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Unformatted text preview: 64 CHAPTER 5. MULTIPLEPARTICLE SYSTEMS of the symmetric and antisymmetric states are quite different, though they look qualitatively the same. 5.2.5 Variational approximation of the ground state 5.2.6 Comparison with the exact ground state 5.3 TwoState Systems 5.3.1 Solution 2statea Question: The effectiveness of mixing states was already shown by the hydrogen molecule and molecular ion examples. But the generalized story above restricts the “basis” states to be orthogonal, and the states used in the hydrogen examples were not. Show that if ψ 1 and ψ 2 are not orthogonal states, but are normalized and produce a real and positive value for ( ψ 1  ψ 2 ) , like in the hydrogen examples, then orthogonal states can be found in the form ¯ ψ 1 = α ( ψ 1 − εψ 2 ) ¯ ψ 2 = α ( ψ 2 − εψ 1 ) . For normalized ψ 1 and ψ 2 the CauchySchwartz inequality says that ( ψ 1  ψ 2 ) will be less than one. If the states do not overlap much, it will be much less than one and ε will be small....
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This note was uploaded on 11/13/2011 for the course PHY 4458 taught by Professor Garvin during the Fall '11 term at University of Florida.
 Fall '11
 GARVIN

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