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Wigner Phase-Space Distribution As a Wave Function

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Wigner Phase-Space Distribution As a Wave Function PHYSICAL REVIEW A 88, 052108 (2013) Wigner phase-space distribution as a wave function Denys I. Bondar,* Renan Cabrera,† Dmitry V. Zhdanov,‡ and Herschel A. Rabitz§ Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA (Received 17 December 2012; revised manuscript received 5 August 2013; published 11 November 2013) We demonstrate that the Wigner function of a pure quantum state is a wave function in a specially tuned Dirac bra-ket formalism and argue that the Wigner function is in fact a probability amplitude for the quantum particle to be at a certain point of the classical phase space. Additionally, we establish that in the classical limit, the Wigner function transforms into a classical Koopman–von Neumann wave function rather than into a classical probability distribution. Since probability amplitude need not be positive, our findings provide an alternative outlook on the Wigner function’s negativity. DOI: 10.1103/PhysRevA.88.052108 PACS number(s): 03.65.Ca, 03.65.Sq, 03.67.Ac I. INTRODUCTION II. BACKGROUND In his seminal work [1], Wigner defined the combined Around the time the Wigner distribution was conceived, distribution of the quantum particle’s coordinate and momen- Koopman and von Neumann [27–30] (for modern develop- tum in terms of the wave function. Since then, the Wigner ments and applications, see Refs. [30–44]) recast classical function has played a paramount role in the phase space mechanics in a form similar to quantum mechanics by formulation of quantum mechanics [2–4], is a standard tool introducing classical complex valued wave functions and for establishing the quantum-to-classical interface [2,5–7], representing associated physical observables by means of com- and has a broad range of applications in optics and signal muting self-adjoint operators. In particular, it was postulated processing [8,9] as well as quantum computing [10–16]. that the wave function |(t) of a classical particle obeys the Techniques for the experimental measurement of the Wigner following equation of motion: function are also developed [7,17–19]. Despite its ubiquity, the d pˆ Wigner function is haunted by the obscure feature of possibly i |(t)=Lˆ |(t), Lˆ = λˆ x − U (xˆ)λˆ p, (1) being negative. Wigner [20] demonstrated that his function is dt m the only one satisfying a reasonable set of axioms for a joint [x,ˆ λˆ x ] = [p,ˆ λˆ p] = i, probability distribution. This feature of the Wigner function (2) x,ˆ pˆ = x,ˆ λˆ = p,ˆ λˆ = λˆ ,λˆ = . has been a subject of numerous interpretations [13,21–23], [ ] [ p] [ x ] [ x p] 0 including the development of a mathematical framework for Without loss of generality one-dimensional systems are con- handling negative probabilities [24,25]. The Wigner function’s sidered throughout. Since the self-adjoint operators represent- negativity was also associated with the exponential speedup in ing the classical observables of coordinate xˆ and momentum pˆ quantum computation [14,16]. commute, they share a common set of orthogonal eigenvectors In this paper we provide insight into the negativity by |px such that 1 = dpdx |pxpx|. In the KvN classical advocating the following interpretation: The Wigner func- mechanics, all observables are functions of xˆ and pˆ.The tion is a probability amplitude for a quantum particle to expectation value of an observable Fˆ = F (x,ˆ pˆ) at time t is be at a certain point of the classical phase space, i.e., (t)|Fˆ|(t). The probability amplitude px|(t) for a the Wigner function is a wave function analogous to the classical particle to be at point x with momentum p at time t Koopman–von Neumann (KvN) wave function of a classical is found to satisfy particle. The remainder of the paper is organized as follows: The ∂ p ∂ ∂ + − U (x) px|(t)=0. (3) necessary background on the KvN classical mechanics and ∂t m ∂x ∂p operational dynamical modeling [26] is reviewed in Sec. II. The basic equations, on which our interpretation rests, are This is the evolution equation for the classical wave function in = ˆ =− = derived in Secs. III to V. A connection between these equations the xp representation, where xˆ x, λx i∂/∂x, pˆ p, and ˆ =− and quantum mechanics in phase space is established in λp i∂/∂p in order to satisfy the commutation relations (2). Sec. VI. Conclusions are drawn in Sec. VII. Utilizing the chain rule and Eq. (3), we obtain the well known classical Liouville equation for the phase space probability distribution ρ(x,p; t) =|px|(t)|2, ∂ p ∂ ∂ + − U (x) ρ(x,p; t) = 0. (4) ∂t m ∂x ∂p *[email protected] †[email protected] Newtonian trajectories emerge as characteristics of either ‡[email protected]; Present address: Department Eq. (3) or (4). Hence, the essential difference between KvN and of Chemistry, Northwestern University, Evanston, Illinois 60208, Liouville mechanics lies in weighting individual trajectories USA (see Fig. 1): Arbitrary complex weights underlying the classi- §[email protected] cal wave function | can be utilized in KvN mechanics (3), 1050-2947/2013/88(5)/052108(6)052108-1 ©2013 American Physical Society BONDAR, CABRERA, ZHDANOV, AND RABITZ PHYSICAL REVIEW A 88, 052108 (2013) [ˆx , pˆ ]=i κ [ˆx, pˆ]=0 q q [ˆx, pˆ]=i 0 κ 1 FIG. 1. (Color online) The conceptual difference between (a) Li- ouville and (b) KvN classical mechanics. In the Liouville mechanics [Eq. (4)], classical particles, moving along Newtonian trajectories, are FIG. 2. (Color online) Schematic flow showing the derivation of tagged by positive weights according to the probability distribution quantum mechanics, classical mechanics, and quantum mechanical ρ. In the KvN mechanics [Eq. (3)], classical particles, following the phase space representation within the operational dynamical model- same trajectories, are tagged by arbitrary real (or complex) weights ing proposed in [26]. representing the KvN classical wave function |. whereas only positive weights having probabilistic meaning and limκ→0 Hˆ qc = h¯ Lˆ . See Fig. 2 for a pictorial summary of are permitted in Liouville mechanics (4). these derivations presented in [26]. Note that the classical wave function and the classical A crucial point for our current analysis is that this unified probability distribution satisfy the same dynamical equation, wave function |κ (t is dropped henceforth) in the xp which reflects the physical irrelevance of the phase of a representation is proportional to the Wigner function W (see classical wave function. Ref. [26]), In recent work [26], we put forth operational dynamical √ modeling as a systematic theoretical framework for deducing px|κ = 2πhκW¯ (x,p), equations of motion from the evolution of average values. (7) First, starting from the Ehrenfest theorems [45], we obtained dy y y W(x,p) = ρ x − ,x + eipy/(¯hκ); the Schrodinger¨ equation if the momentum and coordinate 2πhκ¯ κ 2 2 operators obeyed the canonical commutation relation, and the moreover, KvN equation (1) if the momentum and coordinate operators commuted. Then, applying the same technique to the Ehrenfest √ xλ | = hκρ¯ (u,v), theorems, p κ κ ∂ (¯hκ)2 ∂2 ∂2 d ihκ¯ − − − U(u) + U(v) ρ (u,v) = 0, 2 2 κ m κ (t)|xˆq |κ (t)=κ (t)|pˆq |κ (t), ∂t 2m ∂v ∂u dt (8) (5) = − = + d u x hκλ¯ p/2,v x hκλ¯ p/2. (t)|pˆ | (t)= (t)|−U (xˆ )| (t), dt κ q κ κ q κ with a generalization [xˆq ,pˆq ] = ihκ¯ (0 κ 1) and demanding a smooth classical limit κ → 0, we established III. DERIVATION OF MAIN RESULTS the existence of the uniquely defined operator Hˆ qc such that Since xˆ and λˆ p are commuting self-adjoint operators, the following resolution of the identity is valid in the Hilbert phase d | =Hˆ | ih¯ κ (t) qc κ (t) , space H (i.e., a vector space with the scalar product | dt xp h¯ 1 hκ¯ of functions of two variables): Hˆ = pˆλˆ + U xˆ − λˆ qc x p ˆ = | | | ∈H m κ 2 1Hxp dxdλp xλp xλp , xλp xp , (9) 1 hκ¯ | = | ˆ | = | − U xˆ + λˆ , xˆ xλp x xλp , λp xλp λp xλp ; κ 2 p likewise, xˆq = xˆ − hκ¯ λˆ p/2, pˆq = pˆ + hκ¯ λˆ x/2, (6) ˆ = | | | ∈H 1Hxp dλx dp λx p λx p , λxp xp , where xˆq and pˆq represent the quantum coordinate and (10) pˆ|λx p=p|λxp, λˆ x |λxp=λx|λx p. momentum, respectively, xˆ, pˆ, λˆ x , and λˆ p are the same classical operators as in Eq. (2), and κ denotes the degree of A consequence of the commutators [x,ˆ λˆ x ] = [p,ˆ λˆ p] = i is quantumness or commutativity: κ → 1 corresponds to quan- ipλp −ixλx tum mechanics whereas κ → 0 recovers classical mechanics, λx p|xλp=e /(2π). 052108-2 WIGNER PHASE-SPACE DISTRIBUTION AS A WAVE ... PHYSICAL REVIEW A 88, 052108 (2013) Let Hq denote the Hilbert space of single particle wave Comparing Eqs. (8) and (13), we conclude that functions (i.e., a vector space with the scalar product || √ | = = || || = of functions of one variable), which differs from the Hilbert xλp κ hκ¯ xq u ρˆκ xq v . (14) phase space Hxp defined above. We denote by xˆq and pˆq ([xˆq ,pˆq ] = ihκ¯ ) two pairs of operators: one acting on Hxp [as H Utilizing Eqs. (9), (11), and (14), we obtain defined in Eq. (6)], the other acting on q , | = dxdλ |xλ xλ | xˆ ||x = x ||x , pˆ ||p = p ||p , κ κ p κ p p κ q q q q q q q q ˆ = || || = || || = ∗ 1Hq dxq xq xq dpq pq pq , hκ¯ dxdλp ρκ (u,v)ρκ (u,v) ix p /(¯hκ) e q q = || || || || xq ||pq = √ , ||xq ∈ Hq , ||pq ∈ Hq .
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