Editing Path integral formulation
Jump to navigation
Jump to search
The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.
Latest revision | Your text | ||
Line 63: | Line 63: | ||
:<math>T^{\mathrm {rotational}} = \sum_{i=1}^{d^{\mathrm {rotational}}} \frac{\hat{L}_i^2}{2\Theta_{ii}}</math> | :<math>T^{\mathrm {rotational}} = \sum_{i=1}^{d^{\mathrm {rotational}}} \frac{\hat{L}_i^2}{2\Theta_{ii}}</math> | ||
where <math>\hat{L}_i</math> are the components of the angular momentum operator, and <math>\Theta_{ii}</math> are the moments of inertia. | where <math>\hat{L}_i</math> are the components of the angular momentum operator, and <math>\Theta_{ii}</math> are the moments of inertia. For a rigid three dimensional asymmetric top the kernel is given by (<ref>[http://dx.doi.org/10.1103/PhysRevLett.77.2638 M. H. Müser and B. J. Berne "Path-Integral Monte Carlo Scheme for Rigid Tops: Application to the Quantum Rotator Phase Transition in Solid Methane", Physical Review Letters '''77''' pp. 2638-2641 (1996)]</ref> Eq. 5): | ||
= | |||
:<math>\rho(\omega,\omega'; \beta/P) = \sum_{JM\tilde{K}} \left( \frac{2J+1}{8\pi^2}\right) d_{MM}^J (\tilde{\theta'} ) | |||
\cos \left[ M(\tilde{\phi}' + \tilde{\chi}') \right] \left| A_{\tilde{K}M}^{JM} \right|^2 \exp \left( - \frac{\beta}{P} E_{\tilde{K}}^{JM} \right)</math> | |||
where <math>\omega</math> are the [[Euler angles]], <math>d_{MM}^J </math> is the [[Wigner D-matrix]] and <math>E_{\tilde{K}}^{JM}</math> are the eigenenergies. | |||
==Computer simulation techniques== | ==Computer simulation techniques== | ||
The following are a number of commonly used [[computer simulation techniques]] that make use of the path integral formulation applied to phases of condensed matter | The following are a number of commonly used [[computer simulation techniques]] that make use of the path integral formulation applied to phases of condensed matter |