Quantum and Semiclassical Trajectories

This presentation explores an alternate quantum framework in which the wavefunction Ψ(t, x) plays no role. Instead, quantum states are represented as ensembles of real-valued probabilistic trajectories, x(t, C), where C is a trajectory label. Quantum effects arise from the mutual interaction of different trajectories or “worlds,” manifesting as partial derivatives with respect to C. The quantum trajectory ensemble x(t, C) satisfies an action principle, leading to a dynamical partial differential equation (via the Euler-Lagrange procedure), as well as to conservation laws (via Noether’s theorem). An earlier, non-relativistic version of the trajectory-based theory turns out to be mathematically equivalent to the time-dependent Schroedinger equation [1–5], though it can be derived completely independently [3,4]. On the other hand, the relativistic generalization (for single, spin-zero, massive particles) [6,7] is not equivalent to the Klein-Gordon (KG) equation—and in fact, avoids certain well-known issues of the latter, such as negative (indefinite) probability density. The new relativistic quantum trajectory equations could in principle be used in quantum chemistry calculations, and otherwise could lead to new physical predictions that could be validated or refuted by experiment. [1] Bouda, A.; From a mechanical Lagrangian to the Schroedinger equation: A modified version of the quantum Newton law, Int. J. Mod. Phys. A, 2003, 18, 3347–3368. [2] Holland, P.; Computing the wavefunction from trajectories: particle and wave pictures in quantum mechanics and their relation, Ann. Phys., 2005, 315, 505–531. [3] Poirier, B.; Bohmian mechanics without pilot waves, Chem. Phys., 2010, 370, 4–14. [4] Schiff, J.; Poirier, B.; Communication: Quantum mechanics without wavefunctions, J. Chem. Phys., 2012, 136, 031102. [5] Poirier, B.; The many interacting worlds approach to quantum mechanics, Phys. Rev. X, 2014, 4, 040002. [6] Poirier, B.; Trajectory-based theory of relativistic quantum particles, 2012, arXiv:1208.6260 [quant-ph]. [7] Tsai, H.-M.; Poirier, B.; Exploring the propagation of relativistic quantum wavepackets in the trajectory-based formulation, J. Phys., 2016, 701, 012013.

In my talk, I will review the current state of research in the field of relativistic quantum chemistry. My presentation will be based on our monograph on this topic that recently appeared in its second edition [1]. Most of the talk will deal with the derivation of electrons-only Hamiltonians for low-energy physics, i.e., for chemistry. However, I will also consider our recent results on four-component pre-Born-Oppenheimer calculations with an all-particle wave function expanded in terms of antisymmetrized products of Gaussian geminals [2,3]. [1] M. Reiher, A. Wolf, Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science, 2nd revised and extended edition, Wiley-VCH, Weinheim, ISBN: 9783527334155, 2015 [2] B. Simmen, E. Matyus, M. Reiher, Relativistic Kinetic-Balance Condition for Explicitly Correlated Basis Functions, J. Phys. B 48, 2015, 245004 [3] A. Muolo, E. Matyus, M. Reiher, Generalized constraints for the rigorous elimination of the global translation in explicitly correlated Gaussian functions, J. Chem. Phys., 148, 2018, 084112

The energy-level structure of the one-electron molecules H2+ is known with extraordinary precision from ab initio quantum-chemical calculations that include all known effects. The uncertainties of the most recent calculations are dominated by the uncertainty of the proton-to-electron mass ratio (see, e.g., [1]). Unfortunately, hardly any high-resolution spectroscopic results have been reported on H2+, because it lacks a permanent electric dipole moment and thus has no pure rotational and vibrational spectra. Microwave electronic transitions between the weakly bound levels of the ground X+ 2 $\sum$ g+ and first excited A+ 2 $\sum$ u+ electronic states of H2+ have been observed by Carrington and coworkers (see, e.g., [2]) and the rotational and hyperfine structures of selected levels have been measured [3-5]. This talk will report on the precise experimental determination of the energy-level structure of H2+, HD+ and D2+ in two distinct spectral regions by Rydberg-series extrapolation: the region of the low-lying rovibrational levels (v+=0,1, N+=0-6) and the region near the dissociation limit X+ + Y(1s), with X,Y = H,D. In the first region, we determined the spin-rovibrational structure (including the hyperfine structure in ortho H2+) of the X+ state with sub-MHz accuracy. In the second region, we made the first observation, and also performed the first complete calculations, of the shape and Feshbach resonances in H2+ [6,7], HD+ and D2+, from which we derive elastic-collision and radiative-association cross sections for the corresponding proton-atom collisions. Somewhat unexpectedly, we found that the fourth bound state of the A+ state of H2+, i.e., the v+=1, N+=0 state, the existence of which has been predicted theoretically, actually does not exist. Consequently, the scattering length of the H+ + H(1s) collision must be revised. The proper treatment of the spectrum of H2+ near the dissociation limit must explicitly include the g/u symmetry mixing induced by the hyperfine interaction in close-coupling nonadiabatic calculations. [1] V.I. Korobov, L. Hilico and J.-P. Karr, Phys. Rev. Lett. 118, 223001 (2017) [2] A. Carrington, C.A. Leach, A.J. Marr, R.E. Moss, C.H. Pyne and T.C. Steimle, J. Chem. Phys. 98, 5290 (1993) [3] K. B. Jefferts, Phys. Rev. Lett. 23, 1476 (1969) [4] Ch. Haase, M. Beyer, Ch. Jungen and F. Merkt, J. Chem. Phys. 142, 064310 (2015) [5] A. Osterwalder, A. Wüest, F. Merkt and Ch. Jungen, J. Chem. Phys. 121, 11810 (2004) [6] M. Beyer and F. Merkt, Phys. Rev. Lett. 116, 093001 (2016) [7] M. Beyer and F. Merkt, J. Mol. Spectrosc. 330, 147 (2016)

We discuss the foundation of the Trojan Horse Method (THM) within the Inclusive Non-Elastic Breakup (INEB) theory. We demonstrate in the DWBA limit of the three-body theory of the INEB cross section, the direct part, of a two-step character, becomes, with appropriate approximations and redefinitions, similar in structure to the one-step THM cross section. Application to one-nucleon halo nuclei are discussed. The discussion can be extended to four-body reactions involving a three-fragment projectile collisions. Such projectiles, as $^9$Be (= $^4$He + $^4$He + n), include two-nucleon halo nuclei, such as $^6$He ($^4$He + 2n) or $^{17}$Ne ($^{15}$O + 2p).

While direct reactions with activation barriers are now routinely calculated, reactions going through a long-lived intermediate complex are much more difficult to study [1,2]. Complex-forming reactions are often barrierless and are thus relevant to the field of cold and utra-cold chemistry [2]. With decreasing temperature, wave effects become increasingly important and may dominate the collisional behavior at ultralow temperatures. Capture theories (close-coupling expansion with boundary conditions applied in the reactant channel) are often used to study complex-forming reactions [3,4]. The Langevin capture model is often used to describe barrierless reactive collisions. At very low temperature, quantum effects may alter this simple capture image and dramatically affect the reaction probability. In this talk, we use the trajectory-ensemble reformulation of quantum mechanics without wavefunctions recently proposed by Poirier and coworkers [5,6] to compute adiabatic-channel capture probabilities and cross-sections for the reaction Li + CaH$(v=0,j=0) \rightarrow$ LiH + Ca at low and ultra-low temperatures. The captured quantum trajectory takes full account of tunneling and quantum reflection along the radial collision coordinate. Our approach turns out to be very fast and accurate, down to extremely low temperatures. [1] H. Guo, Rev. Phys. Chem. 31, 1 (2012). [2] M. T. Bell, and T. P. Softley, Mol. Phys. 99, 107 (2009). [3] D.C. Clary and J.P. Henshaw, Faraday Discuss. Chem. Soc. 84, 333 (1987). [4] E.J. Rackham, T. Gonzalez-Lezana, and D.E. Manolopoulos, J. Chem. Phys. 119, 12895 (2003). [5] B. Poirier, Chem. Phys. 370, 4 (2010). [6] G. Parlant, Y.-C. Ou, K. Park, and B. Poirier, Comp. Theor. Chem. 990, 3 (2012).

Molecular dynamical processes fall in a regime that is near the classical limit, and trajectory-based classical molecular dynamics often provides an accurate approach to their simulation. Important cases arise, however, when manifestly nonclassical effects must be incorporated, such as when quantum tunneling and transitions between electronic states are an essential component the dynamics. In this talk, we review recent work on describing the nonclassical aspects of molecular processes using an enhanced classical dynamics based on "quantum trajectories". First, we describe an approach to simulating quantum tunneling based on "entangled trajectories", where the nonlocality of quantum mechanics is incorporated via a breakdown of the statistical independence of trajectory realizations which holds in the classical limit. In addition, we report our recent work on stochastic trajectory methods for simulating molecular dynamics with electronic transitions, which capture the mixed quantum and classical aspects of the problem using ensembles of trajectories that undergo stochastic hops between coupled quantum electronic states while representing classical-limit nuclear dynamics. Fundamental issues of energy conservation and decoherence will be discussed in the context of quantum trajectories, and connections made with the local and nonlocal hidden variable theories that underlie our understanding of the foundations of quantum theory.

We review the mathematics of semiclassical quantum dynamics and discuss the current mathematical understanding of thawed and frozen Gaussian approximations as well as the Wigner phase space method.

Very often waves travel through media with spatially smooth (on the scale of a wavelength) random but modest phase and group velocity variations. Two regimes resulting from such propagation have received much attention: first, the swallowtail catastrophes (the first focal zone of a imperfect lenses), and much later on, the statistical regime of momentum diffusion. A third intermediate regime is often overlooked but important to many fields, one that we term branched flow, the name recalling the distinctive branched pattern that catch the eye. By now there are many diverse contexts where branched flow is known or suspected to play a key role. Here we explain the phenomenon, give examples that span diverse scales and fields, and focus on the intriguing differences between classical and quantum flow in the same potential field.

Multiconfigurational Gaussian wavepacket expansions whose evolution is defined by the Dirac-Frenkel variational principle [1], provide a unique bridge between the multiconfiguration time-dependent Hartree (MCTDH) method [2,3] and trajectory-based representations. In practical applications, the Gaussian-based G-MCTDH approach [4], its variational multiconfigurational Gaussian (vMCG) [5] variant and recently developed multi-layer versions [6] have proven versatile tools for the explicit representation of system-bath dynamics as well as on-the-fly calculations. Furthermore, a hybrid quantum-classical variant of the G-MCTDH approach has been formulated [7], which takes a subset of coordinates to the classical limit, leading to a multiconfigurational Ehrenfest (MCE) type dynamics. Here, we will review the current status and prospects of these methods, along with their extensions towards a statistical description. [1] C. Lubich, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, European Mathematical Soc. (2008). [2] H.-D. Meyer, U. Manthe, and L. S. Cederbaum, Chem. Phys. Lett. 165, 73 (1990), M. H. Beck et al., Phys. Rep. 324, 1 (2000). [3] B. Kloss, I. Burghardt, C. Lubich, J. Chem. Phys. 146, 174107 (2017), M. Bonfanti and I. Burghardt, arXiv:1802.01058 [physics.chem-ph]. [4] I. Burghardt, H.-D. Meyer, L. S. Cederbaum, J. Chem. Phys. 111, 2927 (1999). [5] G. W. Richings et al., Int. Rev. Phys. Chem., 34, 265 (2015). [6] S. Roemer, M. Ruckenbauer, I. Burghardt, J. Chem. Phys., 138, 064106 (2013). [7] S. Roemer, I. Burghardt, Mol. Phys. 111, 3618 (2013).

Dissipative and stochastic dynamics are analyzed within the Bohmian mechanics framework in two different scenarios. First, nonlinear quantum-classical transition wave equations are used and applied in the Caldirola-Kanai and Kostin approaches. In particular, dissipative quantum tunneling through an inverted parabolic barrier in the presence or not of an electric field is studied by means of scaled trajectories, displaying a continuous or gradual description for the transition process going from the purely quantum or Bohmian to classical regime. And second, in the Kostin approach, Bohmian stochastic trajectories are used to characterize the two well-known regimes in any diffusion problem, the ballistic and Brownian regimes. The so-called Bohmian-Brownian motion is presented and discussed for the quantum surface diffusion of light adsorbates.

It is well known that both classical and quantum mechanical systems are described as Hamiltonian systems: finite-dimensional one for the former, and infinite-dimensional for the latter with respect to appropriate symplectic structures. I will show how to exploit such geometric structures to formulate semiclassical dynamics from the symplectic-geometric point of view. Particularly, I will explain how one can formulate the dynamics of semiclassical wave packets from the symplectic/Hamiltonian point of view.

Semi-classical approximations are approximations to quantum theory where part of the system is treated classically and part of the system is treated quantum mechanically. I will discuss how Bohmian mechanics allows for different classical-quantum couplings than those available in standard quantum theory, leading to approximations schemes better than, say, the mean field approximation. I will outline a general recipe to obtain such Bohmian semi-classical approximations. As a particular application, the Jaynes-Cummings model will be discussed.

Looking at the trajectories of quantum particles can give us new explanations for a number of symmetries in quantum mechanics, including: global phase symmetry, time reversal, Galilean boosts, and permutation symmetry. In this talk I will argue that the explanations of these symmetries that are available within a trajectories-only approach to quantum mechanics are superior to those that can be given within a wave function plus trajectories approach.

The bound state of a stationary and separable quantum system consists of steady-state combinations of two unbound --hence the tunnel effect-- scattering processes propagating in opposite directions with equal amplitudes. There exists actually at any energy level an infinity of such pairs of opposite scattering processes (or "matter fluxes") which build up the SAME real-valued Schrödinger eigenstate. This fundamental degree of freedom in their choice echoes both the gauge invariance of the Schrödinger equation and the Heisenberg uncertainty principle. When taking the singular mathematical limit of a vanishing such matter flux, Born's postulate yields for any real-valued Schrödinger eigenstate the normalized time interval spent by the system about any point of its phase space, i.e. its classical probabilistic distribution.

The de Broglie - Bohm pilot-wave theory is a candidate explanation of the quantum formalism which resolves orthodox QM's measurement and ontology problems. One of its nice features is that it allows the definition of sub-system (e.g., single-particle) pilot-waves / wave functions. It is thus possible to rigorously derive, in the context of the pilot-wave theory, an approximation scheme in which the N-particle configuration space wave function is replaced by N single-particle (physical space) pilot-waves. We will explain this so-called "small entanglement approximation" and then discuss preliminary efforts to push beyond it by including additional fields (on physical space) which reproduce (some of) the effects of entanglement. The work should be of interest both to those interested in the ontology of quantum mechanics (e.g., the question of how a field on 3N-dimensional configuration space can be understood physically) as well as those with more practical concerns about developing efficient numerical approximation schemes for many-particle quantum systems.

Several years ago, we developed a formulation of the time-dependent Schrodinger equation (TDSE) using complex-valued classical trajectories. The method has a number of appealing features: 1) it has a simple and rigorous derivation from the TDSE with a precisely formulated approximation; 2) it treats classically allowed and classically forbidden processes on the same footing; 3) it allows the introduction of arbitrary time-dependent external fields into the dynamics seamlessly and rigorously. This talk will have three parts. 1) a review of the method and previous applications, including to barrier transmission, bound state dynamics and nonadiabatic dynamics; 2) two new methodological advances that open the door to multidimensional systems and long time dynamics, respectively; 3) applications made possible by these new advances, including long time wavepacket revivals and preliminary results on strong field attosecond tunneling ionization.

To evaluate vibronic spectra beyond the Condon approximation, we extend the on-the-fly ab initio thawed Gaussian approximation by considering the Herzberg-Teller contribution due to the dependence of the electronic transition dipole moment on nuclear coordinates. The extended thawed Gaussian approximation is tested on electronic absorption spectra of phenyl radical and benzene: Calculated spectra reproduce experimental data and are much more accurate than standard global harmonic approaches, confirming the significance of anharmonicity. Moreover, the extended method provides a tool to quantify the Herzberg-Teller contribution: we show that in phenyl radical, anharmonicity outweighs the Herzberg-Teller contribution, whereas in benzene, the Herzberg-Teller contribution is essential, since the transition is electronically forbidden and Condon approximation yields a zero spectrum. Surprisingly, both adiabatic harmonic spectra outperform those of the vertical harmonic model, which describes the Franck-Condon region better. Finally, we provide a simple recipe for orientationally averaging spectra, valid beyond Condon approximation, and a relation among the transition dipole, its gradient, and nonadiabatic coupling vectors.

I will present an idea to approximate (non-integrable) dynamics by integrable dynamics. This is done by defining regions in phase space where a certain interaction dominates over another one rendering the effective “Dominant interaction Hamiltonian” (DIH) integrable, if the subdominant interaction is neglected. Since this is an approach which requires locality in phase space classical trajectories are constructed switching between governing DIHs whenever they enter a relevant phase space domain. We will discuss 2D electron-atom scattering and high harmonic generation for which an extension to the semiclassical quantum regime is presented.