Semi classical behaviour of Schrödinger's dynamics revivals of wave packets on

Olivier Lablée

Description

Semi-classical behaviour of Schrödinger's dynamics : revivals of wave packets on hyperbolic trajectory Olivier Lablée 7 April 2010 Abstract The aim of this paper is to study the semi-classical behaviour of Schrödinger's dynamics for an one-dimensional quantum Hamiltonian with a classical hyperbolic trajectory. As in the regular case (elliptic trajectory), we prove, that for an initial wave packets localized in energy, the dynamics follows the classical motion during short time. This classical motion is periodic and the period Thyp is order of |ln h|. And, for large time, a new period Trev for the quantum dynamics appears : the initial wave packets form again at t = Trev. Moreover for the time t = pq Trev a fractionnal revivals phenomenon of the initial wave packets appears : there is a formation of a finite number of clones of the original wave packet. Schrödinger's dynamics, revivals of wave packets, semi-classical analysis, hy- perbolic trajectory, Schrödinger operator with double wells potential. Contents 1 Introduction 1 1.1 Context and motivation . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• initial state


• wave packets


• semi-classical regime


• time scale


• ?0 ?


• dimensional quantum


• co-ro


• quantum dynamics

Sujets

• Wave packet
• Time scale
• Quantum dynamics

Informations

• Publié par: pefav
• Nombre de lectures: 8
• Langue: English

Extrait

Semi-classical behaviour of Schrödinger’s dynamics : revivals of wave packets on hyperbolic trajectory

Olivier Lablée

7 April 2010

Abstract The aim of this paper is to study the semi-classical behaviour of Schrödinger ’s dynamics for an one-dimensional quantum Hamiltonian with a classical hyperbolic trajectory. As in the regular case (elliptic trajectory), we prove, that for an initial wave packets localized in energy, the dynamics follows the classical motion during short time. This classical motion is periodic and the periodT hy p is order of | lnh |  for large time, a new period. And, T rev  the initial wave packets formfor the quantum dynamics appears : again att=T rev . Moreover for the timet= p q T rev a fractionnal revivals phenomenon of the initial wave packets appears : there is a formation of a finite number of clones of the original wave packet. Schrödinger’s dynamics, revivals of wave packets, semi-classical analysis, hy- perbolic trajectory, Schrödinger operator with double wells potential.

Contents

1 Introduction 1.1 Context and motivation . . . . . . . . . . . . . . . . . . . . . . . 1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Paper organization . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Quantum dynamics and autocorrelation function 2.1 The quantum dynamics . . . . . . . . . . . . . . . . . . . . . . . 2.2 Return and autocorrelation function . . . . . . . . . . . . . . . . 2.3 Strategy for study the autocorrelation function . . . . . . . . . .

3 The context of hyperbolic singularity 3.1 Link between spectrum and geometry : semi-classical analysis 3.2 Hyperbolic singularity . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Spectrum near the singularity . . . . . . . . . . . . . . . . . . . .

4

Some preliminaries 4.1 Partials autocorrelations functions . . . . . . . . . . . . . . . . . 4.2 Choice of initial state . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The setΔ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 2 2

3 3 3 4

4 4 5 6

8 8 9 15

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Order 1 approximation : hyperbolic period 16 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.2 Definition of a first order approximation time scale . . . . . . . . 18 5.3 Periodicity of the principal term . . . . . . . . . . . . . . . . . . . 20 5.4 Geometric interpretation of the period . . . . . . . . . . . . . . . 20 5.5 Comparison between hyperbolic period and the time scale  0, | ln(h) | α  21 5.6 Behaviour of autocorrelation function on a hyperbolic period . 21

6 Second order approximation : revival period 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Definition of a new time scale . . . . . . . . . . . . . . . . . . . . 6.3 Full revival theorem . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Fractional revivals theorem . . . . . . . . . . . . . . . . . . . . . 6.5 Explicit values of modulus for revivals coefficients . . . . . . . . 6.6 Comparison between time scale approximation, hyperbolic pe- riod and revivals periods . . . . . . . . . . . . . . . . . . . . . . .

1 Introduction

1.1 Context and motivation

24 24 27 28 31 37

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ForP h a pseudo-differential operator (hereh>0 is the semi-classical parame- ter), and forψ 0 an inital state the quantum dynamics is governed by the famous Schrödinger equation :

ih ∂ ψ∂( t t = ) P h ψ(t).

In this paper, we present a detailed study, in the semi-classical regimeh→0, of the behaviour of Schrödinger’s dynamics for an one-dimensional quantum Hamiltonian P h :D ( P h ) ⊂L 2 ( R ) →L 2 ( R ) with a classical hyperbolic trajectory : the principal symbolp∈ C ∞ (R 2 ,R)of P h has a hyperbolic non-degenerate singularity. Dynamics in the regular case and for elliptic non-degenerate singularity have been the subject of many research in physics[Av-Pe], [LAS], [Robi1], [Robi2], [BKP],[Bl-Ko]and, more recently in mathematics[Co-Ro],[Rob], [Pau1],[Pau2], [Lab2]. The strategy to understand the long times behaviour of dynamics is to use the spectrum of the operatorP h  the regular case, the. In spectrum ofP h is given by the famous Bohr-Sommerfeld rules (see for example [He-Ro], [Ch-VuN],[Col8]) : first approximation, the spectrum of inP h in a compact set is a sequence of real numbers with a gap of sizeh. The classical trajectories are periodic and supported on elliptic curves. In the case of hyperbolic singularity we have a non-periodic trajectory sup- ported on a ”height” figure (see figure 2). The spectrum near this singular- ity is more complicated than in the regular case. Y. Colin de Verdière and B. Parisse give an implicit singular Bohr-Sommerfeld rules for hyperbolic sin- gularity ([Co-Pa1],[Co-Pa2]and[Co-Pa3] quantization formula is too). This implicit for using it directly in our motivation. In[Lab3]we have an explicit

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description of the spectrum for an one-dimesional pesudo-differential operator near a hyperbolic non-degenerate singularity.

1.2 Results

With above description, we propose a study of quantum dynamics for large times ( ≫ |lnh |) a localized initial state, at the begining the. We prove that for dynamics is periodic with a period equal toT hy p =C | lnh | (see corollary 5.12). This periodT hy p corresponds to the classical Hamiltonian flow period. Next for large time scale, a new periodT rev of the quantum dynamics ap- pears : this is the revivals phenomenon (like in regular case[Co-Ro],[Rob], [Pau1],[Pau2], [Lab2]). Fort=T hy p the packet relocalize in the form of a quantum revival. We have also the phenomenon of fractional revivals of initial wave packets for timet= p q T rev , with q p ∈Q is a formation of a finite number of: there clones of the original wave packetψ 0 with a constant amplitude (see theorems 6.18, 6.19 & 6.20) and differing in the phase plane from the initial wave packet by fractions p q T hy p (see theorem 6.15).

1.3 Paper organization

The paper is organized as follows. In section 2 we give some preliminaries about the strategy for analyse the dynamics of a quantum Hamiltonian. In this section we define a simple way to understand the evolution oft7→ψ(t)by the autocorrelation function :

c(t):= |h ψ(t),ψ 0 i H | . In section 3 we describe the hyperbolic singularities mathematical context; we also recall the principal theorem of[Lab3]. This theorem provides the spectrum of the operatorP h near the singularity. 4 is devoted to define an initial Section wave packetsψ 0 localized in energy. In part 5 we prove that the quantum dy- namics follows the classical motion during short time (see corollary 5.12). This classical motion is periodic and the periodT hy p is order of | lnh | . In the last part (part 6) we detail the analysis of revivals phenomenon, see theorem 6.7 for full- revival theorem and see theorem 6.15 for fractionnal-revivals phenomenon.

2

Quantum dynamics and autocorrelation function

2.1 The quantum dynamics

For a quantum HamiltonianP h :D ( P h ) ⊂ H → H,His a Hilbert space, the Schrödinger dynamics is governed by the Schrödinger equation :

∂t) ih ∂ ψ( t =P h ψ(t). With the functional calculus, we can reformulate this equation with the unitary groupU(t) = n e −i h P h o t∈R . Indeed, for a initial stateψ 0 ∈ H, the evolution

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given by :

ψ(t) =U(t)ψ 0 ∈ H.

2.2 Return and autocorrelation function

We now introduce a simple tool to understand the behaviour of the vectorψ(t) : a quantum analog or the Poincaré return function.

Definition.The quantum return functions of the operatorP h and for an initial stateψ 0 is defined by : r(t):= h ψ(t),ψ 0 i H ; and the autocorrelation function is defined by :

c(t):= | r(t) | = |h ψ(t),ψ 0 i H | . The previous function measures the return on initial state. This function is the overlap of the time dependent quantum stateψ(t)with the initial state ψ 0 . Since the initial stateψ 0 is normalized, the autocorrelation function takes values in the compact set[0, 1]. Then, if we have an orthonormal basis of eigen- vectors ( e n ) n∈N : P h e n =λ n (h)e n

with λ 1 (h)≤λ 2 (h) ≤  ≤ λ n (h)→+∞; we get, for all integern  e −i h P h  e n =  e −i h λ n (h)  e n . So for a initial vectorψ 0 ∈D(P h )⊂ H, let us denote by(c n ) n∈N = (c n (h)) n∈N the sequence ofℓ 2 (N)given(c n ) n =π ( ψ 0 ) , whereπis the projector (unitary operator) :  H →ℓ π:   ψ7→<ψ, 2 e ( n N > ) H . Then, for allt≥0 we have e −i h P h  ψ(t) =U(t)ψ 0 = 
n∈ ∑ N c n e n ! = ∑ c n e −i h λ n (h) e n . n∈N So, for allt≥0 we obtain

r(t) = ∑ | c n | 2 e −i h λ n (h) ;c(t) = ∑ | c n | 2 e −i h λ n (h) . n∈N  n∈N 

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