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Corrected a couple of small things
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QMC.org
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QMC.org
@ -276,10 +276,10 @@ end function psi
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applied to the wave function gives:
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$$
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\Delta \Psi (\mathbf{r}) = \left(a^2 - \frac{2a}{\mathbf{|r|}} \right) \Psi(\mathbf{r})
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\Delta \Psi (\mathbf{r}) = \left(a^2 - \frac{2a}{\mathbf{|r|}} \right) \Psi(\mathbf{r})\,.
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$$
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So the local kinetic energy is
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Therefore, the local kinetic energy is
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$$
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-\frac{1}{2} \frac{\Delta \Psi}{\Psi} (\mathbf{r}) = -\frac{1}{2}\left(a^2 - \frac{2a}{\mathbf{|r|}} \right)
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$$
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@ -563,7 +563,7 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
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If the space is discretized in small volume elements $\mathbf{r}_i$
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of size $\delta \mathbf{r}$, the expression of $\langle E_L \rangle_{\Psi^2}$
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becomes a weighted average of the local energy, where the weights
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are the values of the probability density at $\mathbf{r}_i$
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are the values of the wave function square at $\mathbf{r}_i$
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multiplied by the volume element:
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$$
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@ -580,7 +580,7 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
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*** Exercise
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#+begin_exercise
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Compute a numerical estimate of the energy in a grid of
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Compute a numerical estimate of the energy using a grid of
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$50\times50\times50$ points in the range $(-5,-5,-5) \le
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\mathbf{r} \le (5,5,5)$.
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#+end_exercise
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@ -783,7 +783,7 @@ gfortran hydrogen.f90 energy_hydrogen.f90 -o energy_hydrogen
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*** Exercise
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#+begin_exercise
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Add the calculation of the variance to the previous code, and
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compute a numerical estimate of the variance of the local energy in
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compute a numerical estimate of the variance of the local energy using
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a grid of $50\times50\times50$ points in the range $(-5,-5,-5) \le
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\mathbf{r} \le (5,5,5)$ for different values of $a$.
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#+end_exercise
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