Finite Elements and Schrodinger

Recall from chap. 2 that often in the solution of differential equations, useful strategies are constructed on the basis of the weak form of the governing equation of interest in which a differential equation is replaced by an integral statement of the same governing principle. In the previous chapter, we described the finite element method, with special reference to the theory of linear elasticity, and we showed how a weak statement of the equilibrium equations could be constructed. In the present section, we wish to exploit such thinking within the context of the Schrodinger equation, with special reference to the problem of the particle in a box considered above and its two-dimensional generalization to the problem of a quantum corral. 

Our preliminary exercise will state the problem in one dimension, with the generalization to follow shortly. The one-dimensional Schrodinger equation may be written 

In the present case, the weak form is obtained by multiplying eqn (3.32) by some test function and integrating the first term by parts. For the purposes of concreteness in the present setting, we imagine a problem for which the domain of interest runs from the origin to point x = a, and for which the wave functions vanish at the boundaries. We note that the Schrodinger equation could have been stated alternatively as the Euler-Lagrange equation for the functional 

The finite element strategy is to replace this functional with a function of the N unknown values i/ (not to be confused with wave functions related to the hydrogen 

Specifically, as with our earlier treatment of the finite element method, we imagine shape functions Ni (x) centered at each node, with the subscript i labeling the node of interest. It is then asserted that the wave function is given as 

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