Bohm’s mechanics was built for explaining individual results in measurements, and mainly for getting rid of the enigmatic reduction postulate. Its main idea is that particles have at any time definite positions and velocities. An additional axiom is that particles follow continuous trajectories that admit the first derivative in time, the velocity. In the quantum theory, if the position of a quantum object is well-defined at some time, a Δt time later the object may be found anywhere in space, so, the velocity defined as Δx/Δt is completely undefined. This incompatibility is regarded in standard quantum theory as nature’s property. The disagreement between quantum and Bohm’s mechanics is particularly strong in wave-like phenomena, e.g. interference. For a particle traveling through an interference fringe, Bohm’s velocity formula shows a dependence of the time-of-flight on the fringe length. Such a dependence is not supported by the quantum theory. Thus, for deciding which prediction is correct one has to measure times-of-flight. But this is a problem. If one detects a particle at two positions and records the detection times, the time difference is meaningless, because the first position measurement disturbs the particle’s Bohm velocity (if exists). This text suggests a way around: instead of measuring positions and times, the particles are raised to an excited, unstable level, by passing them through a laser beam. The unstable level will decay in time, s.t. the density of probability of the excited atoms will indicate the time elapsed since excitation. For comparing the Bohmian and quantum predictions, this text proposes in continuation to send the beam of excited particle upon a mirror. Bohm’s velocity leads to anomalies in the reflected wave.
Bohm’s mechanics was built for explaining individual results in measurements, and mainly for getting rid of the enigmatic reduction postulate. Its main idea is that particles have at any time definite positions and velocities. An additional axiom is that particles follow continuous trajectories that admit the first derivative in time, the velocity. In the quantum theory, if the position of a quantum object is well-defined at some time, a Δt time later the object may be found anywhere in space, so, the velocity defined as Δx/Δt is completely undefined. This incompatibility is regarded in standard quantum theory as nature’s property. The disagreement between quantum and Bohm’s mechanics is particularly strong in wave-like phenomena, e.g. interference. For a particle traveling through an interference fringe, Bohm’s velocity formula shows a dependence of the time-of-flight on the fringe length. Such a dependence is not supported by the quantum theory. Thus, for deciding which prediction is correct one has to measure times-of-flight. But this is a problem. If one detects a particle at two positions and records the detection times, the time difference is meaningless, because the first position measurement disturbs the particle’s Bohm velocity (if exists). This text suggests a way around: instead of measuring positions and times, the particles are raised to an excited, unstable level, by passing them through a laser beam. The unstable level will decay in time, s.t. the density of probability of the excited atoms will indicate the time elapsed since excitation. For comparing the Bohmian and quantum predictions, this text proposes in continuation to send the beam of excited particle upon a mirror. Bohm’s velocity leads to anomalies in the reflected wave.
Bohmian mechanics, also known as pilot-wave theory or de Broglie-Bohm theory, is a formulation of quantum mechanics whose fundamental axioms are not about what observers will see if they perform an experiment but about what happens in reality. It is therefore called a "quantum theory without observers," alongside with collapse theories and many-worlds theories and in contrast to orthodox quantum mechanics. It follows from these axioms that in a universe governed by Bohmian mechanics, observers will see outcomes with exactly the probabilities specified by the usual rules of quantum mechanics for empirical predictions. Specifically, Bohmian mechanics asserts that electrons and other elementary particles have a definite position at every time and move according to an equation of motion that is one of the fundamental laws of the theory and involves a wave function that evolves according to the usual Schrödinger equation. Bohmian mechanics is named after David Bohm (1917-1992), who was, although not the first to consider this theory, the first to realize (in 1952) that it actually makes correct predictions.
Bohmian mechanics, also known as pilot-wave theory or de Broglie–Bohm theory, is a formulation of quantum mechanics whose fundamental axioms are not about what observers will see if they perform an experiment but about what happens in reality. It is therefore called a “quantum theory without observers,” alongside with collapse theories and many-worlds theories and in contrast to orthodox quantum mechanics. It follows from these axioms that in a universe governed by Bohmian mechanics, observers will see outcomes with exactly the probabilities specified by the usual rules of quantum mechanics for empirical predictions. Specifically, Bohmian mechanics asserts that electrons and other elementary particles have a definite position at every time and move according to an equation of motion that is one of the fundamental laws of the theory and involves a wavefunction that evolves according to the usual Schr\\"odinger equation. Bohmian mechanics is named after David Bohm (1917–1992), who was, although not the first to consider this theory, the first to realize (in 1952) that it actually makes correct predictions.
Bohmian mechanics, also known as pilot-wave theory or de Broglie-Bohm theory, is a formulation of mechanics whose fundamental axioms are not about what observers will see if they perform an experiment but about what happens in reality. It is therefore called a quantum theory without observers, alongside with collapse theories and many-worlds theories and in contrast to orthodox mechanics. It follows from these axioms that in a universe governed by Bohmian mechanics, observers will see outcomes with exactly the probabilities specified by the usual rules of mechanics for empirical predictions. Specifically, Bohmian mechanics asserts that electrons and other elementary particles have a definite position at every time and move according to an equation of motion that is one of the fundamental laws of the theory and involves a wave function that evolves according to the usual Schr\odinger equation. Bohmian mechanics is named after David Bohm (1917-1992), who was, although not the first to consider this theory, the first to realize (in 1952) that it actually makes correct predictions.
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.
Short introduction to Bohmian mechanics Sheldon Goldstein in a Foundations of Physics paper [1] indicates (p. 341): “Since macroscopic objects are normally regarded as built out of microscopic constituents …there can be no problem of macroscopic reality per se in Bohmian mechanics.” Goldstein in the same paper remarks (p. 342) that this | ψ | 2 has “a status very much the same as that of a thermodynamic equilibrium.” Basil Hiley the closest collaborator of David Bohm, says the following (Hiley [2] (p. 2)): “What Bohm (1952a; 1952b) did was to show how to retain a description of all the usual properties of a classical world and yet remain completely within the quantum formalism.” The Bohm (1952) [3] [4] references in that quote refer to the original work of Bohm in which he sets out the basics of the Bohmian mechanics. As we will see later in the next section of this chapter, the appearance of an “additional term” in the Hamilton-Jacobi equation, is the hallmark of Bohmian mechanics and it is very often termed the “quantum potential.” Hiley [2] (p. 2) remarks that since the Bohmian momentum is a well-defined function of position and time, an ensemble of trajectories can be found when the quantum potential is non-zero. A unique, classical path is found when the quantum potential is zero. From the outset, we hope the reader can savor the beauty of this simple but very powerful result. Bohmian mechanics indeed shows this very gentle transition from quantum mechanics to classical mechanics via the value of the quantum potential.