Quantum Mechanics with Trajectories

Anastasios Kyprianidis

J S Briggs

Abstract The two‐dimensional paraxial equation of optics and the two‐dimensional time‐dependent Schrödinger equation, derived as approximations of the three‐dimensional Helmholtz equation and the three‐dimensional time‐independent Schrödinger equation, respectively, are identical. Here the free propagation in space and time of Hermite–Gauss wave packets (optics) or harmonic oscillator eigenfunctions (quantum mechanics) is examined in detail. The Gouy phase is shown to be a dynamic phase, appearing as the integral of the adiabatic eigenfrequency or eigenenergy. The wave packets propagate adiabatically in that at each space or time point they are solutions of the instantaneous harmonic problem. In both cases, it is shown that the form of the wave function is unchanged along the loci of the normals to wave fronts. This invariance along such trajectories is connected to the propagation of the invariant amplitude of the corresponding free wave number (optics) or momentum (quantum mechanics) wave packets. It is shown that the van Vleck classical density of trajectories function appears in the wave function amplitude over the complete trajectory. A transformation to the co‐moving frame along a trajectory gives a constant wave function multiplied by a simple energy or frequency phase factor. The Gouy phase becomes the proper time in this frame. Key Points This paper builds a bridge between quantum mechanics (QM) and classical optics in that the identity of the paraxial equation of optics and the Schrödinger equation of QM is shown, the Bohmian trajectories of QM are defined in optics, the Gouy phase of optics is defined in QM and given a new interpretation and The space and momentum wave functions are equivalent along a trajectory.

Elliott Tammaro, Sriram Gundla, Arunabh Pratik +1 more

Quantum mechanics abandoned the classical notion of a particle trajectory, yet trajectories remain conceptually appealing for resolving foundational issues in quantum mechanics. Bohmian mechanics offers one route to associating trajectories with quantum processes, but it requires explicit non-locality to remain consistent with EPR-type experiments, placing it in tension with special relativity. We propose an alternative approach: rather than deriving trajectories from a quantum guidance equation, we search for classical trajectories whose radiated frequency spectra match those of quantum processes such as atomic transitions. Using the Liénard-Wiechert potentials, we compute the electric and magnetic fields generated by a point charge along a given trajectory, numerically solving the retarded-time and from these fields obtain the power spectrum of emitted radiation. By fitting the parameters of a chosen family of trajectories, we identify "equivalent spectrum" trajectories that reproduce a target frequency distribution, including ones of quantum mechanical origin. We discuss the implications of this method and propose future work to determine whether equivalent spectrum trajectories belong to a family governed by a differential equation, which would constitute a quantum-equivalent equation of motion.

Axel Schild

Quantum hydrodynamics is a formulation of quantum mechanics based on the probability density and flux (current) density of a quantum system. It can be used to define trajectories which allow for a particle-based interpretation of quantum mechanics, commonly known as Bohmian mechanics. However, quantum hydrodynamics rests on the usual time-dependent formulation of quantum mechanics where time appears as a parameter. This parameter describes the correlation of the state of the quantum system with an external system -- a clock -- which behaves according to classical mechanics. With the Exact Factorization of a quantum system into a marginal and a conditional system, quantum mechanics and hence quantum hydrodynamics can be generalized for quantum clocks. In this article, the theory is developed and it is shown that trajectories for the quantum system can still be defined, and that these trajectories depend conditionally on the trajectory of the clock. Such trajectories are not only interesting from a fundamental point of view, but they can also find practical applications whenever a dynamics relative to an external time parameter is composed of "fast" and "slow" degrees of freedom and the interest is in the fast ones, while quantum effects of the slow ones (like a branching of the wavepacket) cannot be neglected. As an illustration, time- and clock-dependent trajectories are calculated for a model system of a non-adiabatic dynamics, where an electron is the quantum system, a nucleus is the quantum clock, and an external time parameter is provided, e.g. via an interaction with a laser field that is not treated explicitly.