triple occupation of the box, where each energy level is singly occupied. Use the box eigenfunctions to determine the density. Compare the energies you obtain with the exact energies. (b) For a large occupation number >> 1, obtain an expression for the exact total kinetic energy of the particles in the box.

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a To estimate the kinetic energy using the approximate density function... View the full answer