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Fund Quantum Mechanics Lect & HW Solutions 94

Fund Quantum Mechanics Lect & HW Solutions 94 - 76...

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76 CHAPTER 5. MULTIPLE-PARTICLE SYSTEMS electrons around the same nucleus, the remaining C = 4 Slater determinants can be written out explicitly to give the two-particle states ψ S 1 = ψ l ψ r ↑↑ − ψ r ψ l ↑↑ 2 ψ S 2 = ψ l ψ r ↑↓ − ψ r ψ l ↓↑ 2 ψ S 3 = ψ l ψ r ↓↑ − ψ r ψ l ↑↓ 2 ψ S 4 = ψ l ψ r ↓↓ − ψ r ψ l ↓↓ 2 Note that the Hamiltonian does not involve spin, to the approximation used in most of this book, so that, following the techniques of section 5.5, an inner product like H 23 = ( ψ S 2 | S 3 ) can be written out like H 23 = 1 2 ( ψ l ψ r ↑↓ − ψ r ψ l ↓↑| H ( ψ l ψ r ↓↑ − ψ r ψ l ↑↓ ) ) = 1 2 ( ψ l ψ r ↑↓ − ψ r ψ l ↓↑| ( l ψ r ) ↓↑ − ( r ψ l ) ↑↓) and then multiplied out into inner products of matching spin components to give H 23 = 1 2 ( ψ l ψ r | r ψ l ) − 1 2 ( ψ r ψ l | l ψ r ) = L. The other 15 matrix coefficients can be found similarly, and most will be zero.
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