Abstract Classical nonhamiltonian dynamics, driven by external ‘curl forces’ (which are not the gradient of a potential) is extended to the quantum domain. This is a generalisation of the two-stage Madelung procedure for the quantum Hamiltonian case: (i) considering not individual trajectories but families of them, characterised by their velocity and density fields (both functions of position and in general time); and (ii) adding the gradient of the quantum potential to the external curl force. Curl forces require the velocity field to have nonzero vorticity, so there is no underlying singlevalued wavefunction. Two explicit examples are presented. A possible experiment would be the motion of small particles with complex polarisability, influenced by forces from optical fields.
Abstract Classical nonhamiltonian dynamics, driven by external ‘curl forces’ (which are not the gradient of a potential) is extended to the quantum domain. This is a generalisation of the two-stage Madelung procedure for the quantum Hamiltonian case: (i) considering not individual trajectories but families of them, characterised by their velocity and density fields (both functions of position and in general time); and (ii) adding the gradient of the quantum potential to the external curl force. Curl forces require the velocity field to have nonzero vorticity, so there is no underlying singlevalued wavefunction. Two explicit examples are presented. A possible experiment would be the motion of small particles with complex polarisability, influenced by forces from optical fields.
Abstract A class of Newtonian forces, determining the acceleration F (x, y) of particles in the plane, is F =(Re F(z), Im F(z)), where z is the complex variable x + iy. Curl F is non-zero, so these forces are nonconservative. These complex curl forces correspond to completely integrable Hamiltonians that are anisotropic in the momenta, separable in z and z * but not in x and y if the curl is nonzero. The Hamiltonians can be quantised, leading to unfamiliar wavefunctions, even for the (non-curl) isotropic harmonic oscillator. The formalism provides an alternative interpretation of the analytic continuation of one-dimensional real Hamiltonian particle dynamics, where trajectories are known to exhibit intricate structure (though not chaos), and is a Hermitian alternative to non-Hermitian quantisation.
A classical limit of quantum dynamics can be defined by compensation of the quantum potential in the time-dependent Schrödinger equation. The quantum potential is a non-local quantity, defined in the trajectory-based form of the Schrödinger equation, due to Madelung, de Broglie, and Bohm, which formally generates the quantum-mechanical features in dynamics. Selective inclusion of the quantum potential for the degrees of freedom deemed "quantum," defines a hybrid quantum/classical dynamics, appropriate for molecular systems comprised of light and heavy nuclei. The wavefunction is associated with all of the nuclei, and the Ehrenfest, or mean-field, averaging of the force acting on the classical degrees of freedom, typical of the mixed quantum/classical methods, is avoided. The hybrid approach is used to examine evolution of light/heavy systems in the harmonic and double-well potentials, using conventional grid-based and approximate quantum-trajectory time propagation. The approximate quantum force is defined on spatial domains, which removes unphysical coupling of the wavefunction fragments corresponding to distinct classical channels or configurations. The quantum potential, associated with the quantum particle, generates forces acting on both quantum and classical particles to describe the backreaction.
The approach of defining quantum corrections on nuclear dynamics of molecular systems incorporated approximately into selected degrees of freedom, is described. The approach is based on the Madelung-de-Broglie-Bohm formulation of time-dependent quantum mechanics which represents a wavefunction in terms of an ensemble of trajectories. The trajectories follow classical laws of motion except that the quantum potential, dependent on the wavefunction amplitude and its derivatives, is added to the external, classical potential. In this framework the quantum potential, determined approximately for practical reasons, is included only into the "quantum" degrees of freedom describing light particles such as protons, while neglecting with the quantum force for the heavy, nearly classical nuclei. The entire system comprised of light and heavy particles is described by a single wavefunction of full dimensionality. The coordinate space of heavy particles is divided into spatial domains or subspaces. The quantum force acting on the light particles is determined for each domain of similar configurations of the heavy nuclei. This approach effectively introduces parametric dependence of the reduced dimensionality quantum force, on classical degrees of freedom. This strategy improves accuracy of the quantum force and does not restrict interaction between the domains. The concept is illustrated for two-dimensional scattering systems, where the quantum force is required to reproduce vibrational energy of the quantum degree of freedom.
The quantum trajectory dynamics in imaginary time with the momentum-dependent quantum potential [ J. Chem. Phys. 2010, 132, 014112], which can be used for calculations of low-lying energy eigenstates or for Boltzmann evolution, is extended to many dimensions. Evolution of a nodeless initial wave function represented in exponential form is described in terms of a trajectory ensemble. In the Lagrangian frame of reference the trajectories evolve according to the classical equations of motion under the influence of the inverted classical potential and the quantum potential defined by the gradient of the trajectory momenta. The initial wave function is sampled by the trajectories randomly, which makes the approach scalable to high dimensions. Another practical feature is approximate computation of the quantum potential and force from the global least squares fit of the trajectory momenta in a small basis. The importance sampling and the Eulerian frame-of-reference formulation, compensating for divergent dynamics in bound potentials, are introduced to reduce the number of trajectories in multidimensional applications. The zero-point energies are computed for model and chemical (H 2 O and SO 2 ) systems.
Using the classical trajectory we quantize the motion of a pointlike particle of mass m and energy 1/2${\mathit{mx}\mathit{\ifmmode \dot{}\else \.{}\fi{}}}_{0}^{2}$ under the influence of the force Fn=-\ensuremath{\alpha}\ensuremath{\Vert}x\ifmmode \dot{}\else \.{}\fi{}${\mathrm{\ensuremath{\Vert}}}^{\mathit{n}}$, \ensuremath{\alpha}\ensuremath{\gtrsim}0, n=1,2,.... Such forces are frictional only for x\ifmmode \dot{}\else \.{}\fi{}\ensuremath{\gtrsim}0 and antifrictional for x\ifmmode \dot{}\else \.{}\fi{}0 and can be simulated through the gradient of a one-parameter class of energy-dependent potentials Vn(x;x${\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}_{0}$), which increase monotonically in x. This permits one to construct a class of Lagrange and Hamilton functions that do not contain any integrating functions. We then proceed to quantization and study the quantum scattering problem assuming that Fn acts on a finite interval of the x axis only. The stationary problem can be solved analytically for all n, and quantitative examples are shown for n=2. The time-dependent scattering problem is discussed with emphasis on Ehrenfest's theorems and on linear damping n=1. When the quantum-mechanical probability for reflection is small the method provides an approximation to the motion of a wave packet through a dissipative medium, with velocity-power forces Fdn=-\ensuremath{\alpha}\ensuremath{\Vert}x\ifmmode \dot{}\else \.{}\fi{}${\mathrm{\ensuremath{\Vert}}}^{\mathit{n}\mathrm{\ensuremath{-}}1}$x\ifmmode \dot{}\else \.{}\fi{}. \textcopyright{} 1996 The American Physical Society.