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Fund Quantum Mechanics Lect & HW Solutions 82

Fund Quantum Mechanics Lect & HW Solutions 82 - in...

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64 CHAPTER 5. MULTIPLE-PARTICLE 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 Two-State Systems 5.3.1 Solution 2state-a 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
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Unformatted text preview: in the form ¯ ψ 1 = α ( ψ 1 − εψ 2 ) ¯ ψ 2 = α ( ψ 2 − εψ 1 ) . ±or normalized ψ 1 and ψ 2 the Cauchy-Schwartz inequality says that a ψ 1 | ψ 2 A will be less than one. If the states do not overlap much, it will be much less than one and ε will be small. (If ψ 1 and ψ 2 do not meet the stated requirements, you can always rede²ne them by factors ae i c and be − i c , with a , b , and c real, to get states that do.) Answer: The inner product of ¯ ψ 1 and ¯ ψ 2 must be zero for them to be orthogonal: α 2 a ψ 1 − εψ 2 | ψ 2 − εψ 1 A = 0...
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