Quantum Trajectory Methods
The hydrodynamic picture of quantum mechanics, first developed fully by Bohm following earlier ideas, has recaptured attention in recent years as an alternative to the conventional Schrodinger description. The main appeal of Bohm’s approach is its formulation in terms of “trajectories”, which allows a classical-like visualization of quantum mechanical events. In fact, the Bohmian wavefunction has a form very closely related to the time-dependent semiclassical approximation. The main difference is the presence of a “quantum potential”, which is given by the local curvature of the instantaneous density. The presence of this nonclassical potential field leads to an interdependence of the quantum trajectories.
The most serious practical difficulty in the use of quantum trajectories is the need for concurrent evaluation of the quantum force needed to update the density. In time-dependent semiclassical theory quantum interference effects arise from cross terms corresponding to distinct classical trajectories with fixed boundary conditions. Unlike in classical dynamics, it can be shown that Bohmian trajectories cannot cross in position space. As a consequence, Bohm’s wavefunction consists of a single term. The repulsive force necessary to prevent crossing events originates in the quantum potential. Accurate self-consistent determination of the rugged quantum potential and density poses a numerical challenge and often renders the solution unstable. Successful calculation of Bohmian trajectories in bound anharmonic systems has been possible only by combining the hydrodynamic equations with a direct solution of the Schrodinger equation.
The hydrodynamic and semiclassical formulations account for quantum mechanical effects in strikingly different ways. Bohm’s expression for an expectation value can be written in an initial value representation, where the quantum mechanical phase is entirely absent. By contrast, in the analogous semiclassical expression off-diagonal phase differences between trajectories in the forward and backward propagation steps are entirely responsible for quantum interference. In fact, elimination of such phase differences through forward-backward, linearization, or similar stationary phase approximations produces expressions that are incapable of accounting for quantum interference. It is thus intriguing that Bohm’s fully quantum mechanical method can assume a similar quasiclassical form.
How does the Bohmian method capture quantum mechanical interference in the absence of phase difference factors associated with multiple-bounce trajectories? We have addressed this question by examining the quantum potential and the evolution of Lagrangian fields in a model bound anharmonic oscillator. Our analysis showed that quantum interference manifests itself directly as a spatial variation of the density surrounding kinky trajectories that result from steep forces operating in regions where the corresponding classical solution exhibits focal points or caustics. The deviations of quantum trajectories from the underlying classical solutions are extremely severe in the vicinity of such points, and this behavior represents the leading source of instability in the Bohmian methodology. These features of the hydrodynamic approach, which constitute the hallmark of quantum interference and are ubiquitous in bound nonlinear systems, represent a major source of instability, making the integration of the Bohmian equations extremely demanding in such situations. Classical caustics are less common in barrier problems and entirely absent from the dynamics of quadratic Hamiltonians (except at times that are multiples of a half period) and thus the numerical difficulties encountered in such systems are not severe.
We have shown that the quantum force in Bohm’s formulation of quantum mechanics can be related to the stability properties of the given quantum trajectory. In turn, the evolution of the stability properties is governed by higher order derivatives of the quantum potential, leading to an infinite hierarchy of coupled differential equations whose solution specifies completely all aspects of the dynamics. Neglecting derivatives of the quantum potential beyond a certain order allows truncation of the hierarchy, leading to approximate Bohmian trajectories that provide an accurate description of tunneling phenomena. Use of the method in conjunction with the quantum initial value representation discussed above allows the use of Monte Carlo methods for sampling the trajectory initial conditions.
We have also pointed out that the hydrodynamic formulation of quantum mechanics lends itself as a powerful tool for solving the diffusion equation. We introduced a wavefunction repartitioning methodology that prevents imaginary-time trajectory crossing events and thus leads to stable evolution, overcoming the numerical obstacles that characterize Bohm’s formulation in real time. Use of our approximate technique that focuses on stability properties to solve Bohm’s equations in imaginary time allows determination of the energies of low-lying eigenstates from a single quantum trajectory.
Finally, we have derived a novel fully quantum mechanical formulation of time correlation functions based on quantum trajectories integrated along a forward-backward time contour. Unlike its semiclassical analogue, the new expression involves a smooth integrand amenable to Monte Carlo integration with a readily available position space density. Further, the quantum potential governing the motion of the quantum forward-backward trajectories is weak and well-behaved, facilitating numerical integration.
- Y. Zhao and N. Makri, “Bohmian vs. semiclassical description of interference phenomena”, J. Chem. Phys. 119, 60-67 (2003)
- N. Makri, “Forward-backward quantum dynamics for time correlation functions”, J. Phys. Chem. A 108, 806-812 (2004)
- J. Liu and N. Makri, “Monte Carlo Bohmian dynamics from trajectory stability properties”, J. Phys. Chem A 108, 5408-5416 (2004)
- J. Liu and N. Makri, “Bohm’s formulation in imaginary time: Estimation of energy eigevalues”, Mol. Phys. 103, 1083-1090 (2005)