Description

Derive the Φ-normal ordered Hamiltonian:

Be sure to explicitly show your work.
Question 1b (3 pts). Explicitly define fpq and qualitatively explain what the term PfpqN[a†paq] is.
pq
Question 1c (3 pts). Qualitatively explain how the Φ-normal ordered Hamiltonian HN differs from the standard Hamiltonian H. Be sure to address the significance of H − HN in the context of particle-hole isomorphism. Also be sure to address why HN = Hc, where Hc is the correlation Hamiltonian if we choose the reference determinant to be the Hartree-Fock result.
Xfpq hΦbj|N[a†paq]|Φai i pq hΦ|H|Φabcijki hΦ|H|Φbji
hΦai |H|Φai i hΦai |H|Φabcijki
Question 3 (3 pts). Qualitatively explain the purpose of Wick’s Theorem in 1-2 sentences. Then, explain why the generalized Wick’s theorem is useful in 1-2 sentences.
Question 4 (2 pts). Qualitatively explain particle-hole isomorphism and why it is useful in 2-3 sentences.

Question 5 (3 pts). Qualitatively explain the difference between apa†p and apa†p. Be sure to address:

• the type of contraction involved in each term
• the result in each case
• the qualitative meaning of the result
Question 6 (2 pts). Explain the difference between N[apa†ra†qas] and n[apa†ra†qas]. Be sure to address:
• which formalism each term belongs to
• the result in each case

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