Variation method ground state eigenfunctions

As a simple application of the variation method to determine the ground-state energy, we consider a particle in a one-dimensional box. The Schrodinger equation for this system and its exact solution are presented in Section 2.5. The ground-state eigenfunction is shown in Figure 2.2 and is observed to have no nodes and to vanish at x = 0 and x = a. As a trial function 0 we select 0 = x(a — x), 0 X a... 

As a normalized trial function 0 for the determination of the ground-state energy by the variation method, we select the unperturbed eigenfunction r2) of the perturbation treatment, except that we replace the atomic number Zby a parameter Z ... 

The coefficients Ck are determined using the variational principle. This gives rise to a set of secular equations and a secular determinant very similar to those encountered in the Cl method (17). The solutions to this secular problem provide approximate eigenvalues and eigenfunctions to both the ground and excited states of the system. 

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