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--- public:projects:pathintegrals [2012/06/14 12:31] – wikiadmin
+++ public:projects:pathintegrals [2012/06/16 15:25] (current) – oschuett
@@ Line 1: / Line 1: @@
-====== Ab-initio path integral molecular dynamics and momentum densities ======
 {{ :public:projects:mol_0000.gif?nolink |Quantum-nuclei-simulation isomorphic to P coupled classical systems}}
@@ Line 11: / Line 10: @@
 <latex> $\begin{align*}
-Z = \text{Tr} \left[ \left(e^{-\frac{\beta}{P}\hat H}\right)^P\right] = \int d^{3N}R \ \ \langle\mathbf{R}|e^{-\frac{\beta}{P} \hat H} \dots e^{-\frac{\beta}{P} \hat H} |\mathbf{R} \rangle \ \ \text{with}\  Z \in \mathbb{N} .
+Z = \text{Tr} \left[ \left(e^{-\frac{\beta}{P}\hat H}\right)^P\right] = \int d^{3N}R \ \ \langle\mathbf{R}|e^{-\frac{\beta}{P} \hat H} \dots e^{-\frac{\beta}{P} \hat H} |\mathbf{R} \rangle \ \ \text{with}\  P \in \mathbb{N} .
 \end{align*} $ </latex>
-The new aspect is that with a modification of the conventional path integral scheme, it is possible to express not only quantities in real space (**R**-space), but also momentum densities. A complete derivation of this modified path integral formalism would exceed the space available here, but it can be shown that the momentum density of a nucleus <latex>$i$</latex> can be expressed as:
+The new aspect is that with a modification of the conventional path integral scheme, it is possible to express not only quantities in real space (**R**-space), but also momentum densities. A complete derivation of this modified path integral formalism would exceed the space available here, but it can be shown that the momentum density of a nucleus can be expressed as:
 <latex> $\begin{align*}
 n(\mathbf{k}) = & \int d^3d_2 \dots d^3k_N\ | \Psi(\mathbf{k_1}=\mathbf{k}, \dots \mathbf{k_N})|^2\\
-= & \frac{1}{(2\pi)^3} \int d^3 R_1\, d^3R'_1 \ e^{-i\mathbf{k}(\mathbf{R}_1 - \mathbf{R}'_1)/\hbar} n(\mathbf{R}_1, \mathbf{R}'_1)
+= & \frac{1}{(2\pi)^3} \int d^3 R_1\, d^3R'_1 \ e^{-i\mathbf{k}(\mathbf{R}_1 - \mathbf{R}'_1)/\hbar}\, n(\mathbf{R}_1, \mathbf{R}'_1)
 \end{align*}$ </latex>