English (pdf)
Article in xml format
Article references
Automatic translation
Send this article by e-mail
Cited by SciELO 
Cited by Google 
Similars in
SciELO 
Similars in Google 
Permalink
Introduction
The macroscopic objects obey Newton's equations of motion, where is the position of the particle in time and is the force acting on the particle. By giving values for both position and velocity in a time it is possible to calculate the trajectory of an object and as a result it is possible to predict the position and velocity at a later time.
The article is organized as follows; in the first part the terms of the Schrödinger-Langevin equation are presented, in the second part we implement the proposed solution, as a linear exponential function in , in addition we find the equations of movement, and finally a discussion of this result is made.
At the molecular and atomic level objects obey the laws of quantum mechanics, in this way Newton's laws are not strictly valid, the unique concept of trajectory is no longer literally valid. This is the fundamental basis of quantum mechanics, which is not local, each component of the system has influence on the movement of any other part which is reciprocal.
In this paper we take the Schrödinger-Langevin equation. To study processes of diffusion and dissipation at the quantum level, we generally start from the Schrödinger-Langevin equation in the study of the Brownian movement (Kostin 1972), which contains two new terms, which include a friction constant and the other one ends a random potential. Taking a Coulomb potential and find a possible solution to the Schrödinger-Langevin equation, comparing the interpretations according to quantum mechanics and the casual interpretation of Bhom (Bohm D, 1996) by quantum trajectories.
The article is organized as follows; in the first part the terms of the Schrödinger-Langevin equation are presented, in the second part we implement the proposed solution, as a linear exponential function in , in addition we find the equations of movement, and finally a discussion of this result is made.
Materials and Methods
The Schrödinger-Langevin equation
In quantum mechanics any physical system can be associated with a Hermitic operator
In the Langevin equation (Chia-ChunChou, 2017) of motion, the Hamiltonian operator includes the expression of the potential V L associated with quantum friction processes; the Schrödinger equation in one dimension is given by (Kostin 1972):
where (Kostin 1972):
VR is the potential associated to the random force (in this work it is taken as null), the term W(t) that originates from the potential associated with friction, can be removed by introducing the transformation:
making the corresponding substitutions, you get to:
also
The equation (3) is the one we use to study the behavior of a particle of mass m in a Coulomb potential.
Solution assuming Coulomb potential
Assuming a Coulomb potential V = -qEx = - kx; k = qE, the equation (3) takes the form:
Assuming as solution of the equation (4) of the form:
performing the first time derivative
of the second derivative with respect to time, we obtain
Equating the imaginary and real parts, the system of equations is obtained:
Assuming the function A(x; t) = G(x)F(t) (separable variables); of the equation (6)
Where a is constant. Then:
How A(x;t) = G(x)F(t) and considering that
substituting in (7) we obtain:
Equation of coefficients in x on both sides that leads to:
This is a first order differential equation for a (1), it is found that
Independent terms:
Integrating you get
d constant; Solving for F (t)
where C is constant. Then the A function takes the form:
Taking the equation (2), (5), and (13). You finally get the function
With α1(t),α2(t) given by the equations (10) and (11). Assuming a suggested interpretation of quantum theory in terms of hidden variables, type Bohm (Bohm D. y B. J. &. B. J. S. Hiley, 1996. Bohm D. 1952).
Where R(x; t); S(x; t) are real functions, then we have:
taking the equation found in (14) and (16) substituting in (4) you get:
According to (Bohm D. (1952)):
The equation of movement is:
Results
Equation 19 allows to make the representation of a quantum system as the spatial configuration that evolves in time, as a trajectory under the action of the wave function, this is the main objective of the De Broglie-Bohm theory (or theory pilot wave (Avanzini, 2016)). However, its standard formulation refers to the statistical set of its possible trajectories. The statistical set is introduced to establish the exact correspondence (Born's rule) between the probability density in the spatial configurations and the quantum distribution, that is, the squared modulus of the wave function. The pilot wave theory allows a formally self-consistent representation of quantum systems as the trajectory of a particle.
Equation 19 gives us the first result of the position of the particle as a function of time. Now from (Kostin 1972), where derive a Schrödinger equation for a Brownian particle interacting with a thermal environment. The function W(t) of the equation (4) has the form:
The integrals raised in equation 20 have as a solution
Without considering the interpretation of Bohm (16), we find the equation of time evolution of the expected value of the position x(t),
Discussion
In this paper the authors find a solution to the Schrödinger-Langevin equation, which includes terms of friction, is an article in which a linear exponential solution is proposed in, according to the expected behavior in the movement equation.
The solution obtained in the equation (19) contains a series of constants that must be implemented according to the initial conditions that are given according to the experimental setup (Torres del Castillo G.F. 2016) Analogously, the solution obtained in the equation (21) which has a series of constants associated, which does not allow directly compare the two solutions.
Generally, in the literature there are studies of the behavior of the harmonic oscillator, in addition to the oscillator in an electric field, here we look for an expression in which only the contribution of an electric field is considered, since when considering the case of the oscillator a slip is obtained of energy levels.
Conclusions
The model used to solve the Schrödinger-Langevin equation, allowed to find the expected values of position and moment of a particle in a Coulomb potential, the temporal dependence of such solutions is made explicit, which allows finding the path equations of the particles.
Our results constitute an additional reason to formulate hydrodynamical versions of quantum evolution equations (Hatifi 2019). Although hydrodynamics is a natural language in this context, the mathematics is compatible with other theories of continuum mechanics, for example, elasticity (Holland 2005).
The trajectories of particles in a fluid can be considered analogous to full-wave ray theory in the limit of geometric optics. The last (classical) limit is obtained in circumstances where the internal potential and force are negligible compared to the external potential and force of the body, respectively. The result is a fluid dynamic representation of the classical Hamilton-Jacobi theory. It should be taken into account that this limit describes a continuous set of trajectories that do not interact moving in the potential V (as discussed in Holland 1996), instead of only one. The equation (21) allows to calculate possible quantum trajectories of the Broglie Bohmm type (Mariya 2020), as a classical limit, only the initial path conditions should be considered and the equation allows to determine the temporal evolution of the system.
