A quantum potential $V(x,t)$ of $\delta$-function type is introduced, to describe the inertial motion of a particle. Quantum-mechanically, it is in a bound state, though classically one seems to be free. The motion of the object (micro- or macroscopic) takes place according to the Huygens-Fresnel principle. The new position of the object (wave front) plays the role of the secondary sources that maintain the propagation. The mean value of the potential energy is $-mc^{2}$. We found that the de Broglie - Bohm quantum potential is the difference between the bound energy $E = - mc^{2}/2$ from the stationary case and our potential $V(x,t)$.
A quantum potential V(x,t) of δ-function type is introduced, to describe the inertial motion of a particle. Quantum-mechanically, it is in a bound state, though classically one seems to be free. The motion of the object (micro- or macroscopic) takes place according to the Huygens-Fresnel principle. The new position of the object (wave front) plays the role of the secondary sources that maintain the propagation. The mean value of the potential energy is −mc2. We found that the de Broglie - Bohm quantum potential is the difference between the bound energy E=−mc2/2 from the stationary case and our potential V(x,t).
A quantum potential V (x, t) of δ-function type is introduced, to describe the motion of a particle. Quantum-mechanically, it is in a bound state (potential trap of Hojman and Asenjo [6]), though classically one seems to be free. The motion of the object takes place according to the Huygens-Fresnel principle. The new position of the object (wave front) plays the role of the secondary sources that maintain the propagation. We found that the de Broglie-Bohm quantum potential is the difference between the constant bound energy E = −mc 2 /2 from the stationary case and our potential V (x, t).
Non-relativistic de Broglie–Bohm theory describes particles moving under the guidance of the wave function. In de Broglieʼs original formulation, the particle dynamics is given by a first-order differential equation. In Bohmʼs reformulation, it is given by Newtonʼs law of motion with an extra potential that depends on the wave function—the quantum potential—together with a constraint on the possible velocities. It was recently argued, mainly by numerical simulations, that relaxing this velocity constraint leads to a physically untenable theory. We provide further evidence for this by showing that for various wave functions the particles tend to escape the wave packet. In particular, we show that for a central classical potential and bound energy eigenstates the particle motion is often unbounded. This work seems particularly relevant for ways of simulating wave function evolution based on Bohmʼs formulation of the de Broglie–Bohm theory. Namely, the simulations may become unstable due to deviations from the velocity constraint.
Following de Broglie, Bohm, and others, it is assumed that quantum mechanics may be interpreted causally and that the $\ensuremath{\psi}$ function plays the role of a generating function for particle trajectories. By arguing that the $\ensuremath{\psi}$ function should not be interpreted as a probability amplitude, a new method for generating particle trajectories is postulated. The four-momentum of a scalar particle is assumed to be given as the gradient of an unspecified function $F(\ensuremath{\psi})$, where $\ensuremath{\psi}$ is a pure real solution of the Klein-Gordon equation. Since the location of a particle is determined solely by its trajectory, the probability distribution differs from $\ensuremath{\psi}{\ensuremath{\psi}}^{*}$; and therefore, ordinary experimental results differing from the traditional theory may, in principle, be predicted. Particle motion and trajectories are discussed for three examples: a free particle, a particle in a box, and the double slit.
We proposed a new de Broglie-Bohm interpretation, applied it to various cases, and derived the particle dynamics of de Broglie-Bohm mechanics from the wave dynamics of quantum mechanics.De Broglie-Bohm dynamics, constructed from the ground state of quantum mechanics, is often consistent with classical dynamics. Even when the two differ, the discrepancy remains within the range of minute quantum effects. Consequently, for many quantum mechanical ground states, quantum mechanics and de Broglie-Bohm mechanics, linked by the de Broglie-Bohm interpretation, are consistent and free of contradictions.
In classical physics, a potential V(x) acts on a particle at a location x regardless of its velocity which is taken to be constant within dx (even though the particle accelerates). Thus writing potential energy V(x) at a given x is like taking a snapshot of the particle when it is at x. The snapshot is static and does not indicate the dynamics of the problem as the particle may be moving either to the right or left, which is irrelevant in terms of the exact potential energy and even kinetic energy at point x. A quantum particle moving with constant velocity is described by a two-dimensional dynamic probability exp(ipx), with cos(px) representing a static density and sin(px) a time-shifted density to capture one directional motion such that exp(ipx)exp(-ipx)=1, we argue. V(x), however, only acts on the static density (linked to cos(px) and not sin(px)). Thus it seems that bound state quantum mechanics needs to remove the dynamic pieces of free particle wavefunctions exp(ipx) through interference. One might argue that for wavelengths of the order of the system length one does not know if one has exp(ipx) or exp(-ipx) at an x point, but for high energy levels, wavelengths become small with respect to the system length and yet interference is still present. Thus it seems that for the calculation of energy one requires a static average snapshot of the bound particle. Physically, we argue that the dynamic piece is still present i.e.. if one knocks out a bound particle and it is found in the p momentum state, it is represented by exp(ipx). Thus the purpose of the bound state wavefunction is to facilitate the calculation of average energy which is associated with a static snapshot type of density and not necessarily represent all of the dynamical information in the system i.e. the two dimensionality of each exp(ipx). As a result, the bound wavefunction does not show motion to the right and then to the left in the high energy limit.
Classical mechanics usually employs two variables x,t to describe motion in the form of x(t). Derivatives of this function yield velocity, acceleration etc. For a bound system, velocity (momentum) changes from one x to the next, but one does not necessarily need time to describe the scenario as one could use the pair of variables p(x),x instead of x(t),t. An issue which arises is that p(x) has double values as it is positive when the bound particle is moving to the right and negative when it is moving to the left. In such a case, time distinguishes between the two motions which occur at the same spatial point. From the point of view of energy, however, such a degeneracy at x is irrelevant because kinetic energy is pp/2m. Thus, it seems one may remove time from a bound state problem as long as one uses an energy (i.e. energy conservation) picture. As a result, we propose using a p(xi), xi model to describe a bound state with p(xi) (momentum at xi) being different at each xi. In order to calculate average of the absolute value of p at xo we suggest using a Dirac delta function delta(p-p(xo)) i.e. related to {Integral dp p delta(p-p(xo))} . If one expands the delta function in terms of the basis exp(ipx), it seems that quantum mechanics for the bound state arises as a classical steady stream picture in terms of energy conservation, with exp(ipx) representing a steady stream picture of a particle with a constant momentum. We discuss informational probability properties of exp(ipx) which make it the basis of choice.