Variational principles for the Schrodinger equation

Schrodinger [365] defined the kinetic energy functional as the positive-definite form T = jr d3r Vi/C(r) Vt//(r). valid in a finite volume r enclosed by a surface a. When integrated by parts, defining t = — V2, T = fT d3r i// (r)/1//(r) + fa i/C(cr)Vy/(cr) dcr. With the usual boundary conditions for a large volume, 

Defining the functional V = (i/ v f). where u(r) is a local potential for which E = T + V is bounded below, the Schrodinger variational principle requires E to be stationary subject to normalization =. The variation SE induced 

Use of the Rayleigh quotient A[i//-] = (f l-L ir)/(f f) as a variational functional is an alternative to using a Lagrange multiplier to ensure normalization. Assuming that the eigenfunctions ifa) of H are complete and orthonormal in the relevant Hilbert space, (i/f i/0 = IO IVOI2 and l( IVr)l2 — 

By construction, this stationary principle holds for any eigenstate. 

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