Differences

This shows you the differences between two versions of the page.

--- analyticexercises_7_16 [2014/07/16 08:15]
bogner
+++ analyticexercises_7_16 [2014/07/16 09:13] (current)
bogner
@@ Line 1: / Line 1: @@
 ===== Paper and pen exercises (Wednesday 7/16) =====
-  - For a local, spin-independent two-body interaction, derive the coordinate space representation of the HF single particle Schr\"odinger equation and expression for $E_{HF}$ starting from the general expressions in class.
+  * Continue working on the problem set from {{:exercises_secondquantization.pdf|yesterday}}.
+  * For a local, spin-independent two-body interaction, derive the coordinate space representations of i) the HF single particle Schroedinger equation, and ii) the expression for $E_{HF}$ starting from the general expressions in class. Express answers in terms of $\rho({\bf r})$ and $\rho({\bf r}\sigma,{\bf r}'\sigma')$ as in the lecture.
+  * Prove Koopman's theorem, which states that if we approximate the $N-1$ particle eigenstates by $a_i|\Phi^N\rangle$, we can interpret the HF eigenvalues $\epsilon_i$ as removal energies. That is show that
+\begin{equation}
+\epsilon_i = \langle \Phi^{N}|H|\Phi^{N}\rangle - \langle\Phi^{N}|a^{\dagger}_iHa_i|\Phi^{N}\rangle
+\end{equation}
+  * Prove Brillouin's theorem, which states that in the HF basis, the many-body Hamiltonian matrix doesn't connect the $0p0h$ and $1p1h$ sectors:
+\begin{equation}
+\langle \Phi^{a}_{i}|H|\Phi\rangle = 0
+\end{equation}

TalentDFT