Appendix 1: Atomic Units

Transcription

1 Appendix 1: Atomic Units Quantity Atomic Unit SI Equivalent mass mass of electron kg charge charge of proton C angular momentum h J s length (bohr) Bohr orbit radius in H atom m energy (hartree) 1/2 ionization energy of H atom J time period of ground state Bohr orbit in H s velocity velocity of electron in lowest Bohr orbit m/s 1 hartree = 315,775 K = 219,475 cm 1 = ev. 389

2 Appendix 2: Example QTM Program PROGRAM QTM Implicit none! ! This code integrates the POTENTIAL FORM of the quantum! hydrodynamic equations of motion (see equations 4.5 through 4.7! in Chapter 4). Note that the equation of motion for the action! function is integrated, but the quantum force is not used.!! An Eckart potential is used as an example in this code.!! !! For time integration, this code uses the first-order Euler! time-integrator.!! For spatial derivatives, there are calls to a generic! subroutine deriv.! This can be one of many function approximation routines! (such as moving least squares)! Generally, these s require the following specification! and arrays:!! deriv(np,x,f,d1x,d2x)! where...! np (input) the total number of particles (grid points)! x (input) an array of length np containing the grid! point locations! f (input) an array of length np containing the! function values at the grid points! d1x (output) an array of length np containing the first! spatial derivative of function f! d2x (output) an array of length np containing the second! spatial derivative of function f!

3 Appendix 2. Example QTM Program 391 integer(kind=4), parameter :: np = 63, ntime = ! ntime = total time-steps. integer(kind=4) i,j,k real(kind=8) delv(np),c(np),d1x(np),d2x(np),quantum(np),pot(np), x(np),rho(np) real(kind=8) vx(np),phase(np) real(kind=8) dt,x0,beta,energy,conv,am,vb,xb,wx,xmin,xmax,h,pi, anorm,total density real(kind=8) kinetic energy,lagrange! Initial Wave Packet Parameters ! Time-step in au dt = 0.5d0! Center of Gaussian function x0 = 0.d0! Width parameter for Gaussian wave packet beta = 9.d0! Initial translational energy in cm 1 energy = 8000.d0! Conversion factor from cm 1 to a.u. conv = d0! Translational energy in a.u. energy = energy/conv! Mass in a.u. am = 2000.d0! Eckart potential parameters ! Barrier height in cm 1 vb=8000d.0! Barrier height in a.u. vb=vb/conv! Center of barrier in a.u. xb=6.d0! Width parameter in a.u. wx=5.d0! Build a Uniform Initial Spatial Grid ! Minimum grid point xmin = -0.6d0! Maximum grid point xmax = 0.6d0! Particle spacings h = (xmax - xmin)/dble(np-1)! Particle positions do i = 0,np-1 x(i+1) = xmin + i h

4 392 Appendix 2. Example QTM Program! Initialize Wave Packet and Hydrodynamic Fields ! Normalization for an initial Gaussian wave function pi = 4.d0 atan(1.d0) anorm = (2.d0 beta/pi) (1.d0/4.d0)! Build initial wave packet amplitude and density do i =1,np R(i) = anorm exp(-beta ((x(i) - x0) 2)) c(i) =log(r(i)) rho(i) = R(i) 2! Initial particle velocities (all particles are initially the same velocity and 0 divergence)! These velocities are obtained from the equation Etrans = 1/2mv 2. do i = 1,np vx(i) = sqrt(2.d0 energy/am) delv(i) = 0.d0! Initial phase (action function) for each particle do i = 1,np phase(i)= sqrt(2.d0 am energy) x(i)! ! Time Propagation Loop do k = 1,ntime time = dble(k) dt! conversion from au to fs tfs = time d0! Call subroutine for spatial derivatives of the C-amplitude call deriv(np,x,c,d1x,d2x)! Calculate quantum and classical potentials do i = 1,np! Quantum Potential. quantum(i) = -1.d0/(2.d0 am) (d2x(i) + d1x(i) 2)! Classical Eckart Potential. pot(i) = vb sech(wx (x-x0)) 2! Calculate phase (S) using potentials (Quantum Lagrangian: T-(V+Q)) do i = 1,np kinetic energy = 0.5d0 am vx(i) 2 lagrange = kinetic-(pot(i)+quantum(i)) phase(i) = phase(i) + lagrange dt

5 Appendix 2. Example QTM Program 393! Update particle positions do i = 1,np x(i) = x(i) + vx(i) dt! Call subroutine for spatial derivatives of phase call deriv(np,x,phase,d1x,d2x)! Update velocities and probability density do i = 1,np vx(i) = (1.d0/am) d1x(i) delv(i) = (1.d0/am) d2x(i) rho(i) = rho(i) exp(-delv(i) dt)! End Time Propagation end

6 Index Action function; see also Quantum hydrodynamic equations of motion classical, 50 51, 98 complex-valued, 43 44, 46, for decay of metastable state, 201 for decoherence model, 198 quantum, 8, 43, 46, 150, 198, 339, 379 reduced, see Hamilton s characteristic function Adaptive grid; see Grid, adaptive Adaptive s, ; see also Arbitrary Lagrangian Eulerian, Moving path transforms, Equidistribution principle, Artificial viscosity, Hybrid s Adiabatic electronic representation, ; see also Diabatic electronic representation ALE; see Arbitrary Lagrangian Eulerian ALE grid speeds, 170 Analytic approach to quantum trajectories, 1 2, , , Approximate quantum potential, 228 Approximations to quantum force, 21 22, statistical approach; see also Statistical analysis, application to methyl iodide, determination of parameters, 21, expansion of density in Gaussians, 21, 219 fit density using least squares, 21 22, applications, expansion of density in Gaussians, 225 least squares functional, 225 fit log-derivative of density, 22, applications, , global fit, local fit, AQP; see Approximate quantum potential Arbitrary Lagrangian Eulerian, 3, 19, , , 371 Artificial viscosity, 19 20, 167, , 188, 370; see also Adaptive s Atomic units, 389 Attractors in flux maps, 196 Back-reaction problem in mixed quantum-classical dynamics, 300 Back-reaction through Bohmian particle, ; see also Mixed quantum classical dynamics, Configuration space, Phase space Bell, John, 40 Bohm, David, 40 42, 98 Boltzmann distribution, 64 Born Oppenheimer approximation, 204, 308 Brownian motion, 258; see also Osmotic velocity 395

7 396 Index C-amplitude, 43, 150, 201, C-density, 271 Caldeira Leggett equation, 25 26, 254, 266, 268, Carrier phase, 373, 380; see also Counterpropagating wave Car Parrinello technique, CFM; see Covering function Chaos; see also Lyapunov exponent, Power spectrum of trajectory classical, diagnostics, 111 examples, Lagrangian chaos, 111 relationship to wave function nodes, quantum, 63, , 119 Characteristic function; see Hamilton s characteristic function Circulation integral; see Vortices, circulation integral, Stokes s theorem CL equation; see Caldeira Leggett equation Collocation matrix, 131 Complex amplitude, 35, 371, ; see also Problems with propagation of quantum trajectories form for wave function, 35, 377 equations of motion, 377 Compressibility and trajectory dynamics, 100 Compression of trajectories, 14, 105, 162, 164, 292; see also Inflation of trajectories Complete positivity for density operator, 268; see also Lindblad form, Density operator Complex-valued velocity, 327 Computer code for QTM, 93, 95, 188, Configuration space ; see also Mixed quantum classical dynamics, Phase space applications coupled harmonic oscillators, diffractive surface scattering, light particle heavy particle scattering, comparison with phase space, equations of motion, 29, introduction to, 29 Contextuality, 55, 270 Continuity equation classical, 26, 44 quantum, 32, 47, 52, 78, 94, 203, 236, 377 phase space, stationary states, 355 Control theory, 191 Convective term in equations of motion, 79, 81 Coordinates, physical and computational, 192 Copenhagen interpretation, 40, 41, 111 Correlation function force autocorrelation, 257 spatial, 68 quantum cross-correlation, 102, 226, density correlation, 233 decay of metastable state, Counterpropagating wave, 8, 34 35, 355, application to stationary states, carrier wave, 373, 378, 380 example of decomposition, form for wave function, 373 Covering function, 35 36, 161, 371, applications Eckart barrier scattering, two-dimensional scattering, covering functions, 376, 382, 386 total wave function, 35, 375 Cross-validation, 158 C-space transform, 149; see also C-amplitude DAF; see Distributed approximating functionals Damping term, 223 de Broglie Bohm interpretation, 1 2, 41, 42, 59, 111 de Broglie wavelength, 138 Decay of metastable state, 190, , ,

8 Index 397 Decoherence, 18, 190, 251 definition, description of model, quantum trajectory results, stress tensor components, Density functional theory, 36, 134 Density matrix, 6, 26 27, 64, , 255, definition, 64 trajectory equations of motion, polar form, 289 examples of trajectory evolution decay from Eckart barrier, displaced coherent state, Density operator, 64, , 289; see also Lindblad form for density operator Derivative evaluation problem, 14 17, ; see also Problems with propagation of quantum trajectories Derivative propagation, see also Derivative propagation, Trajectory stability Derivative propagation, 22 23, 235, 370, 237, 240, 296, derivation of equations, equations of motion, 23, 239 evolutionary differential equation, 272 examples decay of metastable state, Eckart barrier transmission, form of wave function, 373 hierarchy of equations, implementation, multidimensional extension, phase space dynamics, , problems with, 25, 236, 250, 252, 271 Descriptor for trajectories, 92, 292 Designer grids, 5, 18, 368; see also Grid, adaptive Diabatic electronic representation, ; see also Adiabatic electronic representation Diffuse elements, 125 Diffusion coefficient, 31, 228 Diosi equation, 26, 255, 270, 275 Discrete variable representation, 223 Dissipation, 11, 63, 266, 269, 271, 371 Distributed approximating functionals, 16, 126, 135 application to Eckart barrier scattering, derivative evaluation, 136 discrete convolution, 135 Hermite DAF function, 136 Divergence of velocity field, 96 DLS; see Dynamic least squares fitting Domain functions, 22, 219, Double slit scattering, 164 Dynamic least squares fitting, 16, 125, Dynamical diagonalization of density matrix, 192, 199, 291 Dynamical grid; see Arbitrary Lagrangian Eulerian Dynamical tunneling, quantum trajectory evolution, two-dimensional model system, Dynamic viscosity, 324 Eckart potential, 63, 105, 136, 139, 161, 175, 179, , 225, 229, 231, 243, 249, 294, 377, 382 Ehrenfest force, 302 Ehrenfest mean field ; see Mean field Ehrenfest, Paul, 302 Eikonal equation, 45 Einstein Podolsky Rosen experiment, 40 Electronic nonadiabatic dynamics, 191, , 370; see also Trajectory surface hopping adiabatic and diabiatic representations, hydrodynamic equations of motion, 203, initial conditions on wave function, 208 matrix elements, , 213 model system, quantum trajectory results, Entangled trajectories, 255, 262, EPR experiment; see Einstein Podolsky Rosen experiment

9 398 Index Equidistribution principle, 20, , , 183, 188; see also Monitor function Equivalence principle; see Quantum equivalence principle Ermakov Pinney differential equation, 45 Euler for integration of differential equations, 149, 240 Eulerian frame, 3, 9, 48, 137, 157, , 183, 186, 256 Eulerian picture; see Eulerian frame Expectation maximization, 21, 218, Faraggi and Matone approach to stationary states; see Equivalence principle Feynman, Richard, 98 Feynman path integral, Field equations, 52 53, 58 Finite difference s, 125, Finite element, 17, 126, Fitting of functions; see also Least squares fitting s, Function approximation Fitting density to Gaussians, 21 22, , 225 Fixed lattice; see Eulerian frame Floyd trajectory, 7, 32 33, 375; see also Quantum equivalence principle application to harmonic oscillator, , form for wave function, 357 hidden variables, introduction to, 354 microstates, 358 modified potential, , quantization condition for eigenstates, 360 trajectory equations, 361 Floydian trajectory; see Floyd trajectory Flow compressible, 90, 100 incompressible, 100 kinetic energy, 8 9, 48 momentum, 48, 56 velocity, 47, 78 Fluid elements, 11 Fluctuation-dissipation theorem, 257 Flux-gate, 278 Flux vector maps; see Probability flux Force approximate quantum force, , 370 classical, 2, 9, 56, 90, 195 classical-quantum comparison, 82 hydrodynamic, 28, 78 80, 82, 286, 306 nonlocal in phase space, 274 quantum, 2, 9, 56, 62, 80, 86, 90, 107, 151, 154, 180, 195, 229; see also Quantum potential random, 257 stochastic, 257 total, 9, 57 Frictional effects, 256 Function approximation, ; see also Least squares fitting s, Dynamic least squares, Distributed approximating functionals, Tessellation-fitting, Finite element, Interpolation Gaussian wave packet, 3, 105, 136, 148, 151, 153, 161, 169, 175, 193, 212, 220, 229, 243, 377, 382 Geometric phase, 36 Ginsburg Landau equation, 36 Grid; see also Arbitrary Lagrangian Eulerian adaptive, 161, , 194, 212, 369, Cartesian, 93 computational, 141 correlation, 129 designer, 18 Eulerian; see Eulerian frame Lagrangian; see Lagrangian frame non-lagrangian, 5, moving, 5, 90, physical, 141 phase space, 280 structured, 371 unstructured, 94, 125 Grid paths, , 185 Grid point concentration, 174

10 Index 399 Grid velocity, 168, 170 Guidance equation in pilot wave interpretation, 59 H+H 2 reaction, 226, , 233, Hamilton s principle, 51, Hamilton Jacobi equation classical, 42, 44, 48 52, 210 stationary, 32, , quantum, 7 8, 42, 48, 56 57, 236, 377 Hamilton s characteristic function, , Herman Kluck propagator, 226 Hidden variables, 2, 59 60, , 365 Huisimi distribution, 234, 265 Huisimi equation of motion, 254, 265 Hybrid hydrodynamical Liouville phase space ; see Phase space Hybrid quantum trajectory s, 20 21, 167, , 370 applications double well potential, Eckart barrier scattering, Hydrodynamic force; see Force, hydrodynamic quantum equations of motion; see also Quantum hydrodynamic equations of motion Madelung Bohm derivation, 8, phase space derivation, 62, density matrix evolution, formulation of quantum mechanics, 1 7, wave function propagator, 13, 97 Hydrogen atom, chaos in quantum trajectories, Implicit time integrator, 5, 149 Inflation of trajectories, 14, 105, 162, 292; see also Compression of trajectories Initial value representation for quantum trajectories, 14, Instabilities; see Problems with quantum trajectories Interpolation, 15, 123, 131, 142, 158; see also Function approximation Jacobi, Carl, 100 Jacobi s theorem, 361 Jacobian equation of motion, 100 relation to wave function propagation, 13, trajectory stability matrix, 247 transported volume elements, 13, 99 Kinetic energy, 8 9, 48, 50, 54, 231 Kramers, Hendrick, Kramers equation, 25, 254, 259, 279, 282 Lagrange, Joseph-Louis, 50 Lagrangian Car Parrinello, 134 classical, 50, 52, 98 quantum, 57, 90, 92 Lagrangian frame, 3, 9, 49, 56, 168, 256 Lagrangian manifold, 356, Lagrangian picture; see Lagrangian frame Landau Zener approximation, 209 Langevin, Paul, 258 Langevin equation, 258 Least squares fitting s, 12, 15 16, 21 22, 125, , 200, 212, 225, 228, 231, 237 dynamic ; see Dynamic least squares local rational approximation, 132 moving least squares, 15 normal equations, 15, 130 polynomial consistency, 127 polynomial basis sets, 15, 128, 200 shape functions, 131 shape matrix, 130 summary of features, variational criterion, 15, 129 weighted functions, 15, 129 Lee Scully Wigner trajectories; see Wigner trajectories Leggett, Anthony, ; see also Caldeira-Leggett equation Leibniz theorem, 238, 272 Lindblad form for density operator, 268; see also Density operator Lindblad operators, 269 Liouville, Joseph, 257

11 400 Index Liouville equation, 25, 63, 71, 81, 254, 256, 305 Liouville von Neumann equation, 269 Liouville s theorem, 256 Local data representation, 124 Local modes of vibration, Local rational approximation; see Least squares fitting s Log derivative of density, 22, 219, 228, Log-likelihood in statistical estimation theory, 221; see also Expectation-maximization Lyapunov exponent, 109, , 119 Madelung, Erwin, 43 Madelung Bohm derivation of hydrodynamic equations, 8 10, 40, MARS; see Multivariate adaptive regression splines Multivariate adaptive regression splines, 128 Mean field, 300, 302; see also Mixed quantum-classical dynamics Mean interaction potential, 301 Measurement, quantum, 36, 40 Mesh compression, 194 Meshless s, 124; see also Arbitrary Lagrangian Eulerian Methyl-iodide, bending states, 21, 218, , 234 Metastable state classical phase space, 66 67, 73 74, 77 one-dimensional model, multidimensional model, phase space dynamics, Microstates; see Floyd trajectory Milne s, 45 Mixed quantum classical Bohmian ; see Configuration space Mixed quantum classical dynamics, 27 30, 300, 370; see also Mean field, Configuration space, Phase space, Backreaction through the Bohmian particle Mixed quantum-classical Bohmian ; see Configuration space MLS; see Least squares fitting s Modified Caldeira Leggett equation; see Diosi equation Moment hierarchy, 63, 80, 240 Moments; see also Phase space classical, dissipative dynamics, for probability density, 228 dissipative phase space dynamics, partial moments, 303 pure states, quantum definition, 10 derivation of equations of motion, Eckart barrier scattering, metastable oscillator, properties, Momentum density, 76, 326 Momentum fluctuation (variance), 76, 306 Momentum moments; see Moments Momentum operator, Monitor function, , , 372; see also Equidistribution principle Monte Carlo sampling, 103 Moving path transforms, 5, , 372; see also Equidistribution principle hydrodynamic equations of motion, 19, 168 time-dependent Schrödinger equation, 20, 175 MQCB; see Mixed quantum classical Multiquadric radial basis function, 158 Navier Stokes equations, 30, 76, 326, 328 Negative basins; see Phase space, negative basins Node in wave function; see also Quasi-node, Prenode, Postnode, Node problem, Voritces classification of types, 54

12 Index 401 dynamical tunneling, , 350 noncrossing rules, 104 phase of wave function, 46 quantum chaos, trajectory dynamics, 5, 104, 106, , 179, 182, Node-free wave functions; see Wave function, node-free Node problem; see Problems with propagation of quantum trajectories Noncrossing rules, 13, 90, 104, 155 Nonlocality, 9, 55, 97, 250, 270, 273, 296, 370 Ontological theory, 2, 60 Open quantum system, 254, 256, 259, 266, , Oppenheimer, J. Robert, 42 Osmotic velocity, 30 31, 228, Outer product, 64 Parallel computers, 95 Particle in cell, 120 Particle, 2, 120 Partition of unity, 131, 230 Path integral; see Feynman path integral Phase integral, 45, 356 Phase space; see also Moments, Phase space introduction, classical distribution, 65, 80 83, derivative propagation, dissipative dynamics, ensembles, 41, 66 67, Eulerian grid, 280 hydrodynamic, 65, Lagrangian trajectories, Liouville, negative basins, 70, normalization of density, 47, 256 propagator for density, 274 trajectories classical, 25 26, 65 67, 262 Eckart barrier scattering, metastable well potential, non-lagrangian, Phase space ; see also Mixed quantum-classical dynamics, application to coupled harmonic oscillators, comparison with configuration space, derivation of equations of motion, hydrodynamic force, 306 hydrodynamic subspace, 304 introduction, 28 Liouville subspace, 304 phase space concepts, partial moments, 303 reduced phase space, 303 Photoabsorption spectra, 52 Photodissociation, 17, 126, 142, 158 Pilot wave, 2, 59, 367 PIM; see Point interpolation Point interpolation, 131 Polar form for wave function; see Wave function, polar form Polynomial basis sets, , 200 Polynomial consistency, 127 Polypeptide chain, Postnode, ; see also Prenode and Node Potential energy coupled harmonic oscillators, 307 decoherence model, delta-function, 216 double well, 175 downhill ramp, 148, 156, 211 Eckart; see Eckart barrier harmonic oscillator, 65, 148 Lennard-Jones, 340 metastable well; see Metastable state Morse oscillator, 340, 375 quantum; see Quantum potential square barrier, 164 uphill ramp, Power spectrum of trajectory, , Principle of extreme action; see Hamilton s principle Prenode, ; see also Postnode, Quasi-node, Node Probability current; see Probability flux

13 402 Index Probability density classical, 66, 83, 279, 282 quantum, 83, 385, , 176, 178, , , Probability flux, 47, 75, , 337, , 347, 356, 374 Problems with propagation of quantum trajectories derivative evaluation, 5, 33 34, 171, instabilities and singularities, 148, 166, 372 nodes, 5, 34 35, , , 169, 179, 182, 187, sampling, 14 stiff equations of motion, 5, 149, 163, 166, 376 Propagator wave function, 13, 97 phase space density, 274 QFD; see Quantum fluid dynamics QHEM; see Quantum hydrodynamic equations of motion QSHJE; see Quantum stationary Hamilton Jacobi equation QTM; see Quantum trajectory Quantization rule for eigenstates, 360, 374 Quantized mean field, 317 Quantum equivalence principle, 7, 32 33, ; see also Floyd trajectory coordinate mapping, 363 equivalence postulate, 363 quantum gravity, 366 hidden variables, 365 quantum potential, 364 Schwartzian derivative, 365 Quantum fluid dynamics, 3, 89, 120, Quantum force; see Force, quantum Quantum hydrodynamic equations of motion; see also Hydrodynamic, equations of motion, quantum equations, 2, 18 19, 42, 135, 148, 163, , 236, 372 derivation, 8 10, 42, electronic nonadiabiatic dynamics, force version, 12, 90, 92 potential energy version, 12, 92, 150 Quantum Liouville equation; see Wigner Moyal equation Quantum optics, 63 Quantum potential, 8, 41, 42, 48, 56, 90, , 149, 180, 236, 310, 338, 348, 364, 367, 377; see also Force, quantum approximate, 231 decoherence model, 194 downhill ramp potential, 160 dynamical tunneling, 348 features, 55 free wave packet, 151, 154 interpretation, shape kinetic energy, 54 Quantum stationary Hamilton Jacobi equation; see Hamilton Jacobi equation Quantum trajectory; see Trajectory, quantum Quantum trajectory, 3, 11 13, 89, 90, 92 94, 120; see also Derivative propagation applications, 17 18, anisotropic harmonic oscillator, decay of metastable state, 19, , 216 decoherence model, downhill ramp, Eckart barrier scattering, 17, 105, 106, , electronic nonadiabatic dynamics, free wave packet, 17, reactive scattering, 164 computer code, 93, 95, equations of motion, 8 10, 12, features of, 94 introduction, phase space equations of motion,

14 Index 403 phase space trajectories, wave function synthesis, 12, Quantum vortices; see Vortices Quasi-node, 105, , 163, 187, 348; see also Node, Prenode, Postnode Quenching technique, 134 R-amplitude, 8, 12, 43, 46, 94, , , Radial basis function interpolation, , 161 Reactive scattering, 164, 188, , Reading guide, Reflection, above barrier, 159 Regression analysis, 124 Repellers in flux maps, 197 Resonant tunneling diode, 62 Runge Kutta for integration of differential equation, 149, 161 S; see Action function Sampling problem, 14 Schrödinger equation; see time-dependent Schrödinger equation Schwarzian derivative and quantum potential, 358, 364 Semiclassical quantum mechanics, 4, 14, 36 Separatrix for phase space trajectories, 66 Shape; see also Stress, shape component component of stress tensor, 76, 327 kinetic energy, 9, 54 parameter, 158 simulated annealing, 134 single value decomposition, 130, 132 Smoothed particle hydrodynamics, 124 Space-fixed grid; see Eulerian grid Spatial derivative evaluation problem; see Problems, derivative evaluation Spin, 36 SPH; see Smoothed particle hydrodynamics Spreading of wave packet, 151, 153 Spring constants for adaptive grid, 174 Square billiard problem, 112; see also Chaos, examples Stability matrix, 247; see also Trajectory stability States mixed, 11, 63 64, 86, 269 pure, 11, 62, 64, 86 stationary, 32 33, 45, Stationary states; see also States, stationary, Floydian trajectory, Equivalence principle, Counterpropagating wave Statistical analysis; see also Approximations to quantum force, Expectation- maximization Bayesian statistics, 221 conditional probabilities, 219 forward probability, 220 joint probabilities, 219 posterior probability, Stiff differential equations; see Problems with propagation of quantum trajectories Stochastic quantum mechanics, 327 Stokes s theorem and circulation integral, 334 Streamline, 92 Stress classical hydrodynamics, classical and quantum components, 326, 328 flow component, 327 introduction, quantum, 9, 55, quantum pressure term, shape component, 76, 327 shear component, 323 stress tensor, 324, 328 tensor components for decoherence model, Superposition of wave functions, 191, 357 Superposition principle, 372, 383; see also Counterpropagating wave, Covering function Surface of section in phase space, 114, 116 Surreal Bohm trajectories, 36 SVD; see Single value decomposition

15 404 Index Synthetic approach to quantum trajectories, 2 4, 11; see also Analytic approach to quantum trajectories, Quantum trajectory, Quantum fluid dynamics System bath Hamiltonian, TDSE; see Time-dependent Schrödinger equation Temporal smoother, 175 Tessellation fitting, 16, 126, Thermal distribution; see Boltzmann distribution Thermal energy, 285 Time-dependent Schrödinger equation, 1, 7 8, 20, 43, 46, 52, 119, 301, 311 Trajectory, 92 classical, 4, 49 51, 66 67, 91, 157, density matrix, grid paths, , 185 quantum, 7, 91, , , 118, , , 152, , 159, , , , 243, 314, 347; see also Quantum trajectory, Quantum fluid dynamics phase space, reasons for running quantum trajectories, 3 4 semiclassical, 4 weight for trajectory, 101 TSM; see Trajectory stability Trajectory stability, 24 25, 235, , 370 coordinate sensitivities, 24, 235 derivation of equations, Eckart barrier scattering, problems with, 25, 236, 250, 252 summary of equations, 24, 248 Trajectory surface hopping, ; see also Electronic nonadiabatic dynamics Transition density, 207 Transmission probability energy resolved, 185, 219, 227, , 249, 225 initial value representation, 103 time-dependent, 103, 157, 159, , , 225, 284, 314, 316, 318 Triangulation of grid points, 126, TSH; see Trajectory surface hopping Uncertainty relations, 36 Variance for momentum moments, 75 76, 81, 306 Velocity coupling approximation, 211 Vibrational decoupling for reactive scattering, 188 Vortices, 31 32, 322, circulation integral, 334 quantization of circulation, 335 relationship to wave function nodes, 335 examples H+H 2 reaction, intramolecular electron transfer, right-angled wave guide, scattering from disk, 339, 342 surface scattering, two-dimensional barrier, Wave function adiabatic; see Adiabiatic electronic representation bipolar form, 357, classical, 52 covering function, 373 complex amplitude, 35, 210, counterpropagating components, , 381 diabatic; see Diabatic electronic representation electronic nonadiabatic dynamics, list of forms, momentum space, 69 node-free, 6, polar form, 8, 29, 43, 44, 45, 235 non-polar form, 372 propagation along trajectory, 13, stationary states, 45, 357, 362, trigonometric form, 44, 357 Weight for trajectory; see Trajectory weight Wentzel Kramers Brillouin approximation, 44

16 Index 405 Wheeler DeWitt equation, 105 Wigner, Eugene, Wigner function, 10, 62 63, 65, 68 74, 86, 199, 240, 254 diffraction fringes, 70 equation of motion; see Wigner Moyal equation examples harmonic oscillator, metastable well potential, general properties, introduction, 68 momentum moments, negative basins in phase space, 70, synthesis using trajectory information, Wigner distribution; see Wigner function Wigner energy level spacing statistics, 111 Wigner Moyal equation, 26, 63, 70 71, 254, 260, 305 Wigner trajectories, 260, 262 WKB ; see Wentzel Kramers Brillouin approximation

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