Derivation

Ring Polymer Molecular Dynamics Derivation

physics derivation

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Schrodinger’s equations states that \begin{align} \frac{d}{dt}|\Psi_t \rangle = \frac{-i \cdot \hat H}{\hbar} \cdot |\Psi_t \rangle \end{align} A formal solution to the equation is \begin{align} |\Psi_t \rangle = e^{\frac{-i \cdot \hat Ht}{\hbar}} \cdot |\Psi_0 \rangle \end{align}

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Liouville's Theorem derivation

physics derivation

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In classical mechanics the liouville’s theorem states that the intgeral element \begin{align} \partial p_0 \partial q_0 = \partial p_t \partial q_t \end{align} where \(p_0, q_0\) and \(p_t, q_t\)are the generalized coordinates at time 0 and t, respectively.

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