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 — documentation:basics:expectation_values [2016/10/10 09:40] (current) 2016/10/06 20:35 Maurits W. Haverkort created 2016/10/06 20:35 Maurits W. Haverkort created Line 1: Line 1: + {{indexmenu_n>​4}} + ====== Expectation values ====== + ### + Expectation values are calculated as $\langle \psi_i | O | \psi_j \rangle$. For an $n$ electron wave-function this represents the $3n$ dimensional integral over the position coordinates of all electrons: + + \begin{align} + ​\langle \psi_i | O | \psi_j \rangle = &​\int_{r_1} \int_{r_2} ... \int_{r_n} \psi_i^*(\vec{r}_1,​ \vec{r}_1, ...\vec{r}_n) ​ \\ + \nonumber ​ & O(\vec{r}_1,​ \vec{r}_1, ...\vec{r}_n)\psi_j(\vec{r}_1,​ \vec{r}_1, ...\vec{r}_n) dr_1 dr_2 ... dr_n + \end{align} + + In Quanty the complex conjugates of the first wavefunction is automatically assumed and expectation values are implemented as multiplication of wavefunctions and operators. + ​ + -- expectation values of operators are + -- calculated by multiplying a wavefunction times + -- an operator times a wavefunction + val = psi1 * Opp * psi2 + ​ + ### + + + ===== Index ===== + - [[documentation:​basics:​basis|]] + - [[documentation:​basics:​operators|]] + - [[documentation:​basics:​wave_functions|]] + - Expectation values + - [[documentation:​basics:​eigen_states|]] + - [[documentation:​basics:​spectra|]] + - [[documentation:​basics:​resonant_spectra|]] + - [[documentation:​basics:​fluorescence_yield|]]