We get 8a 3 1 8a 2 let us now differentiate both sides

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Unformatted text preview: H( )un(r, ) n E( ) Let us now differentiate both sides with respect to E( ) d 3r u*(r, ) n (8A-3) . We get H( )un(r, ) d 3ru*(r, )H( ) n un(r, ) d 3ru*(r, ) n H( ) un(r, ) Consider now the first two terms on the right-hand side. Using the eigenvalue equation and its complex conjugate (with hermiticity of H), we see that they add up to E( ) d 3r u*(r, ) n un(r, ) E( ) d 3ru*(r, ) n un(r, ) E( ) d 3ru*(r, )un(r, ) n 0 We are therefore left with E( ) d 3ru*(r, ) n H( ) un(r, ) H( ) (8A-4) The utility of this result is somewhat limited, because it requires knowing the exact eigenvalues and, for the calculation on the right-hand side, the exact eigenfun...
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This document was uploaded on 10/31/2011.

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