Approximation of semiclassical expectation values by symplectic Gaussian wave packet dynamics

Ohsawa, Tomoki

doi:10.1007/s11005-021-01462-6

Cited by 4 publications

(6 citation statements)

References 30 publications

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“…This includes convergence in the L 2 -norm with order O( √ ε) as well as for expectation values of observables, which resemble certain measurable physical quantities of the wave function, with order O(ε 2 ). These estimates extend and improve the observable bound of [26, theorem 3.5] and the result of [30] from the case of vanishing magnetic potential. The different error bounds for semiclassical and variational approximation are summarized in table 1.…”

Section: Contributions Of the Papersupporting

confidence: 85%

“…Our work generalizes the results in [26] in the sense that we treat time-dependent, magnetic Hamiltonians. We also generalize the results of [25,30] from the position and momentum operator to sublinear observables in the sense of assumption 2.1. For the sake of readability, we postpone the proofs to sections 4-7.…”

Section: Resultsmentioning

confidence: 69%

“…Recently in [30], Oshawa has analysed the expectation values of position and momentum for a variational Gaussian wave packet and proved O(ε 3/2 ) accuracy. Our results here generalize and improve this error bound in two ways: First, we allow for general sublinear observables.…”

Section: Further Wave Packet Results For a =mentioning

confidence: 99%

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Variational Gaussian approximation for the magnetic Schrödinger equation ^*

Burkhard,

Dörich,

Hochbruck

et al. 2024

J. Phys. A: Math. Theor.

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In the present paper we consider the semiclassical magnetic Schrödinger equation, which describes the dynamics of particles under the influence of a magnetic field. The solution of the time-dependent Schrödinger equation is approximated by a single Gaussian wave packet via the time-dependent Dirac–Frenkel variational principle. For the approximation we derive ordinary differential equations of motion for the parameters of the variational solution. Moreover, we prove L 2-error bounds and observable error bounds for the approximating Gaussian wave packet.

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Section: Contributions Of the Papersupporting

confidence: 85%

Section: Resultsmentioning

confidence: 69%

See 1 more Smart Citation

Variational Gaussian approximation for the magnetic Schrödinger equation ^*

Burkhard,

Dörich,

Hochbruck

et al. 2024

J. Phys. A: Math. Theor.

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“…where ρ ′ = ∂ρ dx . After the substitution of the density distribution ansatz (equation ( 5)) in equation (6), we obtain the following expressions for the quantum potential and force [53]:…”

Section: Interacting Trajectory Propagationmentioning

confidence: 99%

“…In recent decades, the development of trajectory-based methods have attracted great interest, chiefly due to the appealing scaling properties of different computational implementations as the dimensionality of the system grows larger [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Interacting trajectory representation of quantum dynamics: influence of boundary conditions on the tunneling decay of resonant states

Cruz-Rodríguez¹,

Uranga-Piña²,

Martínez-Mesa³

et al. 2023

J. Phys. B: At. Mol. Opt. Phys.

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We perform quantum trajectory simulations of the decay dynamics of initially localised resonant states. Quantum dynamics is represented by a swarm of interacting trajectories which maps the originally quantum problem into the motion of an equivalent (higher-dimensional) classical system. We address two model problems, in which the decay of the initial resonance leads to either spatially conﬁned or asymptotically free wave-packet dynamics, speciﬁcally on a double well potential and on a potential plain. The traditional choice of ﬁxed boundary conditions in the interacting trajectory representation, set at inﬁnity, is found to have a moderate inﬂuence on the accuracy of the interacting trajectory representation of quantum trajectory dynamics, for the motion on a double well potential, i.e., the results of the trajectory-based scheme are in good correspondence with those obtained via quantum wave-packet propagation up to several fundamental vibrational periods. On the other hand, standard boundary conditions have negligible eﬀect on the interacting trajectory dynamics of a decaying shape resonance, whose predictions reproduce quantum mechanical results at long times.

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Semiclassical perturbations of single-degree–of–freedom Hamiltonian systems I: Separatrix splitting

Ohsawa,

Yagasaki

2024

Journal of Mathematical Physics

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We study semiclassical perturbations of single-degree-of-freedom Hamiltonian systems possessing hyperbolic saddles with homoclinic orbits, and provide a sufficient condition for the separatrices to split, using a Melnikov-type approach. The semiclassical systems give approximations of the expectation values of the positions and momenta to the semiclassical Schrödinger equations with Gaussian wave packets as the initial conditions. The occurrence of separatrix splitting explains a mechanism for the existence of trajectories to cross the separatrices on the classical phase plane in the expectation value dynamics. Such separatrix splitting does not occur in standard systems of Hagedorn and Heller for the semiclassical Gaussian wave packet dynamics as well as in the classical systems. We illustrate our theory for the potential of a simple pendulum and give numerical computations for the stable and unstable manifolds in the semiclassical system as well as solutions crossing the separatrices.

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Approximation of semiclassical expectation values by symplectic Gaussian wave packet dynamics

Cited by 4 publications

References 30 publications

Variational Gaussian approximation for the magnetic Schrödinger equation ^*

Variational Gaussian approximation for the magnetic Schrödinger equation ^*

Interacting trajectory representation of quantum dynamics: influence of boundary conditions on the tunneling decay of resonant states

Semiclassical perturbations of single-degree–of–freedom Hamiltonian systems I: Separatrix splitting

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Approximation of semiclassical expectation values by symplectic Gaussian wave packet dynamics

Cited by 4 publications

References 30 publications

Variational Gaussian approximation for the magnetic Schrödinger equation *

Variational Gaussian approximation for the magnetic Schrödinger equation *

Interacting trajectory representation of quantum dynamics: influence of boundary conditions on the tunneling decay of resonant states

Semiclassical perturbations of single-degree–of–freedom Hamiltonian systems I: Separatrix splitting

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Product

Resources

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Variational Gaussian approximation for the magnetic Schrödinger equation ^*

Variational Gaussian approximation for the magnetic Schrödinger equation ^*