We discuss quantum trajectories from the point of view of Bohm and Wyatt. We find that the theory can be formally written in terms of a field (probability) [P(x,t)], a position [x], and an average momentum [〈p〉] in a density operator which is normalized and Hermitian, but not positive definite. One consequence of this is that the theory obeys, in a natural way, a formal relation ΔxΔp=0 for these averages. We interpret this as a consistency requirement for a trajectory in space–time with a position x and an average momentum which allows a space–time description. We show that a momentum space form can be written in terms of a field (probability) [P(p,t)], a momentum [p], and an average position [〈x〉]. We briefly discuss potential chemical applications of quantum trajectories in the theory of chemical dynamics, kinetics, and local field theory.
We discuss quantum trajectories from the point of view of Bohm and Wyatt. We find that the theory can be formally written in terms of a field (probability) [P(x,t)], a position [x], and an average momentum [〈p〉] in a density operator which is normalized and Hermitian, but not positive definite. One consequence of this is that the theory obeys, in a natural way, a formal relation ΔxΔp=0 for these averages. We interpret this as a consistency requirement for a trajectory in space–time with a position x and an average momentum which allows a space–time description. We show that a momentum space form can be written in terms of a field (probability) [P(p,t)], a momentum [p], and an average position [〈x〉]. We briefly discuss potential chemical applications of quantum trajectories in the theory of chemical dynamics, kinetics, and local field theory.
Bohm theory is a formulation of Quantum Mechanics that characterises the state of a quantum system according to both the wave function, as in the conventional formulation, and the coordinates (positions) of all the particles that evolve in time drawing quantum continuous trajectories. Furthermore, a statistical ensemble of all the possible trajectories, raising from the impossibility to know the initial position of all the particles, establishes the exact correspondence with the traditional Quantum Mechanics. From a computational point of view, Bohm theory has found many applications in Chemical Physics especially to develop new methodologies for solving the Schrödinger equation and to address semi-classical approximations of Quantum Mechanics. From a theoretical point of view, the most appealing feature of Bohm theory is its capability to supply a conceptual map between the quantum formalism and our representation of what a chemical system is. Chemical systems are composed of molecules, but the same idea of molecule requires a specific arrangement in the space of particles, i.e., the nuclei of the atoms. The statistical description of conventional Quantum Mechanics on the basis of wave function alone is insufficient to establish a clear correspondence with such a picture of molecules. Indeed, chemists employ usually Classical Mechanics in order to overcome this drawback of the standard quantum theory. On the other hand, if the particles position is included in the quantum formalism, as Bohm theory does, the map can be defined in a self-consistent way. In other words, Bohm theory appears to be the suitable quantum framework to represent molecules and their motion. The chemical representation of molecular systems finds a natural correspondence with a single Bohm trajectory, since it is always implicitly assumed that molecular components have specific spatial position independently of our knowledge about it. Consequently, we develop a quantum method whose fundamental assumption is that a single Bohm trajectory, i.e., a quantum molecular trajectory, describes the molecular systems and the molecular motion correctly. First of all, we examine the correspondence between a single Bohm trajectory and the conventional Quantum Mechanics, without using the ensemble of trajectories. We verify that such a correspondence exists through numerical simulations and we prove formally that the statistical properties of a single Bohm trajectory explain the probabilistic description of Quantum Mechanics. Once the consistency of this original approach has been established, we investigate the predicted properties. For instance, we take into account the constants of motion (such as the energy) corresponding to the time evolution of the coordinates and the behaviour of simple chemical systems, e.g., the vibrational motion of single molecules interacting with a resonant field. In this way, unexpected features of the molecular motion are found. Secondly, we tackle the challenge of describing many components systems (like the chemical systems in ordinary conditions). As a matter of fact, the computation of the Bohm trajectory and of the wave function is extremely demanding. However, the statistical properties of the Bohm trajectory allow the derivation of stochastic theories for examining the dynamics of open quantum systems, i.e., few molecules (or few degrees of freedom) interacting with their environment (the other molecules). One of the developed stochastic methods correlates the dynamics of the reduced density matrix, for the degrees of freedom of interest, to the evolution of the corresponding Bohm coordinates. In other words, the Bohm equation, determining the set of all the particles velocities according to the full wave function, is replaced with a stochastic one that approximates the velocity of a subset of coordinates according to the reduced density matrix. In such a way, the quantum fluctuations induced by the environment are taken into account. The advantage of this method concerns its capability of describing quantum systems, including open quantum systems, in terms of a quantum trajectory. This could allow the understanding of the molecular motion during a spectroscopical experiment. The possibility of investigating reactive systems, such as conformational changes, is particularly interesting. As a matter of fact, chemical reactions can be completely characterised only through the particles motion and we define the suit- able quantum methodology providing a self-consistent description of the molecular motion.
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.
In the Entropic Dynamics (ED) framework quantum theory is derived as an application of the method of maximum entropy. The particles have well-defined positions but since they follow non differentiable Brownian trajectories they cannot be assigned an instantaneous momentum. Nevertheless, four different notions of momentum can be usefully introduced. We derive relations among them and the corresponding uncertainty relations. The main conclusion is that in ED momentum is not to be interpreted as an attribute of the particles but as a statistical concept associated to the probability distributions.
Despite the quantum nature of molecules, classical mechanics is often employed to describe molecular motions that play a fundamental role in a wide range of phenomena including chemical reactions. This is due to the need of assigning well-defined positions to the atomic nuclei during the time evolution of the system in order to describe unambiguously the molecular motions, whereas quantum mechanics provides information on probabilistic nature only. One would like to employ a quantum molecular trajectory that defines rigorously the instantaneous nuclear positions and, simultaneously, guarantees the conservation of all quantum mechanics predictions unlike the classical trajectory. We argue that such a quantum molecular trajectory can be formally defined and we prove that it corresponds to a single Bohm trajectory. Our analysis establishes a clear correspondence between the statistical properties of the trajectory and the quantum expectation values. The obvious and undeniable benefit is that of dealing with a quantum methodology fully characterizing the molecular motions without any reference to classical mechanics.
Abstract We revisit the Bohr model of the atom through Brownian motion of the electron and the principles of stochastic optimal control. The electron is assumed to have a definite but random position, represented by a single real-valued stochastic process in physical space whose probability density obeys the Fokker–Planck (FP) equation. Because Brownian paths are not differentiable, the process carries two mean drifts, one for each direction of time. We treat the forward drift as the control field, while the backward drift is fixed by the density of the same process. The running cost of the control problem combines the two drifts into a time-symmetric kinetic term, and through the backward drift it inherits a dependence on the density, so the value becomes a functional on density space. Bellman’s dynamic-programming principle requires the control to minimize the expected action from every intermediate time and density onward. The drift therefore emerges as a feedback law on position and density, rather than from the global stationarity of a stochastic action. The resulting law-dependent HJB–FP system reduces to the Schrödinger equation once the standard single-valuedness condition is imposed on the reconstructed phase. For stationary hydrogen states the theory yields explicit drift fields in spherical coordinates and reproduces the standard radial and angular kinetic-energy averages of the quantum operator formalism. Direct trajectory-level simulations of the electron’s stochastic motion around the nucleus show the coordinate distributions converging to the Born marginals and the time-averaged energies reproducing the quantum expectation values. For the 2 p eigenstates with magnetic quantum number m = ± 1 , a phase-driven azimuthal drift makes the simulated trajectories circulate at the analytically predicted rate, and the angular momentum accumulated from the raw trajectory increments converges to exactly L z = m ℏ . The angular-momentum quantization postulated in the Bohr model thus reappears as a property of the simulated stochastic motion.
The eigenvalues and eigenvectors of a normalized gaussian operator do not seem to have been previously considered. I solve this problem for 1-dimensional translational systems. I also address the question as to whether a gaussian operator is a density operator. To answer that question, it is first necessary to be sure what conditions must be satisfied, so a short review of density operators is given. Since position and momentum do not commute in quantum mechanics, it is useful to start with the consequences of the noncommutation, which is generally the Schrödinger-Robertson uncertainty relation, which is also briefly reviewed. It is found that the question of whether a gaussian operator is a density operator is directly tied to this uncertainty relation. Since the Wigner function is the phase space representation of a translational density operator, it is natural to consider the gaussian phase space function associated with a gaussian operator and to compare the phase space and operator properties. Throughout such discussions, the independent parameters in these functions are the first and second moments of position and momentum. The application of this formalism to the free translation and spreading of a gaussian packet is given and shows the formal similarity between classical and quantum behavior, whereas the literature standardly only considers the pure state case (equivalent to a single wavefunction).