Downloaded from orbit.dtu.dk on: Oct 01, 2021 On coherent-state representations of quantum mechanics: Wave mechanics in phase space Møller, Klaus Braagaard; Jørgensen, Thomas Godsk; Torres-Vega, Gabino Published in: Journal of Chemical Physics Link to article, DOI: 10.1063/1.473684 Publication date: 1997 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Møller, K. B., Jørgensen, T. G., & Torres-Vega, G. (1997). On coherent-state representations of quantum mechanics: Wave mechanics in phase space. Journal of Chemical Physics, 106(17), 7228-7240. https://doi.org/10.1063/1.473684 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. On coherent-state representations of quantum mechanics: Wave mechanics in phase space Klaus B. Mo”ller and Thomas G. Jo”rgensen Department of Chemistry, Chemical Physics, Technical University of Denmark, DTU-207, DK-2800 Lyngby, Denmark Gabino Torres-Vega Departamento de Fı´sica, Centro de Investigacio´n y de Estudios Avanzados del IPN, Apartado Postal 14-740, 07000 Me´xico, D.F. Mexico ~Received 2 January 1997; accepted 27 January 1997! In this article we argue that the state-vector phase-space representation recently proposed by Torres-Vega and co-workers @introduced in J. Chem. Phys. 98, 3103 ~1993!# coincides with the totality of coherent-state representations for the Heisenberg-Weyl group. This fact leads to ambiguities when one wants to solve the stationary Schro¨dinger equation in phase space and we devise two schemes for the removal of these ambiguities. The physical interpretation of the phase-space wave functions is discussed and a procedure for computing expectation values as integrals over phase space is presented. Our formal points are illustrated by two examples. © 1997 American Institute of Physics. @S0021-9606~97!02317-9# I. INTRODUCTION Husimi where a quantum state is represented as a distribution function, observables are represented by functions, and the Phase space is a fundamental concept in Hamiltonian equations of motion are of the Liouville type. These repre- mechanics and phase-space formulations of quantum me- sentations, of course, all have a right on their own in the chanics are therefore of great interest in order to compare sense that a quantum problem can be solved entirely within quantum and Hamiltonian mechanics, for finite \ and when the representation, but they can also be obtained from the \ goes to zero. However, due to the uncertainty principle Schro¨dinger representation, in which case the Fock- caution is required when one wants to consider position and Bargmann wave function is obtained from the Schro¨dinger momentum together in quantum mechanics. wave function by a linear map whereas the Wigner and Hu- The usual Schro¨dinger representation of quantum me- simi distributions are bilinear in the Scho¨dinger wave func- chanics diagonalizes the position operator, i.e., it is multipli- tion. However, the Fock-Bargmann and the Husimi represen- cative, a quantum state is represented as a wave function that tations are closely related in that the Husimi distribution depends on the position variable, with the momentum opera- equals the square magnitude of the corresponding Fock- tor being a differential operator, that is, it is non-local. This Bargmann wave function times exp(2uzu2). representation can be Fourier transformed into a momentum Recently, there has been renewed interest in a quantum representation that diagonalizes the momentum operator, a state-vector phase-space representation ~SVPSR!.12–17 Un- quantum state is represented as a wave function that depends like the Fock-Bargmann representation, this SVPSR is for- on the momentum variable, and the position operator be- mulated in terms of the real ‘‘phase-space’’ coordinates q comes non-local. and p and operator mappings are given for the fundamental Many attempts to a phase-space description of quantum operators Q and P. In Ref. 12 Torres-Vega and Frederick mechanics have been made, the most famous ones being due postulate the existence of a complete set of vectors uq,p& 1 to Fock ~his representation has been studied thoroughly by that may serve as a basis such that a quantum state uc& can 2 3 4 Bargmann !, Wigner and Husimi. Recent reviews on the be represented in phase space by an L2(2) wave function Fock-Bargmann representation can be found in Refs. 5 and 6 c(q,p)5^q,puc& and the operators Q and P in this basis and for recent reviews on the Wigner and Husimi represen- take the ~non-local! forms tations see e.g., Refs. 7 and 8. Moreover, Harriman and Casida8 analyze the Wigner and Husimi distributions in the q ] , \1i ۋQ limit of a small \~see also e.g., Refs. 9 and 10!. A semi- S 2 ]pD classical study using the Fock-Bargmann representation is ~1! given, for instance, by Voros.11 p ] . \2i ۋP The Fock-Bargmann representation is a state-vector rep- S 2 ]qD resentation that diagonalizes the non-Hermitian creation op- erator, where a quantum state is represented as an entire With this representation it appears that Torres-Vega and Fre- ~wave! function of a complex variable, and where the anni- derick have obtained a wave-mechanical formulation of hilation operator is non-local. In this representation the equa- quantum mechanics in phase space similar to the usual ones tions of motion are of the Schro¨dinger type. This should be in position and momentum space; for the phase-space wave contrasted to the phase-space representations of Wigner and function the equation of motion is of the Schro¨dinger type, 7228 J. Chem. Phys. 106 (17), 1 May 1997 0021-9606/97/106(17)/7228/13/$10.00 © 1997 American Institute of Physics Downloaded¬12¬Jan¬2010¬to¬192.38.67.112.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright;¬see¬http://jcp.aip.org/jcp/copyright.jsp Mo”ller, Jo”rgensen, and Torres-Vega: Coherent-state representations 7229 and its square magnitude may play the role of a density in The remainder of this paper is organized as follows. In phase space whose dynamics is controlled by a Liouville Sec. II we introduce coherent states as basis vectors in a type equation. However, the SVPSR based on the operator phase-space representation ~PSR!. We show that any mapping Eq. ~1! differs from representations in position and coherent-state representation ~CSR! gives rise to the operator momentum space on one important point—it is not unique. mapping in Eq. ~1!. And, finally, by relating the CSRs to the The same quantum state is represented by an infinite number position and momentum representations we argue—using the of different wave functions in phase space. This may lead to work of Harriman19—that the CSRs are, in fact, the only some difficulties. For instance, it raises the question about representations that give rise to Eq. ~1!. We also discuss the how to interpret the square magnitude of a phase-space wave interpretation of the square magnitude as a phase-space den- function as a density. sity and the possibilities of using this density to calculate Previously Torres-Vega and co-workers have found ex- expectation values as phase-space averages. In Sec. III we amples of phase-space wave functions representing station- consider the possibilities and limitations of doing wave me- ary states for linear and quadratic potentials and analyzed chanics in phase space by solving a Schro¨dinger equation them for certain features; for instance, whether they are sta- based on Eq. ~1!. We examine the implications of the ambi- tionary solutions to the classical Liouville equation.16,17 guity in the solutions to this equation and argue that this The purpose of the present work is to discuss in more ambiguity, in certain cases, may be removed by supplying detail the origin of the operator mapping in Eq. ~1! and ana- the Schro¨dinger equation with an additional differential lyze the implications of our findings on the interpretation of equation. Furthermore, we show how the Schro¨dinger equa- the square magnitude of a phase-space wave function as a tion and the auxiliary equation can be turned into one equa- density and the possibilities of doing wave mechanics in tion in the Glauber coherent-state representation using the phase space using Eq. ~1!. The reason for the ambiguity in a Fock-Bargmann representation. The main points are illus- phase-space representation based on Eq. ~1! is that there ex- trated by examples. We summarize our findings in Sec. IV. ists an infinite number of complete bases parametrized by q and p that give rise the operator mapping in Eq. ~1!.In II. THE COHERENT-STATE REPRESENTATIONS Diracian sense this implies that we are dealing with an infi- nite number of different representations of quantum mechan- Following Klauder and Skagerstam5 and Perelomov6 a ics that are, in principle, independent. However, since they set of coherent states for the Heisenberg-Weyl group is de- all give the same operator mapping the bases defining each fined as the set of states obtained by application of the Weyl representation must be somehow similar.