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Research Archive http://dx.doi.org/10.1108/IJBM-05-2016-0075 View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Hertfordshire Research Archive Research Archive Citation for published version: Dimitris Kakofengitis, Maxime Oliva, and Ole Steuernagel, ‘Wigner's representation of quantum mechanics in integral form and its applications’, Physical Review A, Vol. 95, 022127, published 27 February 2017. DOI: https://doi.org/10.1103/PhysRevA.95.022127 Document Version: This is the Accepted Manuscript version. The version in the University of Hertfordshire Research Archive may differ from the final published version. Users should always cite the published version of record. Copyright and Reuse: ©2017 American Physical Society. This manuscript version is made available under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. Enquiries If you believe this document infringes copyright, please contact the Research & Scholarly Communications Team at [email protected] Wigner's representation of quantum mechanics in integral form and its applications Dimitris Kakofengitis, Maxime Oliva, and Ole Steuernagel School of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield, AL10 9AB, UK (Dated: February 28, 2017) We consider quantum phase-space dynamics using Wigner's representation of quantum mechanics. We stress the usefulness of the integral form for the description of Wigner's phase-space current J as an alternative to the popular Moyal bracket. The integral form brings out the symmetries between momentum and position representations of quantum mechanics, is numerically stable, and allows us to perform some calculations using elementary integrals instead of Groenewold star products. Our central result is an explicit, elementary proof which shows that only systems up to quadratic in their potential fulfill Liouville's theorem of volume preservation in quantum mechanics. Contrary to a recent suggestion, our proof shows that the non-Liouvillian character of quantum phase-space dynamics cannot be transformed away. PACS numbers: 03.65.-w, 03.65.Ta I. MOTIVATION AND INTRODUCTION II. WIGNER'S DISTRIBUTION AND ITS EVOLUTION Wigner's representation of quantum mechanics In Wigner's representation of quantum mechanics [1, in phase-space [1] is equivalent to Heisenberg's, 7] Wigner's phase-space distribution is the \closest Schr¨odinger'sand Feynman's [2]. The description of quantum analogue of the classical phase-space distribu- the time evolution of Wigner's phase-space distribution tion" [8]. It is defined as function W uses Moyal brackets [3], the quantum analog 1 2i py of classical Poisson brackets. The similarity of the Moyal W (x; p; t) = dy %(x − y; x + y; t)e ~ ; (1) π~ form with classical physics explains its popularity. 1 ´ − 2i xs = ds %~(p − s; p + s; t)e ~ ; (2) π~ Moyal's bracket is defined as an infinite series of deriva- ´ here, ~ = h=(2π) is Planck's constant, integrals run tives, which can make it cumbersome to use and also nu- 1 from −∞ to +1: = , and % and% ~ are the den- merically unstable. It has limited applications because it −∞ sity operator in position´ ´ and momentum representation, assumes that the potential can be Taylor expanded. The respectively. integral form of quantum phase-space dynamics [1,4] is Wigner's distribution W is set apart from other an alternative to Moyal's form, it also applies to piecewise quantum phase-space distributions [7] by the fact that or singular potentials and displays symmetries between only W simultaneously yields the correct projections momentum and position representation not obvious when in position and momentum (%(x; x; t) = dp W and using Moyal's formulation only. 2 %~(p; p; t) = dx W ) as well as state overlaps´jh 1j 2ij = We recently showed that in anharmonic quantum sys- 2π~ dx´ dp W1 W2, while maintaining its form (1) tems the violation of Liouville's volume preservation can when´ ´evolved in time. Additionally, the Wigner distri- be so large that quantum phase-space volumes locally bution's averages and uncertainties evolve momentarily change at singular rates [5]. These singularities are of classically [9, 10]. central importance; they are responsible for the genera- In this work we consider one-dimensional conservative tion of quantum coherences. systems in a pure state with quantum-mechanical Hamil- tonians Here, we investigate a recent suggestion by Dali- 2 arXiv:1611.06891v3 [quant-ph] 27 Feb 2017 p^ gault [6], who provided a recipe that might enable us H^(^x; p^) = + V^ (^x) : (3) to \transform away" the violation of Liouville's theorem 2M in anharmonic quantum-mechanical systems. The Wigner function's time evolution arises, in analogy We illustrate the power of the integral form of Wigner's to Eq. (1), from a Wigner-transform [which can be im- representation, which allows us to give an elementary plemented as a fast Fourier transform (FFT)] of the von @%^ ^ proof that Daligault's suggestion amounts to a specific Neumann equation i~ @t = [H; %^] as [1,7] modification that makes the dynamics classical and is in- p 1 2i py compatible with his stated aim of finding a Liouvillian @tW = − dy @x%(x − y; x + y; t)e ~ system that reproduces quantum dynamics. M π~ ˆ i Our proof shows that the singularities, reported in + dy [V (x + y) − V (x − y)] π~2 ˆ Ref. [5], cannot be removed to make quantum phase- 2i py space dynamics divergence-free. × %(x − y; x + y; t)e ~ ; (4) 2 also known as the quantum Liouville equation. where we used Dirac's δ and wrote the density matrix as @n n 0 P ∗ 0 Throughout, we write partial derivatives as @xn = @x . %(x; x ; t) = k PkΨk(x; t)Ψk(x ; t), a statistical mixture If the potential V can be globally Taylor expanded, the of pure states. Additionally, integrals (4) yields the Moyal bracket {{·; ·}} [3] p p @W 1 dxJx = ds %~(p − s; p + s; t)δ(s) = %~(p; p; t) : = ffH;W gg = (H ?W − W? H) (5) ˆ M ˆ M @t i~ (12) 2 ~ −−! −−! = H sin @x@p − @p@x W: (6) ~ 2 Analogously to Eq. (11), the quantum probability cur- rent| ~ in momentum space [14], is Here we use Groenewold star products (?)[11], defined i −−! −−! ~ @x@p−@p@x as f ? g = fe 2 g; the arrows indicate whether 1 V (x + y) − V (x − y) dx Jp = − dy dx derivatives are executed on f or g. ˆ π~ ¨ 2y Equations (4) or (6) can be written as the continuity 2i py × %(x − y; x + y; t)e ~ equation [1] 1 p = p dp0 dp00 V~ ∗(p00 − p0)~%(p00; p0) @tW + r · J = @tW + @xJx + @pJp = 0 : (7) 3 i 2π~ ˆ−∞ ˆ Comparing Eqs. (7) and (4), we identify the Wigner − V~ (p00 − p0)~%(p0; p00) =| ~(p; t) ; (13) current J = Jx, with position component Jp 1 − i px (where V~ (p) = p dx V (x)e ~ ), while 2π p 2i py p ~ J = dy %(x − y; x + y; t)e ~ = W; (8) ´ x Mπ ˆ M ~ V (x + y) − V (x − y) dp J = − dy and momentum component ˆ p ˆ 2y 1 V (x + y) − V (x − y) × %(x − y; x + y; t)δ(y) J = − dy p π ˆ 2y dV ~ = −%(x; x; t) : (14) 2i py dx ×%(x − y; x + y; t)e ~ : (9) If the potential can be Taylor expanded, the explicit form We would like to emphasize that the quantum terms of of the components of Wigner current J in Eq. (6) is [1, Eq. (10) do not contribute in Eq. (14). 12{14] Averaging over Eqs. (12) and (14), reproduces Ehren- fest's theorem [2] 0 0 1 1 2l J = j + @ P (i~=2) 2l 2l+1 A : (10) hp^i dhx^i − (2l+1)! @p W@x V (x) dxdp Jx = = (15) l=1 ¨ M dt Here, with v = p=M , j = W v is the classical and J −j and −@xV are quantum terms. * + dV^ dhp^i dxdp J = − = : (16) ¨ p dx dt III. FEATURES AND APPLICATIONS OF THE INTEGRAL FORM Where applicable, the Moyal bracket formalism [2] yields the same results. Note the various subtleties associated The integral form (4) is more general than the Moyal with the interpretation of Ehrenfest's theorem [9, 10]. expression since it does not rely on V being analytic. A numerical implementation of the integral form can use fast Fourier transforms. In the case of potentials featuring high order Taylor terms, high order numerical IV. WHEN IS QUANTUM MECHANICAL TIME EVOLUTION LIOUVILLIAN? derivatives can render Eq. (10) poorly convergent [14]. In Ref. [14] we showed (note typographical errors in Eqs. (4) and (5) of [14]) that the p projection of Jx To investigate whether quantum phase-space dynamics yields the quantum probability current | in position is Liouvillian we determine the divergence of its quantum space, phase-space velocity field w [6, 13, 15]. w is the quantum analog of the classical velocity field v (Eq. (10)): ~ dpJx = dy %(x − y; x + y; t)@yδ(y) ˆ 2iM ˆ 0 0 1 J 1 1 2l X ~ ∗ ∗ w = = v + @ P (i~=2) 2l 2l+1 A : (17) = Pk (Ψk@xΨk − Ψk@xΨk) = |(x; t) ; (11) W W − (2l+1)! @p W@x V 2iM l=1 k 3 To rephrase the continuity equation (7) in terms of w, Alternatively, the auxiliary field has the trivial form we switch to the Lagrangian decomposition [6, 13, 15] δJ = j − J which subtracts all the quantum parts in Eq. (10), so that the potential assumes the form V + dW = @tW + w · rW = −W r · w : (18) δV = Vharmonic.
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