Path integral formulation: Difference between revisions

Revision as of 17:01, 3 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} }

Phase transitions, quantum dynamics, centroids etc.

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 {{Stub-general}}
+In the path integral formulation the canonical [[partition function]] (in one dimension) is written as (Ref. review #4 Eq. 1)
+:<math>Q(\beta, V)= \int {\mathrm d} x_1 \int_{x_1}^{x_1} Dx(\tau)e^{-S[x(\tau)]}</math>
+where <math>S[x(\tau)]</math> is the Euclidian action, given by (Ref. review #4 Eq. 2)
+:<math>S[x(\tau)] = \int_0^{\beta \hbar} H(x(\tau))</math>
+where <math>x(\tau)</math> is the path in time <math>\tau</math> and <math>H</math> is the [[Hamiltonian]].
+This leads to (Ref. review #4 Eq. 3)
+:<math>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)}</math>
+where the Euclidean time is discretised in units of
+:<math>\varepsilon = \frac{\beta \hbar}{P}, P \in {\mathbb Z}</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.