On Coherent-State Representations of Quantum Mechanics: Wave Mechanics in Phase Space

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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

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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

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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 1997͒ 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 ͑1993͔͒ 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. ͓S0021-9606͑97͒02317-9͔

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 ប and when the representation, but they can also be obtained from the ប 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(Ϫ͉z͉2). 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 ͑SVPSR͒.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 ͉q,p͘ 1 to Fock ͑his representation has been studied thoroughly by that may serve as a basis such that a quantum state ͉␺͘ 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 ␺(q,p)ϭ͗q,p͉␺͘ and the operators Q and P in this basis and for recent reviews on the Wigner and Husimi represen- take the ͑non-local͒ forms tations see e.g., Refs. 7 and 8. Moreover, Harriman and ץ Casida8 analyze the Wigner and Husimi distributions in the q , ϩiប ۋQ pͪץ limit of a small ប͑see also e.g., Refs. 9 and 10͒. A semi- ͩ 2 classical study using the Fock-Bargmann representation is ͑1͒ ץ given, for instance, by Voros.11 p . Ϫiប ۋP qͪץ The Fock-Bargmann representation is a state-vector rep- ͩ 2 resentation that diagonalizes the non-Hermitian creation operator, where a quantum state is represented as an entire With this representation it appears that Torres-Vega and Fre- ͑wave͒ 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,

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 ͑PSR͒. We show that any mapping Eq. ͑1͒ differs from representations in position and coherent-state representation ͑CSR͒ gives rise to the operator momentum space on one important point—it is not unique. mapping in Eq. ͑1͒. 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. ͑1͒. 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. ͑1͒. 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. ͑1͒ 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. ͑1͒. The reason for the ambiguity in a Fock-Bargmann representation. The main points are illus- phase-space representation based on Eq. ͑1͒ 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. ͑1͒.In II. THE COHERENT-STATE REPRESENTATIONS Diracian sense this implies that we are dealing with an infinite 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. In fact, this simi- ͑or displacement͒ operator larity is well-known. Any set of coherent states, as formu- 5 i lated by Klauder and Skagerstam, used as a basis in a D͑q,p͒ϭexp ͑pQϪqP͒ , ͑q,p͒෈RϫR ͑2͒ SVPSR will result in the operator mapping in Eq. ͑1͒.5 The ͭ ប ͮ perhaps most familiar set of coherent states is the Glauber to any normalized vector ͉␹͘. A set of coherent states is coherent states.18 The use of this set as a basis in a SVPSR ͑over-͒complete and may therefore be used as a basis in a gives a phase-space wave function closely related to the state-vector representation of quantum mechanics. The study Fock-Bargmann wave function and the square magnitude of of representations created in this way is the subject of inter- the phase-space wave function equals ͑apart from a constant͒ est in this paper. the Husimi distribution. Therefore, many of the results we A. Properties of the coherent-state representations present in this paper will contain results about the Fock- Bargmann and Husimi representations as a special cases. Fi- For a fixed ͉␹͘ ͑denoted as the fiducial vector in the nally, we will also show that the Wigner formulation of following5͒ the set of coherent states provides a continuous quantum mechanics plays an important role in understanding representation of a quantum state where the expansion coef- a SVPSR based on Eq. ͑1͒. ficients can be interpreted as an L 2(2) wave function in Although from a slightly different point of view, the phase space. We introduce the vectors multitude of state-vector representations giving rise to Eq. ͉q,p;␹͘ϵD͑q,p͉͒␹͘, ͑q,p͒෈RϫR, ͑3͒ ͑1͒ has also been considered by Harriman19 ͑see also an earlier paper of Torres-Vega and Frederick20͒. Harriman, how- where the dependence of the fiducial vector is shown explic- ever, concentrates most of his analysis on the Glauber itly. These vectors define a basis in which the resolution of coherent-state representation. He points out that the set of unity takes the form5 possible phase-space wave functions formed with this choice 2 dqdp of basis only constitutes a subset of L (2), and that this Iϭ ͉q,p;␹͗͘q,p;␹͉. ͑4͒ subset can be characterized by studying certain phase-space ͵ 2␲ប eigenvalue equations. Our analysis generalizes these obser- The vectors ͉q,p;␹͘ may therefore be used to introduce a vations and provides additional insight into the origin of phase-space wave function for the state ͉␺͘ defined as these eigenvalue equations. Furthermore, we describe how ␺ q,p͒ϵ q,p;␹ ␺ . ͑5͒ these eigenvalue equations can be utilized as a tool to per- ␹͑ ͗ ͉ ͘ form wave mechanics in phase space in an unambiguous In the PSRs defined this way, the Weyl operator D(␩,␰) way. plays the role of a translation operator in phase space just

J. Chem. Phys., Vol. 106, No. 17, 1 May 1997

Downloaded¬12¬Jan¬2010¬to¬192.38.67.112.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright;¬see¬http://jcp.aip.org/jcp/copyright.jsp 7230 Mo”ller, Jo”rgensen, and Torres-Vega: Coherent-state representations like the operators exp(i␰Q/ប) and exp(Ϫi␩P/ប) are transla- D†͑q,p͒ϭeipq/͑2ប͒eiqP/បeϪipQ/ប, tion operators in the momentum and position representation, Ϫipq/͑2ប͒ ϪipQ/ប iqP/ប respectively. Using the well-known relation exp Aexp B ϭe e e , ͑11͒ ϭexp(AϩBϩ͓A,B͔/2), which is valid when ͓A,B͔ commutes with respect to p and q, respectively. We then get for any with both A and B, we get from Eq. ͑3͒ fiducial vector ͉␹͘ that n † n D͑␩,␰͉͒q,p;␹͘ϭ͉qϩ␩,pϩ␰;␹͘ei͑␰qϪ␩p͒/͑2ប͒, ͑6͒ ͗q,p;␹͉Q ͉␺͘ϭ͗␹͉D ͑q,p͒Q ͉␺͘, n ץ q where the additional phase factor stems from the non- ϭ ϩiប ͗q,p;␹͉␺͘, pͪץ 2 ͩ commutativity of Q and P. ͑12͒ Using Eq. ͑4͒ it thus follows that the inner product in a ͗q,p;␹͉Pn͉␺͘ϭ͗␹͉D†͑q,p͒Pn͉␺͘, CSR is evaluated as n ץ p ϭ Ϫiប ͗q,p;␹͉␺͘, qͪץ dqdp ͩ 2 ͗␺͉␾͘ϭ ␺␹*͑q,p͒␾␹͑q,p͒, ͑7͒ ͵ 2␲ប which are just the forms anticipated in Refs. 12–17 and 19. Thus for a ͑Taylor-expandable͒ operator ⍀(Q,P) we have that is, the integration measure in a CSR is given by ץ p ץ 2␲ប)Ϫ1dqdp. For a normalized state ͉␺͘ the phase-space q) ⍀ ϩiប , Ϫiប . ͑13͒ۋ⍀͑Q,P͒ qͪץ p 2ץ wave function defined in Eq. ͑5͒ is thus normalized when ͩ 2 integrating over phase space using this measure. At this stage it is important to emphasize that different B. Relation to the position and momentum ͉␹͘ define different SVPSRs. The change from one CSR to representations another is given by ͓using Eq. ͑4͔͒ The phase-space wave function in Eq. ͑5͒ is defined dqЈdpЈ without reference to any other representation and in this way ␺ ͑q,p͒ϭ ͗q,p;␹Љ͉qЈ,pЈ;␹Ј͘␺ ͑qЈ,pЈ͒. ␹Љ ͵ 2␲ប ␹Ј the CSRs qualify as ‘‘true’’ state-vector representations. ͑8͒ However, all state-vector representations are equivalent and a transition from one representation to another may be ac- Setting ͉␹Љ͘ϭ͉␹Ј͘ this relation also shows that complished by use of the resolution of the identity of the ͗q,p;␹Ј͉qЈ,pЈ;␹Ј͘ plays the role of a ␦-function in the co- representation which is departed from. Specifically, a transi- herent state basis ͉q,p;␹Ј͘. This, however, does not imply tion from the CSRs to the position- or momentum-space rep- that the states ͉q,p;␹Ј͘ and ͉qЈ,pЈ;␹Ј͘ are orthogonal.6 resentation is effected by Common to all the CSRs is that a quantum state is repre- 2 dqdp sented as an L (2) function and integrals like ␺͑qЈ͒ϭ͗qЈ͉␺͘ϭ ͗qЈ͉q,p;␹͘␺ ͑q,p͒, ͵ 2␲ប ␹ (2␲ប)Ϫ1͐dqdp␺* (q,p)␾ (q,p) of course exist but they ␹Ј ␹Љ ͑14͒ do not qualify as a quantum-mechanical inner product. In dqdp 5 ˜␺͑pЈ͒ϭ͗pЈ͉␺͘ϭ ͗pЈ͉q,p;␹͘␺ ͑q,p͒, fact, one has ͵ 2␲ប ␹ dqdp where the coherent-state version of the resolution of the ␺* ͑q,p͒␾ ͑q,p͒ϭ͗␺͉␾͗͘␹Љ͉␹Ј͘. ͑9͒ ͵ 2␲ប ␹Ј ␹Љ identity has been used. These integrals may be regarded as scalar products in phase space between the phase-space wave From this we learn that the phase-space representatives function and the eigenstates of Q and P, respectively, in the coherent-state basis defined by ͉␹͘. Similarly, the reverse ␺␹Ј(q,p) and ␺␹Љ(q,p)of͉␺͘cannot have the same func- transition is accomplished by using the position and momen- tional form unless ͉␹Ј͘ϭ͉␹Љ͘. We also learn that ␺␹Ј(q,p) 2 tum version of the resolution of the identity and ␾␹Љ(q,p) are orthogonal in L (2) if the states ͉␺͘ and ͉␾͘ are orthogonal, but we cannot conclude the reverse since Iϭ͐dqЈ͉qЈ͗͘qЈ͉ and Iϭ͐dpЈ͉pЈ͗͘pЈ͉, respectively, i.e., the two fiducial vectors ͉␹Ј͘ and ͉␹Љ͘ might be orthogonal. ␺␹͑q,p͒ϵ͗q,p;␹͉␺͘, Expressions for the operators Q and P in a coherent state representation can be found using the relations5,6 ϭ ͵ dqЈ͗q,p;␹͉qЈ͘␺͑qЈ͒, ץ q D†͑q,p͒Qϭ ϩiប D†͑q,p͒, pͪ ϭ dpЈ͗q,p;␹͉pЈ͘˜␺͑pЈ͒. ͑15͒ץ 2 ͩ ͑10͒ ͵ These integrals can be regarded as scalar products in position ץ p D†͑q,p͒Pϭ Ϫiប D†͑q,p͒. qͪ ͓momentum͔ space between ␺(qЈ) ͓˜␺(pЈ)͔ and a coherentץ 2 ͩ state parametrized by p and q in the position ͓momentum͔ These relations are easily proved by differentiation of the representation. Without specifying the fiducial vector we do, decompositions of course, not have explicit expressions for the transition

J. Chem. Phys., Vol. 106, No. 17, 1 May 1997

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 7231 elements ͗q,p;␹͉qЈ͘ and ͗q,p;␹͉pЈ͘. However, from the From the anticipated form of the fundamental operators properties of the displacement operator we have for any in the phase-space representations and in, say, the position- ͉␹͘ that space representation, Harriman is able to derive a formal expression for the linear map between the phase-space rep- ͗q,p;␹͉qЈ͘ϭ͗qЈϪq͉␹͘*eϪip͑qЈϪq/2͒/ប, ͑16͒ resentations and the position-space representation. A similar expression for the linear map between the phase-space rep- ͗q,p;␹͉pЈ͘ϭ͗pЈϪp͉␹͘*eiq͑pЈϪp/2͒/ប. resentations and the momentum-space representation is ob- These are eigenstates of Q and P in a coherent-state basis tained. Now, the interesting thing is that these formal expres- and they are easily seen to satisfy sions for the linear maps coincide with those for the transition elements Eq. ͑16͒ above. Since the expressions of ץ q ϩiប ͗q,p;␹͉qЈ͘ϭqЈ͗q,p;␹͉qЈ͘, Harriman were derived from ‘‘first principles,’’ we conclude pͪץ 2 ͩ ͑17͒ that the CSRs discussed above are the only ones where the .position and momentum operators take the form Eq. ͑1͒ ץ p Ϫiប ͗q,p;␹͉pЈ͘ϭpЈ͗q,p;␹͉pЈ͘. As mentioned in the introduction, Torres-Vega and qͪץ 2 ͩ co-workers12–17 have previously introduced a phase-space Using the phase-space scalar product, we find that the phase- state-vector representation based on the postulate that there space eigenfunctions of Q and P are ␦-function normalized exists a set of basis vectors ͉⌫͘ϭ͉q,p͘ with the closure re- as they should be. For instance, lation Iϭ͐d⌫͉⌫͗͘⌫͉ where they can define the operators of dqdp Q and P to take the form given in Eq. ͑1͒. With the conclu- ͗qЉ͉q,p;␹͗͘q,p;␹͉qЈ͘, sion above it is clear that Torres-Vega and co-workers do not ͵ 2 ␲ប describe a single representation but rather a class of representations, namely the CSRs, and we can then identify their dqdp basis vector with any of the vectors ͉q,p;␹͘ divided by ϭ ͗␹͉qЈϪq͗͘qЉϪq͉␹͘eip͑qЉϪqЈ͒/ប, ͵ 2␲ប (2␲ប)1/2. This identification provides useful insight into the work of Torres-Vega and co-workers. For instance, it is now evident that the infinite number of fiducial vectors is respon- ϭ␦͑qЉϪqЈ͒͵dq͗␹͉qЈϪq͗͘qЉϪq͉␹͘, sible for the ambiguity in the phase-space wave functions that they observe. Furthermore, one of the key points in their ϭ␦͑qЉϪqЈ͒, ͑18͒ work was that their PSR cannot be obtained from the position- or momentum-space representations, one can only where we have used that ͐dq͗␹͉qЈϪq͗͘qЉ go the other way around, that is, the position- or momentum- Ϫq͉␹͉͘ ϭ͗␹͉␹͘ϭ1. Similarly, we find that qЉϭqЈ space wave functions can be obtained from the phase-space wave function by a projection. As demonstrated above, this dqdp is certainly not true ͓cf. Eqs. ͑14͒ and ͑15͔͒. However, these ͗pЉ͉q,p;␹͗͘q,p;␹͉pЈ͘ϭ␦͑pЉϪpЈ͒. ͑19͒ ͵ 2␲ប procedures require knowledge of the fiducial vector—a concept that was not recognized in their work—and without this knowledge Eqs. ͑14͒ and ͑15͒ are useless. On the other hand, Also by direct evaluation in phase space we find by comparing the Schro¨dinger equation in position, momentum, and phase space Torres-Vega and Frederick12 found dqdp eϪipЈqЈ/ប that the position- and momentum-space wave functions can ͗pЈ͉q,p;␹͗͘q,p;␹͉qЈ͘ϭ , ͑20͒ be obtained from the phase-space wave function by ͵ 2␲ប ͱ2␲ប ‘‘Fourier-like’’ projections. These procedures must therefore be fiducial-vector independent and, in fact, we get from Eq. as expected.21 ͑15͒ that At this stage we can make a connection to the work of Harriman19 in order to show that the CSRs are the only rep- 1 resentations in which the operators Q and P take the form in ipq/͑2ប͒ dpe ␺␹͑q,p͒, Eq. ͑1͒. Harriman assumes the existence of a phase-space 2␲ប ͵ representation where the position and momentum operators take the form given by Eq. ͑1͒. Based on the form of these 1 ϭ dqЈ͗␹͉qЈϪq͗͘qЈ͉␺͘ dpeϪip͑qЈϪq͒/ប, operators, he finds the linear maps between the position- and 2␲ប͵ ͵ momentum-space representations and the phase-space representation. In fact, Harriman finds infinitely many maps of ϭ dqЈ͗␹͉qЈϪq͗͘qЈ͉␺͘␦͑qЈϪq͒, this type from the position- and momentum-space represen- ͵ tations to a phase-space representation, where the fundamen- ϭ͗␹͉q͉͘qϭ0͗q͉␺͘, ͑21͒ tal operators assume the anticipated form. This implies that there is not a single but infinitely many of such phase-space representations. and

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1 representation diagonalizes the position operator. A similar dqeϪipq/͑2ប͒␺ ͑q,p͒, 2␲ប ͵ ␹ argument holds for the momentum-space representation. 1 Now, although considering a normalized phase-space wave ϭ dpЈ͗␹͉pЈϪp͗͘pЈ͉␺͘ dqeiq͑pЈϪp͒/ប, function, thus giving rise to some density in phase space, a 2␲ប͵ ͵ similar physical significance cannot be ‘‘tied’’ to a phase- space wave function in a simple manner. First of all, because ϭ͵ dpЈ͗␹͉pЈϪp͗͘pЈ͉␺͘␦͑pЈϪp͒, in any CSR none of the fundamental operators are diagonal- ized ͑both operators are non-local͒ and, second, because of ϭ͗␹͉p͉͘pϭ0͗p͉␺͘, ͑22͒ the so far unspecified parameter ␹ of the fiducial vector. So, where the fiducial vector only shows up as a constant on the what sort of information does a phase-space wave function right-hand side. This may, however, be zero if the wave offer with respect to its labels? function ͗q͉␹͘ ͓͗p͉␹͔͘ vanishes at qϭ0 ͓pϭ0͔. This dem- To answer this question, let us first briefly examine the onstrates that the information contained in the phase-space structure of such a wave function. In the cases where the wave function about the fiducial vector is in some sense re- fiducial vector is a ground state of any physical harmonic 2 dundant. oscillator the square magnitude ͉␺␹(q,p)͉ is easily seen to 22 23,8 This may be compared to the work of Wl”odarz who be the well-known Husimi function. Hence, for a general demonstrates that the Wigner function may be expressed as fiducial vector the density is of a ‘‘Husimi-type’’ construc- the so-called Ã-product of a phase-space wave function and tion. its complex conjugate. The phase-space wave functions con- As discussed by e.g., Stenholm24 and Royer,25 in their sidered by Wl”odarz may be shown to be wave functions analyses of the Wigner function, a phase-space density re- ͑with rescaled arguments͒ in any CSR, a fact implicitly rec- sulting from a simultaneous measurement of position and ognized by Wl”odarz by identifying the time-evolution equa- momentum is obtained as the convolution of the Wigner tion for the wave functions in his approach with the one put function for the system under consideration with the Wigner forward by Torres-Vega and Frederick.12 Hence, for comput- function of a ‘‘test body’’24 or probe state with negated ar- ing the Wigner function the fiducial-vector information in guments, rather than the system Wigner function itself. This the CSR wave function is again somewhat redundant. convolution, given by In any case, in order to give the phase-space wave function a meaningful interpretation—and for some computa- tional purposes—knowledge of fiducial vector is imperative. 2␲ប ͵ dqЈdpЈW␹͑qЈϪq,pЈϪp͒W␺͑qЈ,pЈ͒, ͑24͒

C. Interpretation of the phase-space wave function where W (q,p) is the Wigner function for the probe state Like any other representation, the CSRs introduce a ␹ and W␺(q,p) is the Wigner function for the system in ques- probability amplitude or wave function, here denoted tion, constitutes a ‘‘fuzzy’’ phase-space density subject to ␺␹(q,p). The ‘‘universal’’ interpretation of such a quantity the uncertainty principle.25 Equation ͑24͒ is seen to equal the is, of course, that its square magnitude gives the probability square magnitude Eq. ͑23͒ when the fiducial vector ͉␹͘ rep- of the state under consideration being in a basis state of the resents the probe state. representation specified by some labels or parameters. In the 2 Thus, ͉␺␹(q,p)͉ is a ‘‘fuzzy’’ phase-space density and, present case, according to Royer,25 this density gives for each pair of la- 2 2 ͉␺␹͑q,p͉͒ ϭ͉͗q,p;␹͉␺͉͘ ͑23͒ bels (q,p) the relative probability that the system is localized in a ‘‘fuzzy’’ neighbourhood of the centre of the displaced gives the probability of the state ␺ being in the coherent ͉ ͘ probe state ͑fiducial vector͒.26 Two things should be noted state q,p;␹ . The labels are here q, p, and ␹. ͉ ͘ about this interpretation. First, we have in the general case The association of a CSR with a PSR assumes that some that physical significance has been attributed to two of the three labels above, namely q and p. The same pertains to the usual position- and momentum-space representations. Moreover, ͗␹͉Q͉␹͘ϭq␹ , ͗␹͉P͉␹͘ϭp␹ , ͑25͒ in these representations the respective wave functions are associated with a probability measure directly related to the involved labels, i.e., in the position-space representation, for and only when the fiducial vector is physically centred, i.e., 2 instance, ͉␺(q)͉ dq gives the probability of observing the only when q␹ϭp␹ϭ0 the label pair (q,p) denotes the centre position of the system between q and qϩdq. In this way, not of the displaced probe state. In general, the centre is just the label q in the position-space wave function is attrib- (qϩq␹ ,pϩp␹). Therefore, the label point (q,p) in a plot of 2 uted a physical significance but also the wave function itself a general density ͉␺␹(q,p)͉ is a ‘‘statement’’ about the is—namely the well-known fact that the square magnitude physical point (qϩq␹ ,pϩp␹). Second, the meaning of a ͉␺(q)͉2 is considered a probability density. ‘‘fuzzy neighbourhood’’ needs to be specified. If the fiducial Such an interpretation of the position-space wave func- vector is a ground state for some harmonic oscillator the 2 tion is formally enabled by the fact that the corresponding density ͉␺␹(q,p)͉ represents the most precise description of

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D. Expectation values using the phase-space density Substituting back into Eq. ͑30͒ and using that 2 ͐dpW␺(q,p)ϭ͉␺(q)͉ for any state ͉␺͘ we arrive at According to Eq. ͑4͒, the expectation value of the opera- n tor ⍀(Q,P) in the state ͉␺͘ can be calculated in any CSR as n k q ϭ ak͗͑qϪQ͒ ͘␹ , ͑32͒ is usually done in other state-vector representations, viz. k͚ϭ0 p ץ dqdp q ͗␺͉⍀͑Q,P͉͒␺͘ϭ ␺*͑q,p͒⍀ ϩiប , where ͗͘␹denotes the expected value with respect to the p 2ץ 2␲ប ␹ ͩ2 ͵ fiducial vector ͉␹͘. Clearly, an must equal unity, whence we are left with a system of n linear homogeneous algebraic ץ Ϫiប ␺␹͑q,p͒. ͑26͒ equations in the coefficients a0 ,...,anϪ1. ’’qͪ The procedure is completely equivalent for ‘‘pureץ However, let us consider the possibility of using the density powers of the momentum operator P, and for the first and itself for calculating expectation values as integrals over second powers we arrive at the mappings phase space.27 To this end we again turn to the Wigner , qϩq ͒2Ϫ͑⌬q͒2͑ۋqϩq , Q2ۋQ phase-space formulation in which the expectation value of ␹ ␹ ␹ ͑33͒ 2 2 2 , pϩp␹͒ Ϫ͑⌬p͒␹͑ۋ qϩq␹ , Pۋthe operator ⍀(Q,P) in the state ͉␺͘ is expressed by the P phase-space integral where the right-hand sides give the functions to be used in

͗␺͉⍀͑Q,P͉͒␺͘ϭ dqdp⍀W͑q,p͒W␺͑q,p͒. ͑27͒ Eq. ͑28͒. Here, x␹ϵ͗X͘␹ is the expected value and ͵ 2 2 2 (⌬x)␹ϵ͗X ͘␹Ϫ͗X͘␹ the variance, both with respect to the The function ⍀W(q,p) is the Weyl-transform of the operator fiducial vector. ⍀(Q,P).7 If we require that the expectation value of A few remarks about these results are appropriate. For a ⍀(Q,P) should be expressed in a similar fashion when us- physically centred MUS fiducial vector Eq. ͑33͒ coincides 2 8 ing ͉␺␹(q,p)͉ as the density it is expected that the function with results of Harriman and Casida. Hence, Eq. ͑32͒ and its representing the observable in this case must depend on the momentum equivalent is a generalization of these authors’ fiducial vector. In fact, one readily finds using Eq. ͑24͒ that results to the use of an arbitrary fiducial vector. Second, expectation value of the operator ⍀(Q,P) in the state ͉␺͘ again for a physically centred MUS fiducial vector the func- may be expressed as tion ⍀␹(qЈ,pЈ) is recovered as the Glauber-Sudarshan

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P-symbol or anti-normally ordered symbol, used in quantum operator mapping Eq. ͑1͒ only depends on the form of the optics,6,7,28 of the operator ⍀. Therefore, for an arbitrary Weyl operator Eq. ͑2͒ and not the fiducial vector itself. fiducial vector ⍀␹(qЈ,pЈ) represents a generalized Two questions now arise: P-symbol in accordance with the interpretation of the CSR- ͑i͒ How do we know to which fiducial vector a particular 2 density ͉␺␹(q,p)͉ as being a generalized Husimi represen- solution of the Schro¨dinger equation belongs? and tation or, to use the quantum optics terminology, a general- ͑ii͒ Is this important for practical purposes? ized Q-representation. This, in turn, reflects the dual nature The second question can be answered immediately when of the Q- and P-representations, also in the generalized 29,30 ‘‘practical purposes’’ is explicitly defined. The questions sense ͑see also Dahl ͒. In view of this fact it should be asked in quantum mechanics are ultimately questions about noted that use of the term ‘‘Husimi representation’’ for the 8 observables or dynamical variables. By ‘‘practical purposes’’ function ⍀␹(qЈ,pЈ) by Harriman and Casida is somewhat we, accordingly, mean the use of wave functions for calcu- misguiding, giving the impression that the Husimi represen- lating expectation values and transition matrix elements. tation is a self-dual PSR. Hence, for two solutions, ␺␹ and ␾␹Ј,ofEq.͑34͒ and Finally, it should be noted that the presented relations presumably different fiducial vectors, ͉␹͘ and ͉␹Ј͘, the ‘‘ma- Eqs. ͑24͒ and ͑29͒ between the generalized Husimi, Weyl- trix elements’’ of an operator ⍀(Q,P) may be computed by Wigner, and Glauber-Sudarshan symbols provide an interest- application of Eq. ͑9͒ as ing insight into the connections between these symbols and ץ p ץ representations—again for the standard versions as well as dqdp q ␺*͑q,p͒⍀ ϩiប , Ϫiប ␾ ͑q,p͒ qͪ ␹Јץ p 2ץ the generalized ones. It is well known31 that in the standard ͵ 2␲ប ␹ ͩ2 cases the Weyl-Wigner symbol may be obtained from the Glauber-Sudarshan symbol by a Gaussian smearing or con- ϭ͗␺͉⍀͑Q,P͉͒␾͗͘␹Ј͉␹͘. ͑35͒ volution and that the Husimi symbol ͑apart from a multipli- Therefore, in order to calculate the matrix element using cative factor͒ is obtained from the Weyl-Wigner symbol by a phase-space wave functions one has to be sure that the two smearing with same Gaussian. Furthermore, we know from wave functions belong to the same CSR. In the calculation of the Husimi symbol that the widths of the smearing Gaussian expectation values this is trivially fulfilled since the same reflect the uncertainties in position and momentum of the wave function is used twice, so to speak. Hence, if only MUS fiducial vector. Since the Husimi function for a MUS expectation values are needed the answer to question ͑ii͒ too is a Gaussian so are the Wigner and Glauber-Sudarshan above is ‘‘no’’; otherwise ‘‘yes.’’ functions. Hence, the smearing Gaussian in the standard for- This brings us back to the first question and the imme- mulations could be any of the three symbols for the MUS diate answer to that is: ‘‘We don’t.’’ An additional criterion fiducial vector. However, from Eqs. ͑24͒ and ͑29͒ we con- is needed in order to assure that the two chosen wave func- clude that, in the general cases, the smearing function is the tions belong to the same CSR. In order to be a useful crite- Wigner function for the fiducial vector. So the Wigner func- rion it must force a condition on the fiducial part of the wave tion of the fiducial vector provides the link between the three function only and not on the state vector part. mentioned symbols of any operator, and in this respect the Such a criterion is provided by requiring that the fiducial Weyl-Wigner representation may be considered the more vector be an eigenstate of a non-degenerate Taylor expand- fundamental one. able operator, say A␹(Q,P), such that ͉␹͘ is uniquely determined from the eigenvalue equation

A␹͑Q,P͉͒␹͘ϭa␹͉␹͘, ͑36͒ III. WAVE MECHANICS IN PHASE SPACE once the eigenvalue a␹ is specified. This equation is clearly In this section we consider the possibility and implica- formulated without reference to any state vector. Upon em- tions of doing wave mechanics in phase space, that is, solv- ploying the properties of the displacement operator, Eq. ͑36͒ ing the equation of motion in a CSR. For a Hamiltonian of may be turned into an equation in the CSR based on the 2 the type H(Q,P)ϭP /(2m)ϩV(Q) the equation of motion fiducial vector ͉␹͘. Explicitly we have governing the dynamics of a phase-space wave function is † the phase-space Schro¨dinger equation, ͓D͑q,p͒A␹͑Q,P͒D ͑q,p͔͉͒q,p;␹͘ 2 ϭA␹͑QϪq,PϪp͉͒q,p;␹͘ϭa␹͉q,p;␹͘, ͑37͒ ץ p 1 ץ iប ␺␹͑q,p͒ϭ Ϫiប qͪ which results in the phase-space eigenvalue equation for anyץ t ͫ2m ͩ 2ץ state ␺ ͘ ͉ ץ q ϩV ϩiប ␺ ͑q,p͒. ͑34͒ ץ p ץ pͪͬ ␹ qץ 2 ͩ A␹* Ϫ ϩiប ,Ϫ Ϫiប ␺␹͑q,p͒ϭa␹*␺␹͑q,p͒. qͪץ p 2ץ The equation is the same for any fiducial vector and the ͩ 2 38 solution of this equation will therefore yield infinite number ͑ ͒ of phase-space wave functions—corresponding to the infinite This equation constitutes the additional requirement. Hence, number of CSRs—describing the same physical state. The for those fiducial vectors which are eigenvectors for some ambiguity of the fiducial vector clearly arises because the operator A␹(Q,P), we may classify the solutions of the

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The two equations set up by Harriman corresponds to on the relation between the boson operators and the ‘‘physi- † A␹ being the annihilator with a␹ϭ0 and A␹ being the Hamil- cal’’ operators Q and P is that ͓a,a ͔ϭ1 and since ͑ii͒ a tonian for harmonic oscillator with a␹ϭ1/2. Fock-Bargmann-type representation can be associated with Of course, the Glauber coherent-state representation is a any Glauber-type representation based on a fiducial vector very important one but as discussed briefly by Klauder and that is an eigenstate of a.6 This enlarges the class of Skagerstam5 other choices of fiducial vector are not only of (q,p)-parametrized CSRs, that may be handled via a FBR, to academic interest. The above procedure has been employed including those based on a not physically centred fiducial in the study of the spectrum of the quartic oscillator ͑see Ref. vector. However, the purpose of introducing the FBR in the 5 and Refs. 65–71 therein͒. The approach taken in these present context is to enable a simple and convenient scheme investigations is that of choosing the fiducial vector to be one for doing wave mechanics in phase space and with reference of the unknown eigenstates of the quartic oscillator. Thus, a to the discussion in the preceding section, (q,p) natural choice of A␹(Q,P) would be the Hamiltonian itself, -parametrized CSRs with a physically centred MUS as fidu- rendering the auxiliary equation Eq. ͑38͒ the conjugate of the cial vector constitute the physically most appealing state- stationary Schro¨dinger equation. The combining of these two vector PSRs. Accordingly, we shall restrict ourselves to the equations leads to differential equations for the eigenfunc- standard procedure. tions where the fiducial vectors are well-known, in the sense Thus, the boson operators are expressed in terms of Q that they too are eigenfunctions of the Hamiltonian and and P in the standard form therefore may be labeled. In this way the Schro¨dinger equa- 1 tion and the auxiliary equation are solved simultaneously. aϭ ͑␭Qϩi␭Ϫ1P͒, In conclusion, to do wave mechanics in phase space in ͱ2ប ͑39͒ an unambiguous way, i.e., for a given fiducial vector accord- 1 ing to the above described scheme, we have to solve two † Ϫ1 a ϭ ͑␭QϪi␭ P͒, phase-space equations—either simultaneously or in a succes- ͱ2ប sive manner. This should be contrasted to wave mechanics in and with the Glauber coherent states given by the usual Schro¨dinger representation. Here, a single equation is sufficient. ͉␣͘ϭD͑␣͉͒0͘, D͑␣͒ϭexp͑␣a†Ϫ␣*a͒, ͑40͒ A. The Fock-Bargmann approach where a͉0͘ϭ0 and where ␣ϭ(␭qϩi␭Ϫ1p)/ͱ2ប, the rela- The question is if it is possible to device a scheme in tionship between wave functions in the Glauber and Fock- phase space where a single equation suffices to solve the Bargmann representations is quantum problem and to fix the representation. Clearly, such 2 ␺͑␣,␣*͒ϵ͗␣͉␺͘ϭexp͑Ϫ͉␣͉ /2͒␺FB͑z͒,zϵ␣*. ͑41͒ a scheme cannot be based on the operator mapping Eq. ͑1͒;a mapping in unique correspondence with a fiducial vector Finally, the boson operators are in the FBR mapped accord- must be chosen. One such mapping has been employed by ing to 32 ץ .Skodje, Rohrs, and VanBuskirk for a MUS fiducial vector z standard FBR. ͑42͒ۋ†a , ۋHowever, in the following we shall devise another scheme a zץ based on the famous Fock-Bargmann representation mentioned in the Introduction. Based on the above considerations we, therefore, pro- The Fock-Bargmann representation ͑FBR͒ constitutes a pose the following scheme for doing wave mechanics in state-vector representation in the complex plane based on a phase space in an unambiguous Schro¨dinger-like manner, mapping of a pair of boson operators (a,a†). In this repre- that is, wave mechanics in a CSR based on a MUS fiducial sentation, quantum mechanics can be performed unambigu- vector. The fiducial vector is completely fixed by a param- ously due to the fact that it diagonalizes the creation operator eter ␭ which, however, may be chosen at will. a†. Hence, wave mechanics in phase space can be performed ͑i͒ Given a Hamiltonian H(Q,P) we can choose a map in an unambiguous manner, similar to the standard Schro¨- between the fundamental operators (Q,P) and the dinger representation, via the FBR if this can be linked to the boson operators (a,a†) according to Eq. ͑39͒ by CSRs discussed so far. specifying the parameter ␭෈R. In this way the Such a link is provided by the representation in the co- Hamiltonian may be re-expressed as herent states of Glauber.33 Being originally formulated in H͑a,a†͒. ͑43͒ۋH͑Q,P͒ terms of the boson operators, this representation is intimately connected with the FBR. At the same time the Glauber rep- ͑ii͒ By this and by using the standard Fock-Bargmann resentation is closely related to the (q,p)-parametrized CSR operator mapping Eq. ͑42͒ we have chosen a specific

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FBR, that corresponds to a physically centred Let us briefly summarize the procedure of determ- (q,p)-parametrized CSR. In other words, the choice ining phase-space eigenfunctions of the linear potential of ␭ fixes the fiducial vector and hence the represen- used by Torres-Vega et al. They introduce the dimension- tation. In this FBR the Schro¨dinger equation is set up less coordinates xϭ␤(qϪ2E/K) and yϭp/(␤ប) where according to ␤ϭ(2mK/ប2)1/3 in order to re-write the stationary phase- space Schro¨dinger equation as † ץ iប ͉␺͘ϭH͑a,a ͉͒␺͘ 2 ץ x ץ t yץ Ϫi ϩ ϩi ␾͑x,y͒ϭ0, ͑47͒ yץ x 2ץ 2 ͬͪ ͩ ͪ ͩͫ ץ z,z͒␺FB͑z͒. ͑44͒ץ/ץiប ␺FB͑z͒ϭH͑ۋ t where ␾(x,y)ϭ␺E(q(x,E;␤),p(y;␤)). A set of solutions toץ this equation is then given in the form Here zϭ(␭qϪi␭Ϫ1p)/ͱ2ប. It is emphasized that the solutions to this equation are unambiguously tied to the FBR 3 2 x,y ϭNeϪixy/2 dteit /3Ϫ␣͑tϪy͒ /2ϩitx, 48 representation automatically selected by the choice of map- ␾͑ ͒ ͵ ͑ ͒ ping between (Q,P) and (a,a†) when the standard Fock- where ␣ is an arbitrary non-negative parameter and N is a Bargmann operator mapping Eq. ͑42͒ is used. normalization constant. This gives rise to the phase-space ͑iii͒ Therefore, a FBR wave function ␺ (z) is uniquely FB eigenfunctions associated with a wave function ␺␹(q,p) in a physically cen- Ϫip͑qϪ2E/K͒/͑2ប͒ tred (q,p)-parametrized CSR with the explicit relation being ␺E͑q,p͒ϭNe ␺ q,p ϭexp Ϫ z 2/2 ␺ z , ␹ ϭ 0 . ͑45͒ ␹͑ ͒ ͑ ͉ ͉ ͒ FB͑ ͒ ͉ ͘ ͉ ͘ 3 2 ϫ dteit /3Ϫ␣[tϪp/͑␤ប͒] /2ϩit␤͑qϪ2E/K͒. ͑49͒ These steps show that a complete set of wave functions ͵ in the class of CSRs with a MUS fiducial vector may be Now, let us see which fiducial vector that leads to this obtained unambiguously by solving the Schro¨dinger equa- phase-space wave function. Starting from, for instance, the tion in the standard FBR. position-space eigenfunctions,34 We conclude this subsection by noting that the unique correspondence between an operator mapping and a fiducial ␤ it3/3ϩit␤͑qϪE/K͒ vector is equivalent to an implicit fulfillment of the auxiliary ͗q͉␺E͘ϭ ͵ dte , ͑50͒ equation Eq. ͑38͒. The power of the Fock-Bargmann repre- 2␲ͱK sentation is that this condition is fulfilled by construction. it is straightforward to show that with the fiducial vector being a MUS centred at qϭϪE/K and pϭ0, which in po- B. Examples sition space takes the form The three standard examples to consider are the free par- ␭2 1/4 ␭2 ͗q͉␹͘ϭ exp Ϫ ͑qϩE/K͒2 , ͑51͒ ticle, the linear potential and the harmonic oscillator. Since ͩ ␲បͪ ͭ 2ប ͮ the free-particle eigenfunctions equal the eigenfunctions of the momentum operator, these are already given as the lower one obtains the phase-space eigenfunctions of Torres-Vega, expression in Eq. ͑16͒. Torres-Vega and co-workers have Zu´ñiga-Segundo, and Morales-Guzmań using the identifica- previously considered several examples of analytic solutions tion ␣ϭប(␤/␭)2. The parameter ␣ therefore depends on to the phase-space Schro¨dinger equation; coherent states and both the physical system and the properties of the fiducial eigenstates of the harmonic oscillator12,16 and, most recently, vector. For a given set of (m,K) and ប, the value of ␣ is thus eigenfunctions of the linear potential.17 In the following we fixes ␭. However, fixation of ␭ only determines the uncer- analyze the linear and quadratic potential in terms of CSRs tainty properties of the fiducial vector; its centering is deter- in order to provide more insight into the previous results. It mined by the energy. This implies that even for a fixed ␣ the turns out that all the wave functions of Torres-Vega and phase-space energy eigenfunctions of Torres-Vega, Zu´ñiga- co-workers are given in CSRs based on MUSs. Segundo, and Morales-Guzmań do not belong to the same representation. In fact, they all belong to different represen- 1. Eigenstates of a linear potential tations! This shows that since the phase-space Schro¨dinger 17 Torres-Vega et al. have recently considered a set of does not contain any information about the fiducial vector solutions to the stationary phase-space Schro¨dinger equation, one has given up the ‘‘control’’ of the representation by .working directly with this equation ץ q 2 ץ p 1 Ϫiប ϩK ϩiប ␺ ͑q,p͒ϭE␺ ͑q,p͒. To ensure that the different eigenfunctions belong to the p E Eץ q 2ץ 2m 2 ͫ ͩ ͪ ͩ ͪͬ same representation one has to find the relationship between ͑46͒ the different eigenstates in abstract Hilbert space and then The reason to take this problem up again is to analyze the choose a single representation for all of them. In abstract findings in Ref. 17 in terms of CSRs and to illustrate one of Hilbert space, the Schro¨dinger equation is the drawbacks of working directly with the phase-space 2 Schro¨dinger equation arising from the lack of ‘‘control’’ of P ϩK͑QϪE/K͒ ͉␺ ͘ϭ0. ͑52͒ the fiducial vector. ͫ2m ͬ E

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Using the well-known relations33 where again ␣ϭប(␤/␭)2. A physically centred MUS is fully specified through ␭ and, therefore, for a given set of the D†͑␩,␰͒QD͑␩,␰͒ϭQϩ␩, ͑53͒ physical parameters ប and ␤ the choice of the value for ␣ is D†͑␩,␰͒PD͑␩,␰͒ϭPϩ␰, a choice of CSR. With the transformations35 this equation can be re-written as z2/2ϩ2͑␣/2͒3/2z ␺0͑z͒ϭe ␾͑w͒, P2 ͑59͒ 2 ϩKQ D͑ϪE/K,0͉͒␺E͘ϭ0. ͑54͒ wϭͱ2␣zϩ ␣/2͒ , ͫ2m ͬ ͑ the equation for ␾(w) becomes the usual differential equa- Therefore, in abstract Hilbert space, ␺ ϭD(E/K,0) ␺ . ͉ E͘ ͉ 0͘ tion for the Airy function, Since ͗q͉D(␩,0)ϭ͗qϪ␩͉, the position-space eigenfunctions with eigenenergy E are simply obtained from the Eϭ0 d2 Ϫw ␾͑w͒ϭ0. ͑60͒ eigenfunction by a displacement of the position coordinate, ͩ dw2 ͪ as seen in Eq. ͑50͒. In any CSR the relation is, using Eq. ͑6͒, Using Eq. ͑45͒ we can, therefore, write the eigenstate of a ϪipE/͑2បK͒ ␺E͑q,p͒ϭ␺0͑qϪE/K,p͒e , ͑55͒ linear potential with Eϭ0 in the CSRs, with fiducial vectors being physically centred MUSs, in terms of the usual Airy which is a displacement in position and a momentum- function as dependent change of phase. Equation ͑55͒ also shows that Ϫ͉͑z͉2Ϫz2͒/2ϩ2͑␣/2͒3/2z 2 square magnitude of the eigenstates in any CSR obey the ␺0͑q,p͒ϭNe ϫ Ai͓ͱ2␣zϩ͑␣/2͒ ͔, ‘‘classical’’ relation between energy and translation along ͑61͒ the position direction in phase space. The wave functions in where, according to the remarks following Eq. ͑58͒, Eq. ͑49͒ do not satisfy Eq. ͑55͒ illustrating that they cannot belong to the same phase-space representation. ␤ ͱ␣/2 zϭ qϪi p. ͑62͒ However, we readily deduce from above analysis the ͱ2␣ ប␤ complete set of phase-space eigenfunctions with the fiducial vector being, for instance, the ground state of a physical Therefore, the relation between z and (q,p) is determined oscillator ͉0͘ with qЈϭpЈϭ0. These are given by once ប, ␣, and ␤ are specified. Thus, the complete set of eigenfunctions can be obtained using Eq. ͑55͒. That the ϪipE/͑2បK͒ ␺E͑q,p͒ϭ␾͓␤͑qϪE/K͒,p/͑␤ប͔͒e ͑56͒ phase-space wave functions in these representations can be or expressed in terms of well-known functions is not evident from previous results. Ϫipq/͑2ប͒ ␺E͑q,p͒ϭNe

3 2 2. The harmonic oscillator ϫ dteit /3Ϫ␣[tϪp/͑␤ប͒] /2ϩit␤͑qϪE/K͒. ͑57͒ ͵ The harmonic oscillator has been treated by Torres-Vega With this choice of the fiducial vector, the square magnitude and co-workers in Refs. 12 and 15 paying special attention to of these wave functions equals the Husimi function. In Ref. the correspondence between the quantum phase-space dy- 17 the square magnitude of ␾(x,y) is studied as ␣ is varied. namics and classical dynamics. In the following we address Having related ␣ to the shape of the fiducial vector through this point from the point of view of CSRs using a MUS as ␭, the findings in Ref. 17 are in accordance with the inter- fiducial vector. In order to obtain the same notation as used pretations in Sec. II C, namely that high resolution is ob- by Torres-Vega and co-workers, we redefine the parameter tained in the position direction for small ␣ ͑where the uncer- characterizing the shape of the fiducial vector according to ␭2ϭ(1ϩ2␣)/(1Ϫ2␣). For simplicity, let the Hamiltonian tainty of the fiducial vector in the position direction is also 2 2 small͒, whereas high resolution is obtained in the momentum be written in a rescaled form, H(Q,P)ϭ(P ϩQ )/2. direction for large ␣. It also only in this limit that the square With the fiducial vector being a physically centred MUS, magnitude of the eigenstates ͑being independent of q and the time-dependent phase-space wave function of a Glauber p) are strictly stationary states of the classical equations of coherent state may be found directly from Ref. 36 as motion. 1ϩ2␣ ␺ ͑q,p͒ϭ͑1Ϫ4␣2͒1/4exp Ϫ ͑qϪq ͒2 Finally, let us demonstrate how ␺0(q,p) with the fidu- t ͫ 4ប t cial vector being a physically centred MUS can be found by solving the stationary FBR Schro¨dinger equation. Following 1Ϫ2␣ i Ϫ ͑pϪp ͒2ϩ ͑qp Ϫpq ͒ the procedure of the preceding subsection we find that the 4ប t 2ប t t FBR wave function satisfy i␣ i 2 d d Ϫ ͑qϪqt͒͑pϪpt͒Ϫ t , ͑63͒ Ϫ2͓2͑␣/2͒3/2ϩz͔ ϩ͓z2Ϫ4͑␣/2͒3/2zϪ1͔ ប 2 ͬ ͭ dz2 dz ͮ provided that the parameters qt and pt satisfy the classical ˙ ˙ ϫ␺0͑z͒ϭ0, ͑58͒ equations of motion qtϭpt and ptϭϪqt , that is,

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FIG. 1. Schematic figure of the ‘‘classical’’ motion of three different coherent states of a harmonic oscillator in different CSRs. In ͑a͒ the coherent states are depicted in the same CSR based on a physically centred MUS fiducial vector. In ͑b͒ the coherent states are shown in CSRs with fiducial vectors coinciding with the coherent states themselves at time zero. See text for details. The arrows indicate the direction of the phase velocity as the states travel along the dashed circles.

q ϭq cos tϩp sin t, at (ϪqЈ,ϪpЈ). For instance, in the CSR where the fiducial t 0 0 ͑64͒ vector is centred at (q0 ,p0) the phase-space wave function ptϭp0 cos tϪq0 sin t. representing a coherent state at time tϭ0 becomes Equation ͑63͒, which is easily seen to satisfy the phase-space 1ϩ2 1Ϫ2 2 1/4 ␣ 2 ␣ 2 Schro¨dinger equation, is the one found in Ref. 15. The ␺0͑q,p͒ϭ͑1Ϫ4␣ ͒ exp Ϫ q Ϫ p phase-space density, as given by the square magnitude of Eq. ͫ 4ប 4ប ͑63͒,is i i␣ Ϫ ͑qp Ϫpq ͒Ϫ qp , ͑66͒ 1ϩ2␣ 2ប 0 0 ប ͬ 2 2 1/2 2 ͉␺t͑q,p͉͒ϭ͑1Ϫ4␣ ͒ exp Ϫ ͑qϪqt͒ ͫ 2ប which gives rise to a Gaussian density centred at the origin. 1Ϫ2␣ But, the initial phase velocity is still given by Ϫ ͑pϪp ͒2 . ͑65͒ v ϭ(p ,Ϫq ) and the density will move along the circle 2ប t ͬ 0 0 0 centred at (Ϫq0 ,Ϫp0). In Fig. 1͑a͒ we have shown, in the The density is thus a Gaussian centred on the classical orbit. standard CSR, contours of three different coherent states at The axes are aligned with the coordinate axes at all times, time tϭ0 along with the direction of their initial phase- with their lengths controlled by ␣. Since the classical dy- velocity vectors, and a sketch of the phase-space orbits they namics in a rescaled oscillator describes rigid rotations about will follow as time evolves. In Fig. 1͑b͒ the same coherent the origin, the density in Eq. ͑65͒ only obeys classical evo- states are shown in the CSRs defined above, that is, the state lution for ␣ϭ0(␭ϭ1). For this value of ␣ the fiducial vector and the fiducial vector being the same. vector is the ground state of the rescaled harmonic oscillator. These examples show how important the knowledge of The physical properties of the wave functions, Eq. ͑63͒, the fiducial vector is when one wishes to interpret the phase- are, of course, independent of ␣. For any ␣ we find, e.g., that space density. Especially, we see that of the set of fiducial 2 2 ͗␺t͉Q͉␺t͘ϭqt , ͗␺t͉P͉␺t͘ϭpt and (⌬Q)t ϭ(⌬P)t ϭប/4. vectors considered here—minimum uncertainty states—only Thus, all the wave functions in Eq. ͑63͒ are coherent states, a single one will result in classical dynamics for the phase- and not squeezed states as stated in Ref. 15, despite the el- space density of a coherent-state in a harmonic potential, liptic contours in phase space of their square magnitudes. namely the fiducial vector being equal to the ground state of Their different shapes do not reflect different uncertainties in the considered harmonic oscillator. As may be deduced from the fundamental operators but different choices of the fidu- Eq. ͑33͒ in Ref. 36, this is true for any PSR eigenstate of the cial vector! harmonic potential; a fact also observed, albeit not ex- To illustrate how the choice of a not physically centred plained, by Torres-Vega and Morales-Guzmań.15 fiducial vector might be confusing, let us consider as fiducial In fact, the square magnitude of any phase-space wave vector a MUS centred at the arbitrary point (qЈ,pЈ). The function will evolve classically in an at most quadratic po- shape density of the coherent states are still Gaussian but it tential if the fiducial vector is chosen to be a physically cen- will be displaced by ϪqЈ ͓ϪpЈ͔ in the position ͓momentum͔ tred MUS with the ‘‘right’’ shape. From the general direction in comparison with the density in Eq. ͑65͒. This ͑fiducial-vector independent͒ equation of motion for the implies that the density will evolve as though it is subject to density12 this is not easily recognized, but with a specific a harmonic potential with the vertex of the parabola centred choice of fiducial vector the knowledge of the fiducial vector

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This observation has spawned several questions: How 1 2 V͑Q͒ϭk0ϩk1Qϩ k2Q , ͑67͒ should such phase-space wave functions be interpreted, how 2 are they related to other representations, how can wave me- the square magnitude of any phase-space wave function in chanics be done in phase space, and how do phase-space this class of CSRs satisfies wave functions evolve in time? Each of these questions have been addressed with particular emphasis on the role played ץ Vץ ץ p ץ 2 c 2 by the fiducial vector. ͉␺t͑q,p͉͒ ϭ Ϫ ϩ ͉␺t͑q,p͉͒ pͪ Clearly, since the square magnitudes of wave functionsץ qץ qץ t ͩ mץ 2 2 2 in the standard CSR, i.e., the one where the fiducial vector is ץ Vc ␭ ץ 1 ϩប Ϫ ͉␺ ͑q,p͉͒2, a minimum-uncertainty state ͑MUS͒ with vanishing expected p tץqץ q2 2mץ 2␭2 ͩ ͪ values of position and momentum, are Husimi functions, ͑68͒ densities in any CSR must be Husimi-like functions. As for the standard Husimi function the interpretation of any CSR where VcϭV(q). Independently of ប, this equation reduces to the Liouville equation for ␭ϭ(mk )1/4. This implies that density as a probability density in phase space is trouble- 2 25 the phase-space density for a free particle and a particle in a some. However, following Royer, a CSR density may be linear potential will only undergo classical evolution in the considered a ‘‘fuzzy’’ density in phase space, at each point limit ␭ 0 ͑as seen above for the stationary states of the (q,p) giving the relative probability of the system being situ- linear potential→ ͒, and for a harmonic oscillator the phase- ated in a ‘‘fuzzy’’ neighbourhood of the centre of space density evolves classically if the fiducial vector is the D(q,p)͉␹͘. Precisely how this ‘‘fuzzy’’ neighbourhood is ground state of the oscillator. In virtue of Ehrenfest’s Theo- shaped and where it is positioned depends on the properties rem and Eq. ͑33͒, the centre of any density based on a physi- of the fiducial vector. Such an interpretation suggests that cally centred fiducial vector will evolve classically in the information about the system may be extracted from images potential Eq. ͑67͒, but only for a specific choice of such a in phase space. However, extreme care should be exercised. fiducial vector the density as a whole will behave For instance, only for physically centred fiducial vectors the classically.37 It is, furthermore, evident from the equation of point (q,p) represents the actual physical point in phase motion of the Husimi function10,32 that if the potential has space. Moreover, the notion of a ‘‘fuzzy’’ neighbourhood non-vanishing derivatives of higher order than two no choice seems to be descriptive only for localized fiducial vectors, of MUS as fiducial vector will result in classical evolution of such as MUSs. For choices of other more ‘‘diffuse’’ fiducial the phase-space density. vectors very little information about the system is provided If the MUS is not physically centred the dynamics in the by the phase-space images. Accordingly, we infer that im- situations described above will still be ‘‘classical-like’’ in ages of CSR densities are descriptive only in the cases where the sense that the density will move along trajectories that the fiducial vector is localized. Still knowledge about the are solutions to Hamilton’s equation, however with a the fiducial vector is crucial as was demonstrated by the ex- Hamiltonian, cf. Eq. ͑33͒, amples in Sec. III.B.. Being ‘‘true’’ state-vector representations, the CSRs ad- 1 mit the possibility of solving a quantum problem directly in H͑q,p͒ϭ ͑pϩp ͒2ϩV͑qϩq ͒. ͑69͒ 2m ␹ ␹ phase space. However, because of the fiducial-vector inde- Thus for a finite ប the CSR will only behave classical- pendence of the mapping of the fundamental operators, e.g., ¨ like for certain potentials and with special choices of the the stationary Schrodinger equation for a given problem as- fiducial vector. Recently, Klauder38 has examined the dy- sumes the same form in any CSR. Accordingly, any solution namics of the CSR density for ប going to zero.39 He demon- to this equation could belong to any CSR; to which we do strated that if the fiducial vector has a vanishing dispersion in not know. As we have discussed, this is not a problem if one this limit the dynamics of any state in any potential will be merely wishes to compute expectation values whereas ex- classical-like in the above sense as ប goes to zero. plicit knowledge about the fiducial vector is required for calculating transition matrix elements. To resolve this ambiguity we have suggested two approaches. Either one may IV. DISCUSSION AND CONCLUDING REMARKS augment the Schro¨dinger equation with an auxiliary equation In the coherent-state representations ͑CSRs͒, as formu- that fixes the fiducial vector. Or one may invoke the Fock- lated by Klauder and Skagerstam,5 wave functions are Bargmann representation. While the former approach neces- L 2(2) functions of the parameters q and p and the funda- sitates two equations to be solved it is applicable for any mental operators Q and P are mapped according to Eq. ͑1͒, fiducial vector that admits the set-up of the auxiliary equa- irrespectively of the fiducial vectors. From these fundamen- tion. Conversely, the Fock-Bargmann representation can

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Downloaded¬12¬Jan¬2010¬to¬192.38.67.112.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright;¬see¬http://jcp.aip.org/jcp/copyright.jsp 7240 Mo”ller, Jo”rgensen, and Torres-Vega: Coherent-state representations only be put in one-to-one correspondence with CSRs based 6 A. Perelomov, Generalized Coherent States and Their Applications: Texts on MUS fiducial vector, but only solution of a single equa- and Monographs in Physics ͑Springer Verlag, Berlin, 1986͒. 7 tion is required. In either scheme knowledge about the fidu- M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 1984 . cial vector is provided. The ‘‘pitfalls’’ encountered upon just ͑ ͒ 8 J. E. Harriman and M. E. Casida, Int. J. Quantum Chem. 45, 263 ͑1993͒. solving the phase-space Schro¨dinger equation and the reso- 9 M. V. Berry, Philos. Trans. R. Soc. London, Ser. A 287, 237 ͑1977͒. lution of the ambiguity obtained by employing the Fock- 10 K. Takahashi, J. Phys. Soc. Jpn. 55, 762 ͑1986͒. Bargmann approach were illustrated in Sec. III B in the case 11 A. Voros, Phys. Rev. A 40, 6814 ͑1989͒. 12 Go. Torres-Vega and J. H. Frederick, J. Chem. Phys. 98, 3103 ͑1993͒. of a linear potential. 13 As the standard Husimi function may be used for com- Go. Torres-Vega, J. Chem. Phys. 98, 7040 ͑1993͒. 14 Go. Torres-Vega, J. Chem. Phys. 99, 1824 ͑1993͒. puting expectation values as integrals over phase space, so 15 Go. Torres-Vega, in Third International Workshop on Squeezed States may any CSR density. This requires a mapping of the opera- and Uncertainty Relations, edited by D. Han, Y. S. Kim, N. M. Rubin, tor in question into a function of the parameters q and p. Y. Shih, and W. W. Zachary ͑NASA, Maryland, 1994͒, pp. 275–280. 16 Unlike the one for the CSRs, this mapping is fiducial-vector Go. Torres-Vega and J. D. Morales-Guzmań, J. Chem. Phys. 101, 5847 dependent and in the cases the operator being ‘‘pure’’ pow- ͑1994͒. 17 Go. Torres-Vega, A. Zu´ñiga-Segundo, and J. D. Morales-Guzmań, Phys. ers of either of the fundamental operators a procedure for Rev. A 53, 3792 ͑1996͒. obtaining these phase-space functions was given. The first 18 R. J. Glauber, Phys. Rev. 130, 2529 ͑1963͒. two powers of both Q and P were stated explicitly and re- 19 J. E. Harriman, J. Chem. Phys. 100, 3651 ͑1994͒. 20 Go. Torres-Vega and J. H. Frederick, J. Chem. Phys. 93, 8862 ͑1990͒. duced to the previously reported results for the usual Husimi 21 representation.8 Apart from merely devicing the operator P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed. ͑Oxford University Press, Oxford, 1958͒. map the procedure also provided insight into the nature of 22 J. J. Wl”odarz, J. Chem. Phys. 100, 7476 ͑1994͒. these generalized Husimi representations and their connec- 23 J. E. Harriman, J. Chem. Phys. 88, 6399 ͑1988͒. tions to the Weyl-Wigner representation. For instance, the 24 S. Stenholm, Eur. J. Phys. 1, 244 ͑1980͒. 25 A. Royer, Phys. Rev. Lett. 55, 2745 ͑1985͒. dual nature of the Husimi representation, also in the gener- 26 alized case, was demonstrated explicitly by the form of the For a discussion of simultaneous measurements of non-commuting observables in terms of the precise mathematical concepts ‘‘fuzzy sets’’ and operator map which, in fact, defines a generalized Glauber- ‘‘fuzzy measures,’’ see e.g., E. Prugoveoˇki, Found. Phys. 3,3͑1973͒ and Sudarshan P-representation. Moreover, it was shown that the E. B. Davies and J. T. Lewis, Commun. Math. Phys. 17, 239 ͑1970͒. function ‘‘linking’’ the Husimi, Weyl-Wigner, and Glauber- 27 This should be compared with, for instance, the position-space represen- Sudarshan representations together is the Wigner function, tation where the square magnitude of the wave function can be used to and not any of the other symbols, of the fiducial vector cho- calculate the expectation value of any operator that is a function of the position operator. This is due to the fact that any such operator is multi- sen. Hence, another instance giving rise to considering the plicative in the position-space representation. Since neither the position Wigner representation the more fundamental density- nor the momentum operator is multiplicative in a CSR it is not obvious operator phase-space representation. whether the square magnitude of the phase-space wave function can be used in a similar way. In conclusion, the present analysis revealed that the 28 phase-space representation of Torres-Vega and P. Meystre and M. Sargent III, Elements of Quantum Optics, 2nd ed. 12–17 ͑Springer-Verlag, Berlin, 1991͒. co-workers coincide with the class of all CSRs, as for- 29 J. P. Dahl, in Classical and Quantum Systems: Foundations and Symme- 5 mulated by Klauder and Skagerstam, that these representa- tries, Proceedings of the II International Wigner Symposium, edited by H. tions are related to the position and momentum representa- D. Doebner, W. Scherer, and F. Schroeck, Jr. ͑World Scientific, Sin- gapore, 1993͒, pp. 420–423. tions in standard Diracian manner, and that the fiducial 30 vector plays a prominent and crucial role in the CSRs that J. P. Dahl, in Conceptual Trends in Quantum Chemistry, edited by E. S. Kryachko and J. L. Calais ͑Kluwer Academic Publishers, Amsterdam, must be recognized when images of CSR densities are to be 1994͒, pp. 199–226. interpreted and when one desires to do wave mechanics in 31 K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1882 ͑1969͒. phase space. 32 R. T. Skodje, H. W. Rohrs, and J. VanBuskirk, Phys. Rev. A 40, 2894 ͑1989͒. 33 ACKNOWLEDGMENTS R. J. Glauber, Phys. Rev. 131, 2766 ͑1963͒. 34 L. D. Landau and E. M. Lifshitz, Quantum Mechanics,Vol.3ofCourse of The authors would like to thank Dr. J. J. Włodarz and Theoretical Physics, 3rd ed. ͑Pergamon Press Ltd., Oxford, 1977͒. 35 Professor J. P. Dahl for careful reading of the manuscript and D. Zwillinger, Handbook of Differential Equations ͑Academic, San Diego, 1989͒. enlightening comments. This research was supported by The 36 K. B. Mo”ller, T. G. Jo”rgensen, and J. P. Dahl, Phys. Rev. A 54, 5378 Danish Natural Science Foundation. ͑1996͒. 37 This should be contrasted with the Wigner density which always behaves 1 V. Fock, Z. Phys. 49, 339 ͑1928͒. classically in the potential Eq. ͑67͒. 38 2 V. Bargmann, Commun. Pure Appl. Math 14, 187 ͑1961͒. J. R. Klauder, in Workshop on Harmonic Oscillators, edited by D. Han, Y. 3 E. Wigner, Phys. Rev. 40, 749 ͑1932͒. S. Kim, and W. Zachary ͑NASA, Maryland, 1993͒, pp. 19–28. 4 K. Husimi, Proc. Phys. Math. Soc. Jpn. 22, 264 ͑1940͒. 39 The analysis of Klauder only involves the explicit ប dependence in the 5 J. R. Klauder and B.-S. Skagerstam, Coherent States ͑World Scientific, equations of motion and is therefore appropriate if the ប dependence of the Singapore, 1985͒. initial phase-space density is ignored. See also Ref. 32.

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