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Phase Space Approach to Quantum Dynamics

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Phase Space Approach to Quantum Dynamics Phase Space Approach to Quantum Dynamics P. LEBŒUF Division de Physique Théorique* Institut de Physique Nucléaire 91406 Orsay Cedex, France : We replace the Schrôdinger equation for the time propagation of states of a quantised two-dimensional spherical phase space by the dynamics of a system of N particles lying in phase space. This is done through factorisation formulae of analytic function theory arising in coherent-state representation, the "particles" being the zeros of the quantum state. For linear Hamiltonians, like a spin in a uniform magnetic field, the motion of the particles is classical. However, non-linear terms induce interactions between the particles. Their time propagation is studied and we show that, contrary to integrable systems, for chaotic maps they tend to fill, as their classical counterpart, the whole phase space. IPNO/TH 91-20 ' Mars 1991 " Unité de recherche des Universités de Paris XI et Paris VI associée au CNRS 1 Introduction In classical mechanics, when an integrable system is perturbed some invariant tori are broken and replaced by small layers where chaotic motion takes place. This process is amplified as the perturbation increases, the way this transition occurs being described by the KAM theorem. We do not have an equivalent quantum mechanical description, and our understanding of the mathematical and physical implications of the system being classically chaotic is still rudimentary [I]. Recently, P. Leboeuf and A. Voros [2] proposed a representation of quantum states particularly well adapted to semiclassical analysis. For compact phase spaces such as the two-dimensional torus or the sphere <S2, wave functions in coherent-state represen- tation exhibit a finite number N of zeros in phase space, and their location completely determines the state of the system. Moreover, they observed that the distribution of zeros for eigenstates of quantised systems reflects, in the semiclassical regime h —> oo, the nature of the underlying classical dynamics: it is one-dimensional for eigenstates of integrable systems while it tends to spread all over phase space in the case of a classically chaotic dynamics. The purpose of this paper is to develop this representation, and consider its dy- namical aspects. Since the zeros completely determine the state of the system, we can interpret quantum mechanics as a (classical) system of N particles - the zeros - lying in phase space. The nature of their interaction and the presence of external fields acting on them will depend on the Hamiltonian defining the system. A stationary state will be associated, in this picture, to an "equilibrium" configuration of the system of particles, and there will be as many of them as stationary states. However, an arbitrary initial configuration of zeros will in general evolve in time, in the same way that an arbitrary initial state | i/»(0) ) will evolve in time according to the Schrodinger equation. In section 3 we re-write the Schrodinger equation in this new picture, and determine the equations of motion governing the time propagation of the zeros. In doing that, we will clarify the nature of their interaction and the possible presence of external fields. For simplicity, we will restrict ourselves to the case of Hamiltonians written in terms of the generators of the SU(2) group - a spin system -, for which phase space is a two-dimensional sphere and the analytic coherent-state representation of Hilbert space consists of ordinary polynomials of degree 2s. A review is given in section 2. Finally, in section 4 we illustrate the results considering two examples. 2 The SU(2) Coherent-State Representation We recall first some basic formulae arising in the context of SU(2) coherent-state rep- resentation. Consider a Hamiltonian H = H(Sz,S+,S~,t) written in terms of the generators of the SU(2) group, satisfying the algebra [S2 ; S±\ — [S. ; S+] = -2hSz. (2.1) Since H commutes with S2, the Hilbert space Ti is (2s + 1 )-dimensional and spanned by the usual discrete basis S. \m•} = mti \m) m = —s, • • •, s (2.2) S2|m) = s(s + l)fc2|m), while the phase space P is a two-dimensional sphere (The Riemann sphere). We hence- forth take h (2.3) so that the radius of the sphere is fixed to one. With this convention, the phase space can then be labelled by polar and azimutal angles (B,(j>). Moreover, the semiclassical limit corresponds now to s —> oo. An alternative representation of Ti in terms of analytic functions is obtained through the spin coherent-states [3] |r) =e«+/*| _5), Z = COt^e* (2.4) having the norm (z\z) = (l + zz)2'. The complex variable z spanning V corresponds to a stereographic projection of the Riemann sphere onto the plane by the north pole. The state I 2) is a minimum-uncertainty packet on V centered at z = (6,tj)) with a width O(hlt2) in both directions, thus coherent-states are the most "classical" quantum-states of angular momentum. They have the property <•!->= (.?„) •*" (2-5) so that the coherent-state decomposition of a state 11/> ) = Sl3 am\m) oîH am (2.6) is a polynomial of degree 2s. It has therefore JV = 2s complex zeros {zk}^... N on the sphere. We now exploit this analytic structure: the standard factorisation formula W = cfc-4 (2-7) where C is a normalisation constant, provides a one-to-one mapping between states | if) ) of H and configurations {zk}k=l...N °f zeros in V. The spin quantum state is therefore uniquely determined (up to a constant factor) by a set {z*} of N — Is points on V. This representation was already known to E. Majorana [4] in 1932. As he has suggested, we may associate to each zero zj. corresponding to a point P on the Riemann sphere a unit vector OP, 0 being the center of the sphere. The quantum state of the spin s particle (which classically corresponds to a single arrow pointing in a definite direction) is then "composed" of a set of Is unit vectors (spins) pointing in arbitrary directions |O.Pjb} . As we show below, a coherent-state is a very particular state for which the 2s spins point in the same direction. In spite of being particularly simple for spin systems, a similar parametrisation by the zeros can be applied to other spaces of entire functions; in particular, this has being done in [2] for the case of the two-dimensional toric phase space. Moreover, an analog factorisation structure has also being exploited in the context of the quantum Hall effect [5,6] and also in two-dimensional electrons in a periodic magnetic field [7]. Some relevant features of this exact parametrisation are [2] (i) the number JV = 2s of zeros is controlled by the value of fi. (or viceversa, cf.Eq.(2.3)), and tends to infinity in the semiclassical limit (i.e., a thermodynamic limit); (ii) each zero constitutes a topologica) defect of ^>(~), since doing a small closed path around a (non-degenerate) zero changes its phase by 2TT; (iii) the Husimi distribution (z|z) (1 + zz)2' has the same zeros as V'(2)- Thus, this positive-definite real function defined on V carries the full quantum information of the state of the system through its zeros. 2 Since ( z \ z0 ) = (1 + zoz) ' [3], a wave-packet | z0 ) has a particularly simple repre- 2 sentation: it is defined by placing the 2s zeros at the single point z = — 2C/|zo| in V, the opposite point to z0 on the sphere. Finally, in this representation, the generators (2.1) have the following realization by differential operators ( z I S21 ip ) = h(zd, — s)ip(z) {z\S+\ip) =hz(2s -zdz)ii>(z) (2.9) 3 The Equations of Motion Our purpose now is to replace the Schrodinger equation ihdt \rp) = H \i/>} for the time evolution of a state | ip(t) ) of Ti. by an equivalent equation for the time evolution of the zeros. The coherent-state representation of the Schrodinger equation can be written ihdMz,t) = H(z,dz)i>{z,t). (3.1) Given the Hamiltonian if as a function of the generators SZ,S+,S-, the symbol H(z,d:) defined as {z\H\ijf) = H(z,dz)tl>{z) is easily obtained using Eqs.(2.9). We will assume that it has being normal ordered, i.e., all the dz have being placed to the right of the z variables. If at a time i the state has a zero at a certain phase space point z — Zk, then the condition ip(zk + Szk,t + St) = 0 implies Sz _ d 4> k t N (3.2) Using Eq.(3.1) we get (3.3) where the function computed on the right hand side must be evaluated at z = Zfc(£), the position of the zero at time t. These equations determine the dynamics of each of the JV zeros. However, it is not fully written in terms of them, since we still use the function ip(z) to calculate zk. Let us now assume a polynomial expansion for the symbol H(z,dz) n n where dz means the nth derivative with respect to z and fn{z) are arbitrary functions depending on the functional form of H. The higher meaningful order n in Eq.(3.4) is n* = 2s since d"il>(z) = 0 for n > n*. The operator dz can be interpreted as a destruction of one zero since it lowers the degree of the polynomial by one (further evaluation of the function at z = zk annihilates the fcth zero). Conversely, multiplication by (z — z0) creates a zero at z = ZQ.
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