Path integral formulation: Difference between revisions

Revision as of 14:20, 6 June 2008

In the path integral formulation the canonical partition function (in one dimension) is written as (Ref. review #4 Eq. 1)

Q(\beta ,V)=\int {\mathrm {d} }x_{1}\int _{x_{1}}^{x_{1}}Dx(\tau )e^{-S[x(\tau )]}

where $S[x(\tau )]$ is the Euclidian action, given by (Ref. review #4 Eq. 2)

S[x(\tau )]=\int _{0}^{\beta \hbar }H(x(\tau ))

where $x(\tau )$ is the path in time $\tau$ and $H$ is the Hamiltonian. This leads to (Ref. review #4 Eq. 3)

Q_{P}=\left({\frac {mP}{2\pi \beta \hbar ^{2}}}\right)^{P/2}\int ...\int {\mathrm {d} }x_{1}...{\mathrm {d} }x_{P}e^{-\beta \Phi _{P}(x_{1}...x_{P};\beta )}

where the Euclidean time is discretised in units of

\varepsilon ={\frac {\beta \hbar }{P}},P\in {\mathbb {Z} }

x_{t}=x(t\beta \hbar /P)

x_{P+1}=x_{1}

and

\Phi _{P}(x_{1}...x_{P};\beta )={\frac {mP}{2\beta ^{2}\hbar ^{2}}}\sum _{t=1}^{P}(x_{t}-x_{t+1})^{2}+{\frac {1}{P}}\sum _{t=1}^{P}V(x_{t})

Phase transitions, quantum dynamics, centroids etc.

@@ Line 9: / Line 9: @@
 where the Euclidean time is discretised in units of
 :<math>\varepsilon = \frac{\beta \hbar}{P}, P \in {\mathbb Z}</math>
+:<math>x_t = x(t \beta \hbar/P)</math>
+:<math>x_{P+1}=x_1</math>
+and
+:<math>\Phi_P (x_1...x_P;\beta)= \frac{mP}{2\beta^2 \hbar^2} \sum_{t=1}^P (x_t - x_{t+1})^2 + \frac{1}{P}  \sum_{t=1}^P  V(x_t)</math>
 ==External links==
 *[http://www.smac.lps.ens.fr/index.php/Programs_Chapter_3:_Density_matrices_and_path_integrals Density matrices and path integrals] computer code on SMAC-wiki.