analyticexercises_7_16
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analyticexercises_7_16 [2014/07/16 09:04] bogner |
analyticexercises_7_16 [2014/07/16 09:13] (current) bogner |
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| ===== Paper and pen exercises (Wednesday 7/16) ===== | ===== Paper and pen exercises (Wednesday 7/16) ===== | ||
| - | - 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. | + | * Continue working on the problem set from {{:exercises_secondquantization.pdf|yesterday}}. |
| - | - Prove Koopman's theorem, which states that we can interpret the HF eigenvalues $\epsilon_i$ as removal energies. That is show that | + | |
| + | * 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} | \begin{equation} | ||
| \epsilon_i = \langle \Phi^{N}|H|\Phi^{N}\rangle - \langle\Phi^{N}|a^{\dagger}_iHa_i|\Phi^{N}\rangle | \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} | \end{equation} | ||
analyticexercises_7_16.1405515885.txt.gz · Last modified: 2014/07/16 09:04 by bogner
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