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documentation:basics:expectation_values [2016/10/10 09:40] (current)
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 +{{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.
 +<code Quanty Example.Quanty>​
 +-- expectation values of operators are 
 +-- calculated by multiplying a wavefunction times 
 +-- an operator times a wavefunction
 +val = psi1 * Opp * psi2
 +</​code>​
 +###
 +
 +
 +===== 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|]]
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