Quantum Moment Research Articles

Semiconductor Simulations Using a Coupled Quantum Drift‐Diffusion Schrödinger–Poisson Model

A coupled quantum drift‐diffusion Schrodinger–Poisson model for stationary resonant tunneling simulations in one space dimension is proposed. In the ballistic quantum zone with the resonant quantum barriers, the Schrodinger equation is solved. Near the contacts, where collisional effects are assumed to be important, the quantum drift‐diffusion model is employed. The quantum drift‐diffusion model was derived by a quantum moment method from a collisional Wigner equation by Degond et al. [J. Statist. Phys., 118 (2005), pp. 625–665]. The derivation yields an $O(\hbar^4)$ approximation of the equilibrium Wigner function which is used as the “alimentation function” in the mixed‐state formula for the electron and current densities at the interface. The coupling of the two models is realized by assuming the continuity of the electron and current densities at the interface points. Current‐voltage characteristics of a one‐dimensional tunneling diode are numerically computed. The results are compared to those from t...

Differentiability of quantum moment maps and G - invariant star products

We study quantum moment maps of G-invariant star products, a quantum analogue of the moment map for classical Hamiltonian systems. Introducing an integral representation, we show that any quantum moment map for a G-invariant star product is differentiable. This property gives us a new method for the classification of G-invariant star products on regular coadjoint orbits of compact semisimple Lie groups.

Some remarks on [formula omitted]-invariant Fedosov star products and quantum momentum mappings

In these notes we consider a slightly generalized Fedosov star product ∗ on a symplectic manifold (M,ω), emanating from the fibrewise Weyl product ∘ and the triple (∇,Ω,s) consisting of a symplectic torsion free connection ∇ on M, a formal series Ω∈νZ2dR(M)[[ν]] of closed two-forms on M, and a certain formal series s of symmetric contravariant tensor fields on M. We prove necessary and sufficient conditions for certain classical symmetries to become symmetries of the star product, only sufficient conditions having been published in special cases when this letter was written (note, however, the different proofs in [S. Gutt, J. Rawnsley, Natural star products on symplectic manifolds and quantum moment maps, 2003. math.SG/0304498 v1]). For a given symplectic vector field X on M, it is well known that LXΩ=[LX,∇] (=LXs)=0 is a sufficient condition for the Lie derivative LX to be a derivation of ∗. We prove that these conditions are in fact necessary ones, also providing a very simple proof for their being sufficient. Moreover, we prove a criterion that has first been presented by Gutt [S. Gutt, Star products and group actions, Contribution to the Bayrischzell Workshop, April 26–29, 2002] (see also [S. Gutt, J. Rawnsley, Natural star products on symplectic manifolds and quantum moment maps, 2003. math.SG/0304498 v1] for a different proof) and which specifies a necessary and sufficient condition for LX to be a quasi-inner derivation. The statement that this condition is a sufficient one dates back to Kravchenko [O. Kravchenko, Compos. Math. 123 (2000) 131]. Applying our results, we find necessary and sufficient criteria for a Fedosov star product to be g-invariant and to admit a quantum Hamiltonian. Finally, supposing the existence of a quantum Hamiltonian, we present a cohomological condition on Ω that is equivalent to the existence of a quantum momentum mapping. In particular, our results show that the existence of a classical momentum mapping in general does not imply the existence of a quantum momentum mapping and thus give a negative answer to Xu’s question posed in [P. Xu, Commun. Math. Phys. 197 (1998) 167].

Natural Star Products on Symplectic Manifolds and Quantum Moment Maps

We define a natural class of star products: those which are given by a series of bidifferential operators which at order $k$ in the deformation parameter have at most $k$ derivatives in each argument. We show that any such star product on a symplectic manifold defines a unique symplectic connection. We parametrise such star products, study their invariance and give necessary and sufficient conditions for them to yield a quantum moment map. We show that Kravchenko's sufficient condition for a moment map for a Fedosov star product is also necessary.

A Note on quantum moment hydrodynamics and the entropy principle

In this Note, we show how a non-commutative version of the entropy extremalization principle allows one to construct new quantum hydrodynamic models. To cite this article: P. Degond, C. Ringhofer, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 967–972.

QUANTUM MOMENT MAPS AND INVARIANTS FOR G-INVARIANT STAR PRODUCTS

We study a quantum moment map and propose an invariant for G-invariant star products on a G-transitive symplectic manifold. We start by describing a new method to construct a quantum moment map for G-invariant star products of Fedosov type. We use it to obtain an invariant that is invariant under G-equivalence. In the last section we give two simple examples of such invariants, which involve non-classical terms and provide new insights into the classification of G-invariant star products.

A New Invariant For G-Invariant Star Products

The purpose of this Letter is to propose an invariant for a G-invariant star product on a G-transitive symplectic manifold which remains invariant under the G-equivalence maps. This invariant is defined by using a quantum moment map which is a quantum analogue of the moment map on a Hamiltonian G-space. On S2 regarded as an SO(3) coadjoint orbit in \( \mathfrak{s}\mathfrak{o}\left( 3 \right)* \), we give an example of this invariant for the canonical G-invariant star product. In this example, there arises a nonclassical term which depends only on a class of G-invariant star products.

Closure conditions for classical and quantum moment hierarchies in the small-temperature limit

Abstract We analyze closure conditions in the small initial temperature limit for classical and quantum mechanical moment hierarchies of corresponding collisionless kinetic equations. Euler equations with a nondiagonal pressure tensor are obtained in the classical case. In the quantum case we consider the cases of fixed and small (scaled) Planck constant and derive Quantum Hydrodynamic equations.

Modeling of quantum effects in ultrasmall HEMT devices

Numerical simulation of ultra-submicrometer high electron mobility transistors based upon a set of quantum moment equations is presented. These provide a first quantum description, based upon the moments of the Wigner distribution function. In HEMTs, the conduction electrons are confined in a narrow conduction channel and the short gate lengths (and small aspect ratio) create different potential barriers across the conduction channel than in a long-channel situation. In these small structures, quantum effects are expected to be prominent. A substantial change in the electron density distribution is found to occur due to the inclusion of these quantum corrections, and the total current in the simulated devices is increased by as much as 10% for a 240-nm gate-length device.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Ballistic phenomena in GaAs MESFETs: modelling with quantum moment equations

The physical effects inherent in the operation of ultra-small devices are based on the fact that the critical length (e.g. the gate length or the depletion length) becomes so small that it approaches the coherence length of the electrons that provide the operation, which suggests that such small devices must be treated as quantum-mechanical objects. The accurate simulation of ultra-small devices requires quantum effects such as tunnelling and quantum repulsion (complementary to barrier penetration) to be included. The authors have developed a numerical model based upon a full quantum description-the moments of the Wigner distribution function. Numerical simulation of ultra-small MESFETs has been carried out using this model. Here, they emphasize the hot carrier effects such as ballistic transport and velocity overshoot. They also investigate the change of the velocity due to the quantum effects.

Simulation of ultra-small GaAs MESFET using quantum moment equations

Ultra-small MESFETs have characteristic lengths comparable to quantum lengths: wavelength, mean free path, etc. In a first attempt to incorporate these quantum lengths, the authors develop a model based upon a set of quantum moment equations obtained from the Wigner function equation-of-motion. Interesting time-dependent current oscillation behavior has been observed when a step voltage is applied to the initial steady state. The oscillation frequency is peaked around 500 GHz, which is related to the plasma response of the carriers in the channel. Quantum effects, such as barrier repulsion and penetration, have been demonstrated in the simulation. These effects modify the electron density distribution and current density distribution both in the channel and near the source. Modifications of the frequency spectrum of the oscillation current due to the quantum effects are obvious. >

Positive elements in the algebra of the quantum moment problem

Let\(\mathfrak{A}\) denote the extended Weyl algebra,\(\mathfrak{A}_0 \subset \mathfrak{A}\), the Weyl algebra. It is well known that every element of\(\mathfrak{A}\) of the formA=ΣBk*Bk is positive. We prove that the converse implication also holds: Every positive elementA in\(\mathfrak{A}\) has a quadratic sum factorization for some finite set of elements (Bk) in\(\mathfrak{A}\). The corresponding result is not true for the subalgebra\(\mathfrak{A}_0 \). We identify states on\(\mathfrak{A}_0 \) which do not extend to states on\(\mathfrak{A}\). It follows from a result of Powers (and Arveson) that such states on\(\mathfrak{A}_0 \) cannot be completely positive. Our theorem is based on a certain regularity property for the representations which are generated by states on\(\mathfrak{A}\), and this property is not in general shared by representations generated by states defined only on the subalgebra\(\mathfrak{A}_0 \).

Non-commutative moment problems

We are concerned with two problems, called the generalized moment problem (GMP) and the quantum moment problem (QMP). Both problems can be considered as non-commutative versions of the classical moment problem et G be a connected Lie group and #7B-E(g) the universal enveloping algebra of the Lie algebra of G. Suppose f is a given linear functional on #7B-E(g) and ν is a fixed unitary representation of G. We then ask: (GMP) When do there exist a unitary representation U of G and a cyclic C ∞ -vector φ for U such that f(x)= , x∈#7B-E(g)? (QMP) When is f of the form f(x)=trdV(x)ρ, x∈#7B-E(g), where ρ is an operator such that dV(x)ρ is of trace class for every x∈#7B-E(g)?

Quantum moment balance equations and resonant tunnelling structures

This study describes the evolution and implementation of a set of quantum balance equations for examining transport im mescoscopic structures

A comparison of classical and quantum mechanical calculations of vibrational energy transfer. III. Use of the forced quantum oscillator and moment methods for quantization of product state distributions

Vibrational transition probabilities calculated using the semiclassical forced quantum oscillator model and the information theoretic moment method were compared with exact quantal results in two model systems. In both cases the moment method was generally more accurate. Under conditions leading to a large distortion of the molecular force constant, the forced oscillator model was found to be in significant error. At low collision energies with two or three open states the moment method was inaccurate, and the forced oscillator model is preferred.