PREFACTOR-FREE FORWARD-BACKWARD SEMICLASSICAL DYNAMICS

Within a differential expression of the Heisenberg operator, the forward and backward evolution can be joined together along a closed time contour. This manipulation leads to a dramatic cancellation of oscillations due to the two individual propagators in the Heisenberg operator and the resulting forward-backward propagator is more tractable to semiclassical approximations. This article gives a detailed description of the forward-backward semiclassical dynamics (FBSD) formalism. The semiclassical propagators, especially those of the initial value representations (IVRs), are briefly discussed. The derivation of the FBSD based on the Herman–Kluk propagator is reviewed. Different FBSD formulations with other semiclassical IVRs are worked out and numerical calculations show that they are also capable of describing quantum dynamics semiquantitatively and all display accuracy similar to the classical Wigner method.

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Gaussian approximation for the structure function in semiclassical forward-backward initial value representations of time correlation functions

The Journal of Chemical Physics ◽

10.1063/1.3271241 ◽

2009 ◽

Vol 131 (22) ◽

pp. 224107 ◽

Cited By ~ 9

Author(s):

Guohua Tao ◽

William H. Miller

Keyword(s):

Structure Function ◽

Correlation Functions ◽

Gaussian Approximation ◽

Time Correlation ◽

Time Correlation Functions ◽

Initial Value ◽

Initial Value Representations

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Hidden Nambu mechanics II: Quantum/semiclassical dynamics

Progress of Theoretical and Experimental Physics ◽

10.1093/ptep/ptz144 ◽

2019 ◽

Vol 2019 (12) ◽

Author(s):

Atsushi Horikoshi

Keyword(s):

Time Evolution ◽

Quantum Dynamics ◽

Degrees Of Freedom ◽

Hamiltonian Dynamics ◽

Extended Phase ◽

Nambu Mechanics ◽

Mechanical Structure ◽

Wave Packet Dynamics ◽

Semiclassical Dynamics ◽

Metastable Potential

Abstract Nambu mechanics is a generalized Hamiltonian dynamics characterized by an extended phase space and multiple Hamiltonians. In a previous paper [Prog. Theor. Exp. Phys. 2013, 073A01 (2013)] we revealed that the Nambu mechanical structure is hidden in Hamiltonian dynamics, that is, the classical time evolution of variables including redundant degrees of freedom can be formulated as Nambu mechanics. In the present paper we show that the Nambu mechanical structure is also hidden in some quantum or semiclassical dynamics, that is, in some cases the quantum or semiclassical time evolution of expectation values of quantum mechanical operators, including composite operators, can be formulated as Nambu mechanics. We present a procedure to find hidden Nambu structures in quantum/semiclassical systems of one degree of freedom, and give two examples: the exact quantum dynamics of a harmonic oscillator, and semiclassical wave packet dynamics. Our formalism can be extended to many-degrees-of-freedom systems; however, there is a serious difficulty in this case due to interactions between degrees of freedom. To illustrate our formalism we present two sets of numerical results on semiclassical dynamics: from a one-dimensional metastable potential model and a simplified Henon–Heiles model of two interacting oscillators.

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Boundary Conditions and Initial Value Lines for Unsteady Homentropic Flow Calculations

Aeronautical Quarterly ◽

10.1017/s0001925900005333 ◽

1970 ◽

Vol 21 (2) ◽

pp. 145-162 ◽

Cited By ~ 2

Author(s):

W. A. Woods ◽

H. Daneshyar

Keyword(s):

Boundary Condition ◽

Analytical Solution ◽

Boundary Conditions ◽

Good Accuracy ◽

Numerical Calculations ◽

Time Dependent ◽

Straight Pipe ◽

Initial Value ◽

Use Of Time ◽

The Difference

SummaryA detailed discussion on the difference between an initial value line and a line characterised by a boundary condition has been presented. Two types of boundaries are described and illustrated. To examine each boundary, several different calculations have been performed for a straight pipe. The results of the numerical calculations are compared with an analytical solution. It is shown that known pressure and velocity at the pipe ends give the most accurate results. Comparisons are also made between several practical types of calculations which give similar findings. The use of time-dependent boundaries can lead to errors as large as 40 per cent in derived results. It is shown that good accuracy can be restored by converting the boundaries into initial value lines. It is concluded that in general no more than one time-dependent boundary should be used in any calculation. Finally it is demonstrated that errors are not revealed by means of pressure diagrams alone.

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