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Starting DMC
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@ -1809,6 +1809,41 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
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:header-args:f90: :tangle dmc.f90
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:END:
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Consider the time-dependent Schrödinger equation:
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\[
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i\frac{\partial \Psi(\mathbf{r},t)}{\partial t} = \hat{H} \Psi(\mathbf{r},t)
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\]
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We can expand $\Psi(\mathbf{r},0)$, in the basis of the eigenstates
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of the time-independent Hamiltonian:
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\[
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\Psi(\mathbf{r},0) = \sum_k a_k\, \Phi_k(\mathbf{r}).
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\]
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The solution of the Schrödinger equation at time $t$ is
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\[
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\Psi(\mathbf{r},t) = \sum_k a_k \exp \left( -i\, E_k\, t \right) \Phi_k(\mathbf{r}).
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\]
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Now, let's replace the time variable $t$ by an imaginary time variable
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$\tau=i\,t$, we obtain
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\[
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-\frac{\partial \psi(\mathbf{r}, t)}{\partial \tau} = \hat{H} \psi(\mathbf{r}, t)
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\]
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where $\psi(\mathbf{r},\tau) = \Psi(\mathbf{r},-i\tau) = \Psi(\mathbf{r},t)$
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and
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\[
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\psi(\mathbf{r},\tau) = \sum_k a_k \exp( -E_k\, \tau) \phi_k(\mathbf{r}).
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\]
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For large positive values of $\tau$, $\psi$ is dominated by the
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$k=0$ term, namely the ground state.
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** TODO Hydrogen atom
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*** Exercise
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