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.
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.
Can a classical charged-particle trajectory be associated with a quantum process through a shared radiation spectrum?This Perspective examines spectral equivalence as a possible basis for such an association.For a prescribed trajectory, the Lieńard-Wiechert potentials provide a classical route to the electromagnetic fields and, through Fourier analysis of the radiation field, a corresponding spectrum.The proposed inverse procedure would adjust parameters within a chosen family of trajectories to reproduce a target spectrum, including one associated with a quantum process.We outline this framework and distinguish the reported numerical field calculations from the spectral fitting that remains to be demonstrated in this article.No fitted trajectory or quantitative spectral comparison is presented here.A spectral match would establish correspondence for the specified observable and comparison conditions; it would not, by itself, establish a unique particle path or a complete account of quantum dynamics.We identify questions concerning existence, uniqueness, physical consistency, and validation that must be addressed before assigning broader significance to equivalent-spectrum trajectories.The immediate objective is to formulate a testable research programme whose conclusions remain proportionate to the evidence obtained.
de Broglie-Bohm theory (dBBT), treating quantum particles as point objects moving along well defined (Bohmian) trajectories, offers an appealing solution of the measurement problem in quantum mechanics; it has, however, problems relating to spin, relativity and lack of proper integration with the Hilbert space based framework. In this work, we present a consistent formalism which has the traditional state-observable framework integrated with the desirable features of dBBT. We adopt ensemble interpretation for the Schrodinger wave function ψ. Given a Schrodinger wave function ψ, we use its value ψ0 at some fixed time (say, t=0) to define the probability measure ∣ψ0∣2dx on the system configuration space M (=Rn). On the resulting probability space M0, we introduce a stochastic process ξ(t) corresponding to the Heisenberg position operator XH(t) such that, in the Heisenberg state ∣ψh⟩ corresponding to ψ0, the expectation value of XH(t) equals that of ξ(t) in M0. This condition leads to the de Broglie-Bohm guidance equation for the sample paths of the process ξ(t) which are, therefore, Bohmian trajectories supposedly representing time-evolutions of individual members of the ψ0-ensemble. Stochastic processes and Bohmian trajectories corresponding to observables with discrete eigenvalues (in particular spin) are treated by extending the configuration space to the spectral space of the commutative algebra obtained by adding appropriate discrete observables to the position observables. Pauli's equation is treated as an example. A straightforward derivation of von Neumann's projection rule employing the Schrodinger-Bohm evolution of individual systems along their Bohmian trajectories is given. Some comments on the potential application of the formalism developed here to quantum mechanics of the universe are included.
Quantum trajectories, originating from the de Broglie–Bohm hydrodynamic description of quantum mechanics, are used to construct time-correlation functions in an initial value representation. The formulation is fully quantum mechanical and the resulting equations for the correlation functions are similar in form to their semiclassical analogs but do not require the computation of the stability or monodromy matrix or conjugate points. We then move to a local trajectory description by evolving the cumulants of the wave function along each individual path. The resulting equations of motion are an infinite hierarchy, which we truncate at a given order. We show that time-correlation functions computed using these approximate quantum trajectories can be used to accurately compute the eigenvalue spectrum for various potential systems.
In a recent publication, Goldfarb et al. have proposed a new trajectory scheme to solve quantum problems: Bohmian mechanics with complex action BOMCA . According to these authors, their contribution represents “a novel formulation of Bohmian mechanics.” In our opinion, rather than a new formulation of Bohmian mechanics, the proposal of Goldfarb et al. consists of a novel quantum computational scheme, which is based on solving the complex trajectory equations resulting from an already known complex version of Bohmian dynamics. On the other hand, Goldfarb et al. also claimed that their numerical scheme describes “fully local complex quantum trajectories,” with the reward of a significantly higher degree of localization than in the conventional Bohmian mechanics by lowering the magnitude of the quantum potential. At least from a theoretical and interpretative point of view, we consider that it is very important to clarify what “fully local quantum trajectories” means. Bohmian mechanics is a nonlocal theory because standard quantum mechanics is a nonlocal theory. This nonlocality manifests via the presence of a quantum potential, and therefore the corresponding quantum trajectories will display features that reflect such nonlocality. Something similar should also occur for any trajectory-based approach that intends to be equivalent to standard quantum mechanics. In this Comment we would like to provide a general discussion on these two aspects arising from the work of Goldfarb et al., which can be of much interest to the Chemical Physics community. In particular, we would like to stress the special interest relying on how the term locality should be employed. The approach of Goldfarb et al. starts with the timedependent quantum Hamilton-Jacobi equation. This equation was considered by Leacock and Padgett in 1983 within the context of the quantum transformation theory as a way to obtain bound-state energy levels of quantum systems with no need to calculate the corresponding eigenfunctions. According to Leacock and Padgett, the quantum Hamilton-Jacobi equation can be either stated as a postulate as done by these authors or derived from the Schrodinger equation through a simple connection formula. In the second case, the quantum action corresponds to the complex phase of the wave function when the latter is expressed as a pure complex exponential function. This relationship allows one to pass straightforwardly from the quantum Hamilton-Jacobi formalism to the standard one in terms of the time-dependent Schrodinger and vice versa . Although both the Ansatz and the formulation used by Leacock and Padgett are similar to those considered in the standard quantum WKB semiclassical approximation, as pointed out by these authors, their approach conceptually differs from the WKB one. Within Bohmian mechanics, energy eigenstates present an inconvenience: they give rise to zero velocity fields. Hence, when particles are described by this type of states, they will remain steady motionless . To overcome this problem, Floyd and Faraggi and Matone developed timeindependent quantum Hamilton-Jacobi-like formulations starting from real bipolar Ansatze, though they did not claim full equivalence with standard quantum mechanics regarding their predictions. Later, John proposed a timedependent complex quantum trajectory formalism based on the same connection formula mentioned by Leacock and Padgett to study the dynamics associated with some simple analytical cases, such as the harmonic oscillator or the step barrier. This “modified de Broglie–Bohm approach to quantum mechanics,” as termed by John, is formally equivalent to the BOMCA formulation of Goldfarb et al. Nevertheless, we would like to specify that, due to the fundamental theoretical nature of John’s contribution, a separate calculation of the wave function is required in order to obtain the corresponding trajectories John makes use of the “guiding wave” idea from Bohmian mechanics . Meanwhile, in the version of Goldfarb et al. the modified de Broglie–Bohm approach is formulated in terms of a closed set of equations in the spirit of quantum hydrodynamical formalisms , which avoids the calculation of the wave functions, thus providing a new, practical use to such an approach. Furthermore, within the approach of Goldfarb et al., the expression for the quantum force happens to be typically much smaller than in standard Bohmian mechanics. On the other hand, we would also like to note that a related formulation has also been used by Chou and Wyatt, though they developed a different, novel numerical scheme to tackle the problem of bound-energy levels and scattering in one dimension. When trying to understand the quantum-classical correspondence from the perspective of a WKB-like formulation of Bohmian mechanics WKB-BM , Sanz and co-workers reached the complex quantum HamiltonJacobi equation the same year as John, but independently. The WKB-BM formulation constitutes an excellent, compreTHE JOURNAL OF CHEMICAL PHYSICS 127, 197101 2007
We present a novel covariant relativistic guidance formula à la de Broglie-Bohm derived from conserved quantities associated with the Lagrangian of the Dirac equation in a 1+1-dimensional spacetime. In line with the standard guidance formula of Bohmian mechanics, this generalized guidance formula incorporates space-time derivatives of the Dirac spinor phases and consistently reproduces relativistic features such as mass-dependent trajectories and zitterbewegung. We formally show that this formulation explicitly reveals how the relativistic quantum potential affects the trajectories. Using numerical simulations, we compare these trajectories with those presented in the usual approaches. Finally, we show how the guidance formula naturally transitions to the standard de Broglie-Bohm theory in the non-relativistic limit and gives additional contributions in the expansions giving new insight into the zitterbewegung in pilot-wave theories.
Bohmian mechanics is a non-relativistic quantum theory based on a particle approach. In this paper, we study the Schrödinger equation with a rapidly oscillating potential and the associated Bohmian trajectory. We prove that the corresponding Bohmian trajectory converges locally in a measure, and the limit coincides with the Bohmian trajectory for the effective Schrödinger equation on a finite time interval. This is beneficial for efficient simulation of the Bohmian trajectories in oscillating potential fields.