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Discrete and Continuous Dynamical Systems - S

Semiclassical limit of Husimi function

Hassan Emamirad, Philippe Rogeon

Published Dec 19, 2012• 0 citations
Publisher

Abstract

We will show that Liouville and quantum Liouville operators $L$ and $L_\hbar$ generate two $C_0$-groups $e^{tL}$ and $e^{tL_h}$ of isometries in $L^2(\mathbb{R}^{2n})$ and $e^{tL_h}$ converges ultraweakly to $e^{tL}$. As a consequence we show that the Gaussian mollifier of the Wigner function, called Husimi function, converges in $L^1(\mathbb{R}^{2n})$ to the solution of the Liouville equation.

DOI
10.3934/dcdss.2013.6.669

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Abstract. We establish the difference between the propagation of semiclassical Wigner functions and classical Liouville propagation. First we re-discuss the semiclassical limit for the propagator of Wigner functions, which on its own leads to their classical propagation. Then, via stationary phase evaluation of the full integral evolution equation, using the semiclassical expressions of Wigner functions, we provide the correct geometrical prescription for their semiclassical propagation. This is determined by the classical trajectories of the tips of the chords defined by the initial semiclassical Wigner function and centered on their arguments, in contrast to the Liouville propagation which is determined by the classical trajectories of the arguments themselves. 1. introduction The Wigner [11] function W is a representation of the quantum density operator ˆρ as a real function on phase space, for a non-relativistic dynamical system with classical-quantum correspondence and n degrees of freedom. Here, all the underlying geometry is euclidean and we can include the Wigner function within the general framework of Weyl representation of operators: Â → A ∈ Ck IR (IR2n), so W(x) = (2π�) −n ρ(x) , where x ≡ (p1,...pn, q1,...qn) denotes a point in phase space. In this context, the quantum evolution of the density operator ˆρt effected by an autonomous external hamiltonian
We establish the difference between the propagation of semiclassical Wigner functions and classical Liouville propagation. First we rediscuss the semiclassical limit for the propagator of Wigner functions, which on its own leads to their classical propagation. Then, via stationary phase evaluation of the full integral evolution equation, using the semiclassical expressions of Wigner functions, we provide the correct geometrical prescription for their semiclassical propagation. This is determined by the classical trajectories of the tips of the chords defined by the initial semiclassical Wigner function and centred on their arguments, in contrast to the Liouville propagation which is determined by the classical trajectories of the arguments themselves.
我们发现一个新κ-参数挤压了协调州的| p ， q > _κ 表示，它拥有行为优良特征，即，它的 Wigner 功能“ s 边缘的分发在”q方向“并且在里面”p方向“是 Gaussian 形式 exp {-κ(q''- q )~ 2 }，并且 exp {( p ”-p)~2/κ},分别地。基于这， | p 的 Husimi 功能， q > _ κ也被获得，它是 Gaussian 拓宽 Wigner 功能的版本。| p， q > _ κ状态为学习 Husimi 操作员的各种各样的性质提供一个好代表性的空格。
We study the properties of the Wigner transform for arbitrary functions in L^2 for hermitian kernels like the so-called density matrices. And we introduce some limits of these transforms for sequences of functions in L^2 , limits that correspond to the semi-classical limit in Quantum Mechanics. The measures we obtain in this way, that we call Wigner measures, have various mathematical properties that we establish. In particular, we prove they satisfy, in linear situations (Schrödinger equations) or nonlinear ones (time-dependent Hartree equations), transport equations of Liouville or Vlasov type.
Le passage de l'equation de schrodinger a l'equation de wigner conduit aux egalites d'invariance, en norme, entre les solutions des problemes de schrodinger, liouville-von neumann et husimi. L'objet de cette these est d'etudier differents aspects de ces egalites, dans le cadre de la theorie de scattering (diffusion). Pour le probleme de wigner, nous etablissons un resultat d'existence et de completude des operateurs d'onde, a partir de resultats concernant ces operateurs pour l'equation de schrodinger. Dans le cas de l'equation de liouville-von neumann, nous avons travaile sur l'espace des operateurs a trace, et nous montrons egalement l'existence des operateurs d'onde, pour les semi-groupes dynamiques quantiques. Enfin, nous montrons que le probleme de husimi est mal pose malgre la positivite de la solution. Nous contournons cette difficulte en faisant appel a la theorie des familles d'existence de delaubenfels, qui est concue pour les problemes mal poses, et tout a fait adaptee a notre cas.