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Corrected a couple of small things

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filippi-claudia 2021-01-30 23:29:19 +01:00 committed by GitHub
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@ -276,10 +276,10 @@ end function psi
applied to the wave function gives: applied to the wave function gives:
$$ $$
\Delta \Psi (\mathbf{r}) = \left(a^2 - \frac{2a}{\mathbf{|r|}} \right) \Psi(\mathbf{r}) \Delta \Psi (\mathbf{r}) = \left(a^2 - \frac{2a}{\mathbf{|r|}} \right) \Psi(\mathbf{r})\,.
$$ $$
So the local kinetic energy is Therefore, the local kinetic energy is
$$ $$
-\frac{1}{2} \frac{\Delta \Psi}{\Psi} (\mathbf{r}) = -\frac{1}{2}\left(a^2 - \frac{2a}{\mathbf{|r|}} \right) -\frac{1}{2} \frac{\Delta \Psi}{\Psi} (\mathbf{r}) = -\frac{1}{2}\left(a^2 - \frac{2a}{\mathbf{|r|}} \right)
$$ $$
@ -563,7 +563,7 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
If the space is discretized in small volume elements $\mathbf{r}_i$ If the space is discretized in small volume elements $\mathbf{r}_i$
of size $\delta \mathbf{r}$, the expression of $\langle E_L \rangle_{\Psi^2}$ of size $\delta \mathbf{r}$, the expression of $\langle E_L \rangle_{\Psi^2}$
becomes a weighted average of the local energy, where the weights becomes a weighted average of the local energy, where the weights
are the values of the probability density at $\mathbf{r}_i$ are the values of the wave function square at $\mathbf{r}_i$
multiplied by the volume element: multiplied by the volume element:
$$ $$
@ -580,7 +580,7 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
*** Exercise *** Exercise
#+begin_exercise #+begin_exercise
Compute a numerical estimate of the energy in a grid of Compute a numerical estimate of the energy using a grid of
$50\times50\times50$ points in the range $(-5,-5,-5) \le $50\times50\times50$ points in the range $(-5,-5,-5) \le
\mathbf{r} \le (5,5,5)$. \mathbf{r} \le (5,5,5)$.
#+end_exercise #+end_exercise
@ -783,7 +783,7 @@ gfortran hydrogen.f90 energy_hydrogen.f90 -o energy_hydrogen
*** Exercise *** Exercise
#+begin_exercise #+begin_exercise
Add the calculation of the variance to the previous code, and Add the calculation of the variance to the previous code, and
compute a numerical estimate of the variance of the local energy in compute a numerical estimate of the variance of the local energy using
a grid of $50\times50\times50$ points in the range $(-5,-5,-5) \le a grid of $50\times50\times50$ points in the range $(-5,-5,-5) \le
\mathbf{r} \le (5,5,5)$ for different values of $a$. \mathbf{r} \le (5,5,5)$ for different values of $a$.
#+end_exercise #+end_exercise