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arXiv:cond-mat/0209347 · The Journal of Chemical Physics

Schrödinger equation for current carrying states

Daniel S. Kosov

Published Apr 8, 2002• 32 citations
arXivHTMLPDF

Abstract

Schrödinger equation with given, a priori known current is formulated. A nonzero current density is maintained in the quantum system via a subsidiary condition imposed by vector, local Lagrange multiplier. Constrained minimization of the total energy on the manifold of an arbitrary current density topology results into a nonlinear self-consistent Schrödinger equation. The applications to electronic transport in two-terminal molecular devices are developed and new macroscopic definition of a molecular current–voltage characteristic is proposed. The Landauer formula for the conductance of an ideal one-dimensional lead is obtained within the approach. The method is examined by modeling of current carrying states of one-dimensional harmonic oscillator.

DOI
10.1063/1.1460857

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arXiv:quant-ph/9710010•Oct 2, 1997•1 citation
Schrödinger equation with given, a priori known current is formulated. A nonzero current density is maintained in the quantum system via a subsidiary condition imposed by vector, local Lagrange multiplier. Constrained minimization of the total energy on the manifold of an arbitrary current density topology results into a nonlinear self-consistent Schrödinger equation. The applications to electronic transport in two-terminal molecular devices are developed and new macroscopic definition of a molecular current–voltage characteristic is proposed. The Landauer formula for the conductance of an ideal one-dimensional lead is obtained within the approach. The method is examined by modeling of current carrying states of one-dimensional harmonic oscillator.
We consider the electron transport properties through fully interacting nanoscale junctions beyond the linear-response regime. We calculate the current flowing through an interacting region connected to two interacting leads, with interaction crossing at the left and right contacts, by using a non-equilibrium Green function technique. The total current at one interface (the left one for example) is made of several terms which can be regrouped into two sets. The first set corresponds to a very generalized Landauer-like current formula with physical quantities defined only in the interacting central region and with renormalized lead self-energies. The second set characterizes inelastic scattering events occurring in the left lead. We show how this term can be negligible or even vanish due to the pseudo-equilibrium statistical properties of the lead in the thermodynamic limit. The expressions for the different Green functions needed for practical calculations of the current are also provided. We determine the constraints imposed by the physical condition of current conservation. The corresponding equation imposed on the different self-energy quantities arising from the current conservation is derived. We discuss in detail its physical interpretation and its relation with previously derived expressions. Finally several important key features are discussed in relation to the implementation of our formalism for calculations of quantum transport in realistic systems.
We regard a current flow through an open one-dimensional quantum system which is determined by a dissipative Schrödinger operator. The imaginary part of the corresponding form originates from Robin boundary conditions with certain complex valued coefficients imposed on Schrödinger’s equation. This dissipative Schrödinger operator can be regarded as a pseudo-Hamiltonian of the corresponding open quantum system. The dilation of the dissipative operator provides a (self-adjoint) quasi-Hamiltonian of the system, more precisely, the Hamiltonian of the minimal closed system which contains the open one is used to define physical quantities such as density and current for the open quantum system. The carrier density turns out to be an expression in the generalized eigenstates of the dilation while the current density is related to the characteristic function of the dissipative operator. Finally a rigorous setup of a dissipative Schrödinger–Poisson system is outlined.
The constrained energy functional, defined by E0(a) =minSa{a〈Ψ‖Ĥ0‖Ψ〉a}, ai=a〈Ψ‖Âi‖Ψ〉a, i=1, ..., M, ‖Ψ〉a∈Sa, with a={ai} a preassigned set of expectation values generated by any state vector ‖Ψ〉a in the class Sa, is shown to satisfy a first order, but nonlinear, partial differential equation. The equation results by replacing the Lagrange multipliers introduced in an earlier paper by the corresponding partial derivatives of the to-be-found energy functional with respect to the constraints. Two alternative expressions for E0(a), one focusing upon the ground state eigenvalue of a modified Hamiltonian operator and the other upon the corresponding eigenvector, lead via standard Rayleigh–Schrödinger perturbation theory to two approximate partial differential equations for the energy functional. Both equations together with the imposed subsidiary conditions lead in turn to the same approximate E0(a). An example based on the linear harmonic oscillator illustrates the concepts presented in the formalism.
A net current flow through an open 1-dimensional Schrödinger-Poisson system is modeled by replacing self-adjoint boundary conditions by dissipative ones. To give a rigorous definition of carrier and current densities the well-known dilation theory of dissipative operators is used where the self-adjoint dilation is regarded as the Hamiltonian of a larger closed system which contains the open one. The carrier density turns out to be performed by the generalized eigenstates of the dilation while the current density is related to the characteristic function of the dissipative operator. A rigorous setup of a dissipative Schrödinger-Poisson system is outlined.
, which are relevant to properly describe phenomena such as molecular dissociation with density-functional theory, to the case in which the electronic wavefunction corresponds to a complex current-carrying state. In such a case, an extra term in the form of a vector potential appears which cannot be gauged away. Both scalar and vector potentials are introduced via the exact factorization formalism which allows us to express the given Schrödinger equation as two coupled equations, one for the marginal and one for the conditional amplitude. The electronic vector potential is directly related to the paramagnetic current density carried by the total wavefunction and to the diamagnetic current density in the equation for the marginal amplitude. An explicit example of this vector potential in a triplet state of two non-interacting electrons is showcased together with its associated circulation, giving rise to a non-vanishing geometric phase. Some connections with the exact factorization for the full molecular wavefunction beyond the Born-Oppenheimer approximation are also discussed.
A nonlinear modification of the Schrödinger equation is proposed in which the Lagrangian density for the Schrödinger equation is extended by terms polynomial in Δmln⁡(Ψ∗/Ψ)Δ^{m}\ln (Ψ^{*}/Ψ)Δmln(Ψ∗/Ψ) multiplied by Ψ∗ΨΨ^{*}ΨΨ∗Ψ. This introduces a homogeneous nonlinearity in a Galilean invariant manner through the phase SSS rather than the amplitude RRR of the wave function Ψ=Rexp⁡(iS)Ψ=R\exp (iS)Ψ=Rexp(iS). From this general scheme we choose the simplest minimal model defined in some reasonable way. The model in question offers the simplest way to modify the Bohm formulation of quantum mechanics so as to allow a leading phase contribution to the quantum potential and a leading quantum contribution to the probability current removing asymmetries present in Bohm's original formulation. It preserves most of physically relevant properties of the Schrödinger equation including stationary states of quantum-mechanical systems. It can be thought of as the simplest model of nonlinear quantum mechanics of extended objects among other such models that also emerge within the general scheme proposed. The extensions of this model to nnn particles and the question of separability of compound systems are studied. It is noted that there exists a weakly separable extension in addition to a strongly separable one. The place of the general modification scheme in a broader spectrum of nonlinear modifications of the Schrödinger equation is discussed. It is pointed out that the models it gives rise to have a unique definition of energy in that the field-theoretical energy functional coincides with the quantum-mechanical one. It is found that the Lagrangian for its simplest variant represents the Lagrangian for a restricted version of the Doebner-Goldin modification of this equation.