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md2011-06-note_Part_10

# md2011-06-note_Part_10 - HK Theorem I For any system of...

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HK Theorem I For any system of interacting particles in V ext , the potential is deﬁned uniquely, except for a constant, by the ground state particle density n 0 ( r ) . Corollary : Since the H is thus fully determined, it follows that many-body wavefunctions for all states are determined. Thus all properties of the system are fully determined by n 0 ( r ) . HK Theorem II A universal functional for the energy E [ n ] in terms of the density n ( r ) can be deﬁned, valid for any external potential V ext ( r ) . The exact ground state energy of the system is the global minimum value of this functional, and the density which minimizes the functional is the exact ground state density n 0 ( r ) . Corollary : The functional E [ n ] alone is suﬃcient to determine the exact ground state energy and density. Notes

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I V ext ( r ) Ψ i ({ r }) Ψ 0 ({ r }) n 0 ( r ) HK I Usually, the Schrödinger equation is solved by following the small arrows above, where the potential
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md2011-06-note_Part_10 - HK Theorem I For any system of...

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