Approximation of Semiclassical Expectation Values by Symplectic Gaussian Wave Packet Dynamics

• URL: http://arxiv.org/abs/2012.05464v2
• Date: Wed, 27 Oct 2021 13:11:26 GMT
• Title: Approximation of Semiclassical Expectation Values by Symplectic Gaussian Wave Packet Dynamics
• Authors: Tomoki Ohsawa
• Abstract summary: This paper concerns an approximation of the expectation values of the position and momentum of the solution to the semiclassical Schr"odinger equation with a Gaussian as the initial condition.
• Score: 0.9137554315375922
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper concerns an approximation of the expectation values of the position and momentum of the solution to the semiclassical Schr\"odinger equation with a Gaussian as the initial condition. Of particular interest is the approximation obtained by our symplectic/Hamiltonian formulation of the Gaussian wave packet dynamics that introduces a correction term to the conventional formulation using the classical Hamiltonian system by Hagedorn and others. The main result is a proof that our formulation gives a higher-order approximation than the classical formulation does to the expectation value dynamics under certain conditions on the potential function. Specifically, as the semiclassical parameter $\varepsilon$ approaches $0$, our dynamics gives an $O(\varepsilon^{3/2})$ approximation of the expectation value dynamics whereas the classical one gives an $O(\varepsilon)$ approximation.

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