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Quantum Trajectories Quantum Trajectories Edited by Keith H. Hughes School of Chemistry University of Wales Bangor Bangor LL57 2UW United Kingdom and G´erard Parlant Institut Charles Gerhardt Universit´eMontpellier 2, CNRS Equipe CTMM, Case Courrier 1501 Place Eug`ene Bataillon 34095 Montpellier France Suggested Dewey Classification: 541.2 ISBN 978-0-9545289-9-7 Published by Collaborative Computational Project on Molecular Quantum Dynamics (CCP6) Daresbury Laboratory Daresbury Warrington WA4 4AD United Kingdom c CCP6 2011 ii Preface v Mixed Quantum/Classical Dynamics: Bohmian and DVR Stochastic Trajectories C. Meier, J. A. Beswick, T. Yefsah 1 Trajectory-Based Derivation of Classical and Quantum Mechanics Bill Poirier 6 The Bipolar Reaction Path Hamiltonian (BRPH) Approach for Multi- Dimensional Reactive Scattering Systems Jeremy B. Maddox and Bill Poirier 9 An Iterative Finite Difference Method for Solving the Quantum Hydro- dynamic Equations of Motion Brian K. Kendrick 13 Kinematic Quantum Trajectories Timothy M. Coffey 20 Complex Trajectories and Dynamical Origin of Quantum Probability Moncy V. John 25 Energy Rays for Electromagnetic Pulses Scattering from Metal- Dielectric Structures Robert E. Wyatt 28 Quantum Dynamics through Quantum Potentials S. Duley, S. Giri, S. Sengupta and P. K. Chattaraj 35 Conceptual Issues, Practicalities and Applications of Bohmian and other Quantum Trajectories in Nanoelectronics John R. Barker 43 Principles of Time Dependent Quantum Monte Carlo Ivan P. Christov 49 Types of Trajectory Guided Grids of Coherent States for Quantum Prop- agation Dmitrii V. Shalashilin and Miklos Ronto 54 iii Accurate Deep Tunneling Description by the Classical Schr¨odinger Equa- tion Xavier Gim´enez and Josep Maria Bofill 62 The Bohmian Model, Semiclassical Systems and the Emergence of Clas- sical Trajectories Alex Matzkin 66 Quantum Trajectories for Ultrashort Laser Pulse Excitation Dynamics G´erard Parlant 69 Quantum Dynamics and Super-Symmetric Quantum Mechanics Eric R. Bittner and Donald J. Kouri 72 Bohmian Trajectories of Semiclassical Wave Packets Sarah R¨omer 79 Quantum Trajectories in Phase Space Craig C. Martens 83 The Semiclassical Limit of Time Correlation Functions by Path Integrals G. Ciccotti 86 Path Integral Calculation of (Symmetrized) Time Correlations Functions S. Bonella 91 On-the-fly Nonadiabatic Bohmian DYnamics (NABDY) Ivano Tavernelli, Basile F. E. Curchod and Ursula Rothlisberger 96 Quantum Many-Particle Computations with Bohmian Trajectories: Ap- plication to Electron Transport in Nanoelectronic Devices A. Alarc´on, G.Albareda, F.L.Traversa and X.Oriols 103 An Account on Quantum Interference from a Hydrodynamical Perspec- tive A. S. Sanz 111 Quantum, Classical, and Mixed Quantum-Classical Hydrodynamics I. Burghardt and K. H. Hughes 116 iv K. H. Hughes and G. Parlant (eds.) Quantum Trajectories c 2011, CCP6, Daresbury Preface This booklet was produced as a result of the CCP6 Workshop on “Quantum Trajectories” held at the Bangor University, UK, between July 12-14, 2010. The workshop was sponsored by the UK Collaborative Computation Project 6 (CCP6) on molecular quantum dynamics. Details of CCP6 and its activities can be found at http://www.ccp6.ac.uk. The main focus of the workshop was the hydrodynamic formulation of quan- tum mechanics. Over the last 10-13 years the hydrodynamic formulation of quantum mechanics has grown into a quantum trajectory methodology with widespread applications in physics and chemistry. Furthermore, theoretical in- vestigations into the hydrodynamic form of quantum mechanics have spawned, or inspired, a number of other trajectory based approaches to quantum mechan- ics. The workshop brought together researchers involved in the computational and numerical development of quantum trajectories and researchers involved in use of quantum trajectories as an insightful and intuitive approach for studying and interpreting a wide range of quantum phenomena in a diverse range of fields. Each speaker was asked to provide a brief article which could be collected into a workshop booklet that reviews their work and the topics covered in their talk. This booklet should be of interest to both the specialist and non-specialist in this field. The booklet covers computational approaches to the topic of quantum trajectories and should serve as a guide to many of the recent developments in this field. The editors would like to thank all those who participated and contributed to the workshop. K. H. Hughes G. Parlant August 2011 v K. H. Hughes and G. Parlant (eds.) Quantum Trajectories c 2011, CCP6, Daresbury Mixed Quantum/Classical Dynamics: Bohmian and DVR Stochastic Trajectories C. Meier, J. A. Beswick and T. Yefsah∗ Laboratoire Collisions, Agr´egats, R´eactivit´e, IRSAMC Universit´ePaul Sabatier, Toulouse, France I. INTRODUCTION In many systems comprising a large number of particles, even though a de- tailed quantum treatment of all degrees of freedom is not necessary, there may exist subsets that have to be treated quantum mechanically under the influ- ence of the rest of the system treated classically. In these cases, mixed quan- tum/classical approaches have to be used to describe its dynamics [1]. The most popular of these are the mean-field approximation [2], the surface hopping trajectories [4, 5] or the methods based on quantum/classical Liouville space representations [6–12]. In the mean-field treatment the force for the classical motion is calculated by averaging over the quantum wavefunction. In the surface hopping scheme the classical trajectories move according to a force derived from a single quantum state with the possibility of transitions to other states. Based on these ideas, Bastida et al [13] have proposed a scheme (MQCS for mixed quantum classical steps) in which the quantum wave function is expanded in a discrete variable rep- resentation (DVR) rather than in the usual finite (spectral) basis representation (FBR). The interpretation of the dynamics of the quantum degree of freedom is provided in terms of stochastic hops from one grid point to another, governed by the time evolution of the wave packet. Another way to mix quantum mechanics with classical mechanics, proposed in refs. [14–20], is based on Bohmian quan- tum trajectories for the quantum/classical connection. In this approach, which was called MQCB (Mixed Quantum/Classical Bohmian) trajectories, the wave packet is used to define de Broglie-Bohm quantum trajectories [21–24] which in turn are used to calculate the force acting on the classical variables. The main ∗ present address: Laboratoire Kastler Brossel, D´epartement de Physique de l’Ecole Normale Sup´erieure, Paris, France 1 difference between MQCB and MQCS lies in the way the force on the classi- cal variables is calculated. In the MQCS method, the force is evaluated using different discrete points of the quantum degree of freedom, while in the MQCB method, it is calculated at the continuous points of the quantum trajectory. Hence, the underlying question is to determine what is the connection between the sequence of grid points obtained by stochastic hopping methods and the de- terministic, continuous quantum trajectory. In his paper Quantum mechanics in terms of discret beables [25], Jeroen C. Vink has shown that indeed a connection exists for a three-point approximation of the kinetic energy operator. In this contribution, we generalize Vink’s formulation to an arbitrary N-point expression. The paper is organized as follows. In Sec. (II) we give a brief review of MQCS method; in Sec. (III) we address the problem of representing a quantum wave packet by stochastic hops between discrete points in space, and under which conditions the sequence of hopping trajectories converge towards a deterministic quantum trajectory. Finally Sec. (IV) gives a general discussion of the relationship between all these treatments. II. STOCHASTIC DVR AND MQCS Considering a Hamiltonian with x being the quantum degree of frredom and X(t) the classical trajectory. Choosing a one-dimensional (diabatic) DVR basis |xni with eigenvalue xn of the position operator, for the quantum degree of X freedom, the wavefunction can be expressed by ψ(x, (t),t) = n cn(t) |xni, where the wave packet Ψ(x,t) propagates with time along the x coordinate, and P is a DVR basis function with eigenvalue xn of the position operator, such that cn(t)= hxn|ψ(t)i = ψ(xn, X(t),t). The values of the time-dependent coefficients can be obtained by solving the time-dependent Schr¨odinger equation using any conventional method (split operator, etc...). Initially the wave packet is a coherent superposition of DVR states. The time evolution of the coefficients of the wave packet in these DVR states is given by dcn i i X = −~ cm(t)hxn|H|xmi = −~ cm(t)hxn|T |xmi + V (xn, (t)) (1) dt m " m # X X where we have used the fact that the non diagonal matrix elements of the Hamil- tonian reduce to the matrix elements of the kinetic energy operator T in the DVR basis and the potential energy matrix is diagonal. The classical degrees of freedom are calculated using the force evaluated at one single DVR point, 1 ∂ (V (x,X)) P˙ = − (2) M ∂X x=xn(t),X=X(t) 2 the dynamics in the quantum subspace is accounted for by the possiblity of quantum jumps between these points. Hence, xn is a sequence of DVR points, forming a stochastic quantum trajectory, defined as follows: Eq.(1) determines the transition probabilities between DVR states. Let us introduce the diagonal 2 2 density matrix elements Pn = |cn| = |ψ(xn, X(t),t)| representing the proba- bility for the system to be in state |xni at time t. The time evolution of Pn(t) is deduced from Eq.(1) can be written as [26] dPn = (Tnm Pm − Tmn Pn) (3) dt m X with ~ ∗ (2) cm cn Tnm = D ℑ if Tnm > 0 (4) M nm |c |2 m (2) 2 2 where we have denoted by Dnm = hxn|∂ /∂x |xmi the matrix elements of the second derivative.
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