Schrodinger-Langevin Equation Research Articles

Another Solution to the Schrödinger-Langevin Equation

Introduction: an alternative solution to the Schrodinger-Langevin equation is presented, where the temporal dependence is explained, assuming a Coulomb potential. Finally, the trajectory equations are found. Objective: in this paper we contribute by presenting a detailed and simple solution of the Schrödinger-Langevin equation for a Coulomb potential. Materials and Methods: using an appropriate ansatz, we solve the Schrödinger-Langevin equation, finding the expected values of position and moment. Results: a simple method was presented to find the expected position and moment values in the Schrödinger-Langevin equation, the ansatz used to find these solutions allows the model to be generalized in a certain way to electric potentials and harmonic oscillators. 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.

Dissipative motion of Gaussian ‎wavepackets: free propagation and ‎transmission through a rectangular ‎barrier

Ignoring thermal fluctuations of the environment, taking into account only its dissipative effects, free propagation of a Gaussian wavepacket is studied in the framework of the linear Caldirola-Kanai (CK) equation and non-linear equations, the Schrodinger-Langevin (SL) equation, known as Kostin equation, and the Schuch-Chung-Hartmann (SCH) equation. By a Gaussian ansatz for the probability density one obtains two equations, one for the evolution of the center of the wave packet which is just the classical Langeving equation and one for the evolution of the width of the wavepacket. This last equation has different forms in different approaches having only an analytic solution in the CK and SCH frameworks. Computations show that for a given friction, the width of the wavepacket increases with time in all approaches. In a given time, it reduces with friction in both CK and SL approaches revealing localization effects of dissipation, while has opposite behavior in the SCH approach. Furthermore, energy expectation value and its time-derivative is computed and compared in all approaches. It is shown that the rate of energy is given by the expectation value of momentum field in the SL framework. Finally, transmission of a Gaussian wavepacket from a rectangular barrier is numerically studied in the context of CK and compared to the non-dissipative case. The transmission decreases considerably with the dissipation.

Schrodinger-Langevin Equation and Ion Transport atNano Scale

Schrodinger-Langevin equation has been constructed for the ion-transport for K-ion channel. The stability of the solutions of this equation has been discussed under various physical situations. This will shed new light on the ion transport at nano-scale as well as the possibility of ion trapping and quantum information processing.

A local coherent-state approximation to system-bath quantum dynamics

A novel quantum method to deal with typical system-bath dynamical problems is introduced. Subsystem discrete variable representation and bath coherent-state sets are used to write down a multiconfigurational expansion of the wave function of the whole system. With the help of the Dirac-Frenkel variational principle, simple equations of motion--a kind of Schrodinger-Langevin equation for the subsystem coupled to (pseudo) classical equations for the bath--are derived. True dissipative dynamics at all times is obtained by coupling the bath to a secondary, classical Ohmic bath, which is modeled by adding a friction coefficient in the derived pseudoclassical bath equations. The resulting equations are then solved for a number of model problems, ranging from tunneling to vibrational relaxation dynamics. Comparison of the results with those of exact, multiconfiguration time-dependent Hartree calculations in systems with up to 80 bath oscillators shows that the proposed method can be very accurate and might be of help in studying realistic problems with very large baths. To this end, its linear scaling behavior with respect to the number of bath degrees of freedom is shown in practice with model calculations using tens of thousands of bath oscillators.

Quantum motion over a finite one-dimensional domain: With and without dissipation

Quantum motion of a single particle over a finite one-dimensional spatial domain is considered for the generalized four parameter infinity of boundary conditions (GBC) of Carreauet al [1]. The boundary conditions permit complex eigenfunctions with nonzero current for discrete states. Explicit expressions are obtained for the eigenvalues and eigenfunctions. It is shown that these states go over to plane waves in the limit of the spatial domain becoming very large. Dissipation is introduced through Schrodinger-Langevin (SL) equation. The space and time parts of the SL equation are separated and the time part is solved exactly. The space part is converted to nonlinear ordinary differential equation. This is solved perturbatively consistent with the GBC. Various special cases are considered for illustrative purposes.

Schr�dinger quantization and variational principles in dissipative quantum theory

Certain phenomenological equations of dissipative quantum theory such as the Caldirola-Kanai equation or Kostin's nonlinear Schrodinger-Langevin equation are considered in the framework of Lagrangian field theory. The role of the Schrodinger quantization procedure in connection with the construction of variational principles for these equations is scrutinized. It is pointed out that incorrect Lagrangians for linearly damped systems have repeatedly been used in the literature. This has led to a considerable amount of confusion and contradictory results. It is demonstrated how a proper use of Schrodinger-quantization yields a correct Lagrangian description — in particular of Kostin's Schrodinger-Langevin equation-whose properties and implications are then investigated. The paper also deals with an application of Noether's theorem, contains some remarks about the introduction of external magnetic fields and points out an analogy in classical hydrodynamics.

Adding dissipation to the Schr�dinger equation from the quantum-potential viewpoint

We discuss methods of adding dissipation to the Schrodinger equation in light of the quantum-potential interpretation of quantum mechanics. This favors nonlinear equations in general and Kostin's Schrodinger-Langevin equation in particular. We give a simple example, the damping of small coherent oscillations.

Derivation of the non-linear Schrödinger equation from stochastic optimal control

Abstract The quantum-mechanical formulation of a class of stochastic optimal control problems is investigated and two versions of non-linear Schrodinger equation are derived. First, a wave-function is introduced which connects the probability density function of a diffusion process with the minimum cost functional of the stochastic optimal control problem associated with the diffusion process. It is then demonstrated that this wave-function satisfies a kind of Schrodinger-Langevin equation in quantum physics. Finally, the probabilistic interpretation of the wave-function is discussed.

Fluid formulation of a generalised Schrodinger-Langevin equation

The author recasts a generalised Schrodinger-Langevin equation into a set of hydro-dynamical equations and draws some analogies with a classical (plasma) fluid theory describing the motion of charged particles in a neutralised background.

Quantum fluid dynamics description of a multiterm potential nonlinear Schrodinger-Langevin equation

Via quantum fluid dynamics the author presents a general method to obtain a solution of the nonlinear Schrodinger-Langevin equation with a multiterm potential for the description of interactions in non-conservative systems. The nonlinear complete set of equations forming the basis of the model are derived. The corresponding solutions are exhibited in terms of auxiliary ordinary differential equations which provide relations between the potential coefficients and other coefficients prescribing the fluid-particle behaviour.

Variations on a Wignerian theme

The Wigner distribution and its equation of motion in the scalar potential case are arrived at in an unusual way. This in turn suggests (a) a departure from the standard Wigner distribution treatment for a charged particle in a magnetic field and (b) a new approach to quantization of nonconservative systems. Suggestion (a) is found to be, like the standard treatment, in agreement with Schrodinger's equation but, unlike it, also satisfies local classical-type conservation laws and employs a distribution which is gauge-invariant rather than merely gauge-covariant. Suggestion (b) gives a clear result only in the case of resistance proportional to velocity, when it agrees with the Schrodinger-Langevin equation; for other dissipative systems a fresh assumption is required, and a proposal in that direction is put forward.

Coherent states and the irreversible motion of a particle coupled to a system of oscillators

A many-body system consisting of a central elastically bound particle coupled to a large number of harmonic oscillators is studied. The co-ordinate of this particle satisfies the equation of motion of a damped harmonic oscillator. To get the quantal description of this damped motion, the many-body wave function for the complete system is written in the coherent-state basis,i.e. as a product of a real stationary wave function and a co-ordinate- and time-dependent phase factor. In order to isolate the motion of the central particle, the stationary wave function is approximated by a product of single-particle wave functions. By calculating the contribution of the variable phase factor to the single-particle wave function, it is shown that the resulting equation for the damped motion of an oscillator is the Schrodinger-Langevin equation.

On the quantization of dissipative systems

A recent paper of Dekker on the quantization of dissipative systems is examined in some detail. It is argued that one can construct a large number of classical equivalent Hamiltonians for damped systems. These can be formally quantized according to Dirac's method, and the resulting equations are mathematically consistent, but yield different eigenfunctions for the same classical system. However, this procedure should be rejected on physical grounds. That is in quantum mechanics, unlike classical dynamics, the definition of the time derivative of a dynamical variable is unique, and is given by the commutator of the proper Hamiltonian (or the energy operator) and that variable. If the proper Hamiltonian is used for the quantization of a damped system, then the quantal equations are inconsistent for the cases where the rate of energy dissipation depends on the velocity of the particle. As an alternative approach to the quantal theory of dissipative phenomena, a generalization of the Hamilton-Jacobi formalism is considered, where the equation for the principle functionS, depends not only on the space and time derivatives ofS, but onS itself. This leads to a new class of damped systems in classical mechanics. The original Schrodinger method of quantization via the Hamilton-Jacobi equation has been applied to this class of dissipative systems, with the result that the wave equation in this case is a solution of a non-linear Schrodinger-Langevin equation. This formulation has no analogue in the Hamiltonian approach, since in the latter, the resulting wave equation is always linear.

Friction and dissipative phenomena in quantum mechanics

Frictional and dissipative terms of the Schrodinger equation are studied. A proof is given showing that the frictional term of the Schrodinger-Langevin equation causes the quantum system to lose energy. General expressions are derived for the frictional term of the Schrodinger equation.