The Hamilton approach to classical dynamics is extended to incorporate quantum effects. Quantization of the Hamilton equations of motion results in a hierarchy of equations that are equivalent to quantum mechanics in the Heisenberg form. Closure of the hierarchy gives approximations to the exact quantum dynamics. A specific dynamics algorithm is presented and tested against model applications that exhibit tunneling and zero point motion effects. The quantized Hamilton approach is found accurate, consistent, flexible, and computationally very efficient.
The Hamilton approach to classical dynamics is extended to incorporate quantum effects. Quantization of the Hamilton equations of motion results in a hierarchy of equations that are equivalent to quantum mechanics in the Heisenberg form. Closure of the hierarchy gives approximations to the exact quantum dynamics. A specific dynamics algorithm is presented and tested against model applications that exhibit tunneling and zero point motion effects. The quantized Hamilton approach is found accurate, consistent, flexible, and computationally very efficient.
The quantization of Hamilton dynamics (QHD) [J. Chem. Phys. 113, 6557 (2000)] that efficiently generalizes classical mechanics to include quantum tunneling and zero-point energy effects is extended to a general position dependent potential. A Taylor series expansion of the potential is considered both around a fixed point and around the moving instantaneous value of the position variable. The equations-of-motion obtained for the moving frame are significantly simpler than for the fixed frame, while still satisfying the classical limit. The number of the QHD variables and the order of the Taylor expansion of the potential constitute two independent approximation parameters. Conservation of the total energy and the Heisenberg commutator relationship is established for the second-order QHD that includes linear and quadratic variables. The formal results are illustrated by examples, including the harmonic oscillator, tunneling in a doublewell potential, and energy exchange between coupled Morse oscillators representing the stretching vibrations of the SPC-F water molecule commonly used in molecular dynamics simulations. QHD provides an improvement over classical mechanics in all cases. The QHD data agree with the exact data in the large ℏ quantum limit and in the ℏ=0 classical limit, with deviations observed for the intermediate ℏ values, depending on the system and initial conditions. QHD provides a quantitative short time approximation. The second-order QHD becomes inappropriate when quantum wave packets split. Higher-order QHD approximations are required in such cases. The second-order QHD is particularly suitable for the description of zero-point energy effects and properly treats moderate tunneling events.
The quantized Hamilton dynamics (QHD) method, which was introduced and developed in J. Chem. Phys. 113, 6557 (2000) to the second order, is extended to the third and fourth orders. The QHD formalism represents an extension of classical mechanics and allows for the derivation of a hierarchy of equations of motion which converge with the quantum-mechanical limit. Here, the second, third, and fourth order QHD approximations are applied to two model problems: the decay of a particle in a metastable cubic potential and the intermode energy exchange observed in the Henon–Heiles system. The QHD results exhibit good convergence with the quantum data with increasing order yet preserve the computational efficiency of classical calculations. The second order QHD approximation already does an excellent job in maintaining the zero-point energy in the Henon–Heiles system and describing moderate tunneling events in the metastable potential. Extensions to higher orders substantially improve the QHD results for deep tunneling and are capable of describing the finer details of energy exchange.
A conceptually simple approximation to quantum mechanics, quantized Hamilton dynamics (QHD) includes zero-point energy, tunneling, dephasing, and other important quantum effects in a classical-like description. The hierarchy of coupled differential equations describing the time evolution of observables in QHD can be mapped in the second order onto a classical system with double the dimensionality of the original system. While QHD excels at dynamics with a single initial condition, the correct method for generating thermal initial conditions in QHD remains an open question. Using the coherent state representation of thermodynamics of the harmonic oscillator (HO) [Schnack, Europhys. Lett. 45, 647 (1999)], we develop canonical averaging for the second order QHD [Prezhdo, J. Chem. Phys. 117, 2995 (2002)]. The methodology is exact for the free particle and HO, and shows good agreement with quantum results for a variety of quartic potentials.
Abstract The quantization of the dynamics of point systems is closely connected with the Hamilton-Jacobi theory of the calculus of variations for simple integrals, the latter being a suitable mathematical formalism for describing the classical laws of point dynamics. This connexion is evident from the fact that, in order to quantize a dynamical system, one has first to know what the pairs of canonically conjugate variables are, and also from the fact that the quantum laws, if expressed in terms of commutation brackets, have exactly the same form as the classical laws, if expressed in terms of Poisson brackets. In dealing with the quantization of the dynamics of continuous media, e.g. the electromagnetic and other fields, which has been developed along different lines, following Heisenberg and Pauli (1929), it seems tempting to try to base the method of quantization on an extended Hamilton-Jacobi theory of the calculus of variations for multiple integrals, the latter being the appropriate formalism for describing most relevant systems of continuum physics. In a previous paper (Weiss 1936), referred to as I, such an attem pt has been made, by extending the notion of pairs of conjugate variables. That attem pt led to quantum relations on an arbitrary space-like section in space-time, provided that one postulated that, if the space-like section becomes especially a space section, the quantum relations should go over into those of Heisenberg and Pauli.
Classical and quantum physics represent two distinct theories; however, quantum physics is regarded as the more fundamental of the two. It is posited that classical mechanics should arise from quantum mechanics under certain limiting conditions. Nevertheless, this remains a challenging objective. In this work, we explore the potential for unifying the dynamics of classical and quantum physics. This discussion does not suggest that classical behavior emerges from quantum mechanics; rather, it demonstrates the exact equivalence between the dynamics of quantum observables and their classical counterparts. It is shown that the Heisenberg equations of motion can be cast in a form that is structurally equivalent to Newton's equations of motion, with ℏ dropping out from the equations. In a generalized analysis, the Heisenberg equations are cast in a form that is identical to the classical Hamilton's equations of motion. This implies that both quantum and classical dynamics are governed by the same form of equations, with the Heisenberg operators substituting the classical observables.
Quantized Hamilton dynamics (QHD) is a simple and elegant extension of classical Hamilton dynamics that accurately includes zero-point energy, tunneling, dephasing, and other quantum effects. Formulated as a hierarchy of approximations to exact quantum dynamics in the Heisenberg formulation, QHD has been used to study evolution of observables subject to a single initial condition. In present, we develop a practical solution for generating canonical ensembles in the second-order QHD for position and momentum operators, which can be mapped onto classical phase space in doubled dimensionality and which in certain limits is equivalent to thawed Gaussian. We define a thermal distribution in the space of the QHD-2 variables and show that the standard beta=1/kT relationship becomes beta'=2/kT in the high temperature limit due to an overcounting of states in the extended phase space, and a more complicated function at low temperatures. The QHD thermal distribution is used to compute total energy, kinetic energy, heat capacity, and other canonical averages for a series of quartic potentials, showing good agreement with the quantum results.