Wigner functions of s waves

Wigner functions of s waves

J. P. Dahl,^1,2S. Varro,^3,2A. Wolf,²and W. P. Schleich²

1Chemical Physics, Department of Chemistry, Technical University of Denmark, DTU 207, DK-2800 Lyngby, Denmark

2Institut für Quantenphysik, Universität Ulm, D-89069 Ulm, Germany

3Research Institute for Solid State Physics and Optics, H-1525 Budapest, P.O. Box 49, Hungary 共Received 15 January 2007; published 11 May 2007兲

We derive explicit expressions for the Wigner function of wave functions inDdimensions which depend on the hyperradius—that is, ofswaves. They are based either on the position or the momentum representation of the s wave. The corresponding Wigner function depends on three variables: the absolute value of the D-dimensional position and momentum vectors and the angle between them. We illustrate these expressions by calculating and discussing the Wigner functions of an elementaryswave and the energy eigenfunction of a free particle.

DOI:10.1103/PhysRevA.75.052107 PACS number共s兲: 03.65.Ca, 02.30.Uu, 42.50.⫺p

I. INTRODUCTION

Quantum phase-space distributions play an important role in various branches of physics ranging from nuclear and par- ticle physics via quantum optics to quantum chaology关1–4兴.

Due to its simplicity and neighborhood to classical phase- space distributions, the Wigner function关5兴stands out most clearly from the wealth of such distributions and has at- tracted a lot of attention. However, most studies of the Wigner function have concentrated on one-dimensional quantum systems. In the present paper we analyze the Wigner function of quantum states in aD-dimensional con- figuration space where the wave function depends only on the hyperradius that is the square root of the sum of squares of the D Cartesian coordinates. By choice, the quantum states in question may refer to a single particle inDdimen- sions or to a system of 2 or more particles, each of which resides inD/ 2 or fewer dimensions关6兴.

A. Whyswaves?

Wave functions which depend on the hyperradius only correspond to waves with angular momentum zero. They are commonly referred to asswaves. However, for a system of particles there also exist wave functions which correspond to a vanishing total angular momentum and do not have the property to depend exclusively on the hyperradius. In order to distinguish theseswaves from the ones with hyperradius dependence we have introduced in Ref.关7兴the namehyper- radial s waves. We emphasize that the present paper deals solely with hyperradial s waves, but in order to keep the notation simple we use throughout the article the name s wave as a shorthand notation for a hyperradials wave.

The importance of our study derives from the close cor- respondence between entanglement and the negative parts of the Wigner function recently established关7兴forswaves de- scribing a system of particles. Indeed, due to the constraint on the dependence of theswave on the coordinates,swaves also describe entangled quantum systems. In a recent paper 关7兴 we have shown that the negative volume of the Wigner function is an excellent measure for the entanglement con- tained in such a state. Moreover, it does not suffer the restric- tions关8兴of the other measures on the number of particles.

Apart from this application of Wigner functions to the field of quantum informationswaves are interesting in their own right. Indeed, the ground state of isotropic quantum sys- tems such as a hydrogen atom 关9兴 or a Bose-Einstein con- densate is described by answave. Here the Wigner function provides a deeper insight into the physics of these states and, in particular, correlations between positionrជand momentum pជ spanning quantum phase space.

Since s waves correspond to angular momentum zero, a naive consideration would make one expect the position and momentum variables rជ ^{and p}ជ to be either parallel or antiparallel—that is, the angle␪betweenrជandpជ⬅បkជto take on only the values ␪= 0 or ␪=␲. However, this classical picture of answave neglects interference. Indeed, theswave is the result of theinterferenceof a continuous superposition 关10兴 of one-dimensional motions corresponding to rជ ^{and p}ជ being parallel or antiparallel. Since thes wave depends on the hyperradius only and is independent of the directions, the individual elementary waves corresponding to the one- dimensional radial motion in all space directions contribute with equal weight. In this sense theswave consists predomi- nantly of interference.

As a consequence of this complex interference of elemen- tary waves now motions which are not in the radial direction emerge. Since they originate from interference, their weight as described by the Wigner function may be negative. Thus, the Wigner function of answave may take on nonvanishing values for angles␪⫽0 or ␪⫽␲ and, in particular, may dis- play domains whereWis negative.

At first sight this behavior is rather unusual. In a classical phase-space approximation of an eigenstate of angular mo- mentum we expect the values of angular momentum to be constrained by a ␦ function to the eigenvalue. However, in the Wigner-function formalism we find that all values of an- gular momentum manifesting themselves in different angles can occur.

This behavior is reminiscent of the properties of the Wigner function of an energy eigenstate in one dimension.

Here a classical picture would suggest a␦function along the classical phase-space trajectory determined by the energy ei- genvalue. However, the quantum description by the Wigner function softens the ␦ function into an Airy function with

oscillations in the phase-space domain circumvented by the classical trajectory.

Phase space is usually described by position and momen- tum variables rជ and pជ, respectively. However, in order to simplify the formalism we express phase space in terms of position and wave vector kជ⬅pជ/ប. In this way the resulting expressions for the Wigner function do not explicitly contain ប.

B. Outline and summary

Our paper is organized as follows. In Sec. II we start from the well-known definition of a Wigner function inDdimen- sions and demonstrate that the Wigner function W corre- sponding to answave can only depend on the absolute val- uesrandkof the position and the wave vectorsrជandkជand the angle␪ ^betweenrជandkជ. We then derive explicit expres- sions forW, either in terms of the position or the wave vector representation of the wave function. Both formulas clearly demonstrate thatWdepends onr,k, and␪only. Moreover,W results from a double integral over a position and an angle variable.

The next two sections illustrate these expressions forW for two specific quantum states. In Sec. III we calculate the Wigner functions of an elementary s wave followed in the Sec. IV by the example of an energy eigenstate of a free particle in D dimensions. Three arguments support this choice:共i兲whereas the elementaryswave illustrates the ex- pression for W in position space, the free particle follows from the complementary representation in wave vector space,共ii兲the elementaryswave has already played a major role in a previous study关7兴on the connection of entangle- ment and negative parts of the Wigner function, and共iii兲the example of a free particle leads to differential equations in phase space 关11兴 which can still be solved analytically in order to test the result obtained from the direct integration of the definition of the Wigner function. In this discussion of the two exemplary Wigner functions we focus on their de- pendence on all three space variables—that is,r, k, and ␪^. Moreover, we analyze in detail the influence of the number D of dimensions. We conclude in Sec. V by presenting a summary and an outlook.

In order to concentrate on the essential ideas while keep- ing the paper self-contained we have included all relevant calculations but have moved them to extended appendixes.

Since our discussion of the Wigner function of an s wave relies heavily on the concept of hyperspherical coordinates, we dedicate Appendix A to a brief summery of this topic.

Here we follow closely the treatment of Ref.关12兴. In Appen- dix B we perform the integrations overD− 2 angles in the original definition of the Wigner function, arriving at the two expressions for the Wigner distribution of ans wave central to the present paper. In order to lay the foundation for the discussion of the Wigner function of the free particle as well as to show that the wave-vector representation of answave only depends onkwe recall in Appendix C the wave vector representation of an s wave. Appendix D is devoted to the evaluation of the two-dimensional integration in the defini- tion of the Wigner function of the elementary s wave. In

Appendix E we finally turn to the example of a free particle inD dimensions. Here we first recall the wave function in position space and with the help of Appendix C we find that the corresponding wave-vector representation involves a ␦ function. We then perform the necessary integrations and ob- tain the Wigner functionWE of a free particle. As a test of this expression we verify that its marginal distribution leads to the probability density in position space. We conclude by deriving the differential equations in phase space determin- ingWE. One set of equations follows from application of the Weyl-Wigner transform to the energy eigenvalue equation.

An additional set of equations emerges from the eigenvalue equation of the square of the angular momentum. Only these two sets together determine uniquely the Wigner function.

II. REPRESENTATION

In this section we show that the isotropy of a D-dimensionals wave with 2艋Dpermits us to express the corresponding Wigner function as a double integral. We de- rive two equivalent representations of the Wigner function:

one is in terms of the position and the other in terms of the wave-vector wave function. For this purpose we first note that the corresponding Wigner function of ansstate depends on three variables only.

A. Three variables only

The crucial ingredient of our proof is the fact that for any orthogonal matrixUwith

UU^T=U^TU=1 共1兲 and

detU= 1, 共2兲

the value of the wave function

␺共兩Urជ兩兲=␺共兩rជ兩兲 共3兲 of ansstate does not change. When we now recall the defi- nition

W共rជ,kជ_{兲 ⬅} ¹

共2␲兲^D

冕

^{dD␰e−ikជ·␰ជ␺*}

冉

^{rជ−12␰ជ}

冊

^␺

冉

^{rជ+12␰ជ}

冊

^共4兲

of the Wigner function and include the identity matrixU^TU

=1in the arguments of the Fourier term,

e^{−ikជ·␰ជ}=e^{−i共Ukជ兲T·U␰ជ}, 共5兲 and in the wave functions,

␺

冉

^{rជ±2␰ជ}

冊

^=␺

冉

^UT

冋

^{Urជ± U2␰ជ}

册 ^冊

^{, 共6兲}

and establish the integration variable

␨ជ_=U␰ជ_{, d}^D␨^=dD␰^, 共7兲 we realize that the value of the Wigner function

W共rជ,kជ兲=W共Urជ,Ukជ兲 共8兲 is independent of a simultaneous rotation ofrជandkជ. Due to this fact, the Wigner function can only depend on three vari-

ables: namely, the absolute valuer⬅兩rជ兩 of the position vec- tor, the absolute value k⬅兩kជ_兩 of the wave vector, and the angle␪⬅arccos关共rជ^·kជ_兲/rk兴 between them.

B. Explicit expressions

In the previous section we have shown that due to the isotropy of the s-wave function, the corresponding Wigner functionW can depend onr, k, and the angle␪ ^{between r}ជ and kជ only. We now derive two completely equivalent ex- pressions for W which bring out this fact explicitly. These two formulas are either in terms of the position or wave- vector representation of thes wave.

1. Position space

We start from the standard definition, Eq. 共4兲, of the Wigner function inDdimensions and note that for answave the wave function␺⁼␺共兩rជ兩兲depends on the hyperradius only.

As a consequence Eq.共4兲takes the form W= 1

共2␲兲^D

冕

^{dD␰e−ikជ·␰ជ␺*共r−兲␺共r+兲, 共9兲}

where we have introduced the abbreviation

r_±⬅

冏

^{rជ±12␰ជ}

冏

⁼

冋

^{rជ2+ 14␰ជ2±rជ·␰ជ}

册

^{1/2. 共10兲}

The special form of the integrand in Eq.共9兲suggests that one perform the integration over theD components of ␰ជ _in hyperspherical coordinates summarized in Appendix A. This method involves as variables the hyperradius␰, the angle␽^, and theD− 2 angles␸1, . . . ,␸D−2. It is now possible to orient the coordinate system of␰ជ relative torជandkជin such a way that it is possible to perform the integration over theD− 2 angles ␸1, . . . ,␸D−2 explicitly. In Appendix B 1 we pursue this approach and derive the expression

W=共2␲兲^{−共D+1兲/2}共ksin␪兲^{−共D−3兲/2}

冕

0

⬁

d␰␰^共D+1兲/2

⫻

冕

0

␲

d␽^{sin共D−1兲/2}␽^J_{共D−3兲/2}共␰^ksin␪^sin␽兲

⫻e^{−ik␰cos␪cos␽}␺^*共r₋兲␺共r₊兲 共11兲 for the Wigner function of ans wave. In hyperspherical co- ordinates the quantitiesr_±defined in Eq.共10兲take the form r_±=

冋

^{r2+14␰2±r␰cos␽}

册

^{1/2. 共12兲}

Equations 共11兲 and 共12兲 bring out most clearly that W only depends onr,k, and the angle␪^betweenrជandkជ—that is,W=W共r,k,␪兲. Moreover, we emphasize that two integra- tions are necessary to obtainW. At first sight one might think that a single integration might suffice since the wave func- tion of answave depends on a single variable. However, the above calculation demonstrates that this suspicion is wrong.

We conclude by noting that strictly speaking these expres- sions are not valid for D= 2, since they emerge from the

expression, Eq.共11兲, which is only defined for 3艋D. Nev- ertheless, we show in Appendix B that the expression, Eq.

共11兲, holds also true forD= 2.

2. Wave-vector space

We now turn to a representation ofWin terms of the wave function ␺^˜ in wave-vector space. For this purpose we start from the definition

W共r,k兲= 1

共2␲兲^D

冕

^{dDqeirជ·qជ␺˜*}

冉

^{kជ−12qជ}

冊

^␺˜

冉

^kជ+12qជ

冊

^共13兲

in terms of the wave function

␺˜共kជ_兲= 1

共2␲兲^D/2

冕

^{dDr␺共rជ兲e−ikជ·rជ 共14兲}

inkជspace.

When we compare the two definitions, Eqs.共4兲and共13兲, of the Wigner function we recognize that the roles ofrជandkជ are interchanged. Therefore, we expect an expression forW in terms of ␺^˜ similar to Eq. 共11兲. In order to derive this formula we first recall in Appendix C that the wave-vector representation of an s wave depends only on the absolute value k⬅兩kជ_兩 of the wave vector. As a consequence the Wigner function of ans wave takes the form

W共rជ,kជ兲= 1

共2␲兲^D

冕

^{dDqeirជ·qជ␺˜*共k−兲␺˜共k+兲, 共15兲}

with

k_±⬅

冏

^kជ±12qជ

冏

⁼

冋

^{kជ2+14qជ2±kជ·qជ}

册

^{1/2. 共16兲}

Again we can express theDintegrations over the wave- vector componentsq₁, . . . ,qDin hyperspherical coordinates.

However, in the choice of the orientation of theqជ-coordinate system we have to take into account the fact that the roles of rជand kជ are interchanged. In Appendix B 2 we perform the integration over theD− 2 angles ␸1, . . . ,␸D−2explicitly and derive the expression

W=共2␲兲^{−共D+1兲/2}共rsin␪兲^{−共D−3兲/2}

冕

0

⬁

dqq^共D+1兲/2

⫻

冕

0

␲

d␽^{sin共D−1兲/2}␽^J共D−3兲/2共qrsin␪^sin␽兲

⫻e^{irqcos␪cos␽}␺^˜*共k₋兲␺^˜共k₊兲 共17兲 for the Wigner function of answave with

k_±=

冋

^{k2+14q⫿kqcos␽}

册

^{1/2. 共18兲}

Again the formula only depends onr,k, and ␪^.

We conclude by noting that again the expression, Eq.

共17兲, contains the caseD= 2.

3. Kernel

We conclude by comparing the two expressions, Eqs.共11兲 and共17兲, for the Wigner function W of answave. First we note that the formal structure of both relations is identical.

The difference occurs in the wave function␺ ^versus ␺^{˜ and} the roles ofrandkare interchanged.

Indeed, we can bring out the similarity even more clearly by introducing the kernel

K共b,␪兩␨^,␩兲 ⬅ 共2␲兲^{−共D+1兲/2}共bsin␪兲^{−共D−3兲/2}␨^共D+1兲/2

⫻sin^{共D−1兲/2}␩^{e−ib␨cos␪cos␽}

⫻J_{共D−3兲/2}共␨^bsin␪^sin␩兲, 共19兲 which casts Eqs. 共11兲 and 共17兲 into the rather symmetric form

W=

冕

0

⬁

d␨

冕

0

␲

d␩K共k,␪兩␨^,␩兲␺^*„s₋共r兲…␺„s₊共r兲… 共20兲 and

W=

冕

0

⬁

d␨

冕

0

␲

d␩K^*共r,␪兩␨^,␩兲␺^˜*„s₋共k兲…␺^˜„s₊共k兲…, 共21兲 where

s_±共s兲 ⬅

冋

^{s2+14␨2±s␨cos␩}

册

^{1/2. 共22兲}

We note that the kernelK depends on two pairs of vari- ables. The first pair consists of the two integration variables

␨^and␩^{. Here}␩corresponds to the angle␽^and␨^represents either the position variable␰or the wave vectorq. The sec- ond pair includes two out of the three phase-space variables.

The angle␪ is always present. However, the first slot ofK indicated by the variablebdepends on the representation of the state. When we start from the wave function␺in position space—that is,␺=␺共r兲—the first argument ofK is the vari- ablekwhich is complementary tor. Therdependence of the Wigner function then results from the wave function ␺ evaluated at the argument s_±共r兲. When we start from the wave-vector representation with ␺^˜=␺^˜共k兲 the first argument of K is r and the k dependence enters through the wave function␺^˜ evaluated at the arguments_±共k兲.

Needless to say both formulas, Eqs. 共20兲 and 共21兲, are completely equivalent. However, one might be more conve- nient to perform the two integrations than the other. Indeed, in Sec. IV we discuss the Wigner function of a free particle inDdimensions. Whereas the calculation in position space is extremely difficult, the integrations in wave-vector space are rather elementary.

C. Marginal distributions

The total number of dimensions of the phase space is 2D consisting of theDcoordinates of rជandDcoordinates ofkជ. However, the Wigner function of answave depends on only three variables: namely,r,k, and ␪.

From the original definitions, Eqs. 共4兲 and 共13兲, of the Wigner function we can easily deduce the marginal distribu- tions

兩␺共rជ兲兩²=

冕

^{dDkW共rជ,kជ兲 ⬅P共rជ兲 共23兲}

and

兩␺共kជ_兲兩²=

冕

^{dDrW共rជ,kជ兲 ⬅P共kជ兲. 共24兲}

Obviously this property must also hold for the Wigner func- tion of ans wave. However, it is not obvious from the ex- pressions, Eqs. 共20兲 and共21兲. We now verify the marginal distributions, Eqs.共23兲and共24兲. For this purpose we substi- tute the Wigner distribution, Eq. 共20兲, into the integral, Eq.

共23兲, which yields P共rជ兲=

冕

0

⬁

d␨

冕

0

␲

d␩M共␨^,␩兲␺^*„s₋共r兲…␺„s₊共r兲…. 共25兲 Here we have introduced the integrated kernel

M共␨^,␩兲 ⬅

冕

^{dDbK共b,␪兩␨,␩兲, 共26兲}

with the integration

冕

^dDb=

冕

0

⬁

dbb^D−1

冕

0

␲

d␪^sinD−2␪

冕

^{d␻ 共27兲}

extending over theD-dimensional space ofb.

Likewise we find from Eq.共21兲 P共kជ兲=

冕

0

⬁

d␨

冕

0

␲

d␩M^*共␨^,␩兲␺^*„s₋共k兲…␺„s₊共k兲…. In Appendix B 3 we derive the expression

M共␨^,␩兲= 1

冑

␲

⌫

冉

^D2

冊

⌫

冉

^{D2− 1}

冊

^{sinD−2␩␦共␨兲 共28兲}

and the␦ function in␨allows us to perform the integration which yields

P共rជ兲= 1

冑

_␲

⌫

冉

^D2

冊

⌫

冉

^D2^{− 1}

冊 ^冕

^{0␲d␩sinD−2␩兩␺共r兲兩2 共29兲}

or

P共kជ兲= 1

冑

_␲

⌫

冉

^D2

冊

⌫

冉

^D2^{− 1}

冊 ^冕

^{0␲d␩sinD−2␩兩␺˜共k兲兩2. 共30兲}

With the help of the integral relation关13兴

冕

0

␲

d␸^sin␯␸⁼

冑

_␲^⌫

冉

^{␯+ 12}

冊

⌫

冉

^{␯+ 2}2

冊

^{, 共31兲}

we arrive at the probability distributions, Eqs.共23兲and共24兲, in position or wave-vector space.

III. ELEMENTARYsWAVE

Our first application of the expression, Eq. 共11兲, for the Wigner function of answave results from the wave function

⌿^共D兲共r兲=N关1 +a共␬r兲²兴

冉

^␬␲²

冊

^{D/4e−共␬r兲2/2, 共32兲}

with the normalization constant

N⬅

冋 冉

^{1 +12aD}

冊

^2+12a2D

册

^{−1/2. 共33兲}

Here ␬ is a real constant with a dimension of an inverse length andais a dimensionless real parameter.

Despite its simplicity the elementaryswave⌿^共D兲contains a wealth of physics. Fora= 0 we find the familiar Gaussian and in the limit of a→±⬁ we arrive at the shell function which has attracted our attention in the context of the time evolution ofswaves 关14兴. Most recently we have also con- centrated on the state emerging for a= −2 /D. In particular, we have shown关7兴that this member of the class of elemen- tary s waves corresponds to a maximally entangled state.

Since the parametera interpolates between these cases, we refer to it as the interpolation parameter.

When we substitute Eq.共32兲into the expression, Eq.共11兲, for the Wigner function in terms of the position wave func- tion and perform the integration over␰^and␽we arrive at the formula

W_a^共D兲=W₀^共D兲共r,k兲共1 +aPa共D兲兲, 共34兲 with the Gaussian

W₀^共D兲共r,k兲=␲^{−De−共␬r兲2−共k/␬兲2} 共35兲 and the polynomial

Pa共D兲⬅㜷−r²−㜷+k²+㜷共r²+k²兲²− 4㜷k²r²sin²␪ 共36兲 defined by the coefficients

㜷±共a,D兲 ⬅2 1 +aD/2 ±a

共1 +aD/2兲²+a²D/2 共37兲 and

㜷共a,D兲 ⬅ a

共1 +aD/2兲²+a²D/2. 共38兲 For the details of the calculation we refer to Appendix D.

A. Discussion

We now briefly discuss the Wigner function in its depen- dence on the phase-space variablesrជandkជand the numberD

of dimensions for special cases of a. First we note that in- deedrជandkជenterW_a^共D兲only throughr,k, and␪in complete agreement with the considerations of Sec. II. Next we recog- nize that W_a^共D兲 is the product of two contributions: 共i兲 the GaussianW₀^共D兲 inr andk and共ii兲 a polynomial P_a^共D兲 multi- plied by the interpolation parametera.

As a consequence the casea= 0 corresponding to a Gauss- ian wave function yields the Gaussian Wigner functionW₀^共D兲. This distribution is always positive. However, whenais non- vanishing reflecting a non-Gaussian state the Wigner func- tion of the elementarys wave can take on negative values.

The origin of this behavior is the polynomialP_a^共D兲. It contains two terms which for 0⬍a are always negative or at most zero:共i兲the second term in Eq.共36兲which is proportional to k²and共ii兲the contribution containing the square of the sine function.

The latter not only involves even powers ofr andk but also the angle␪^betweenrជandkជ. In this context it is worth- while mentioning that this term is always positive or zero.

Indeed, when rជ and kជ are parallel or antiparallel—that is,

␪^{= 0 or}␪⁼␲—this contribution toP_a^共D兲 vanishes. For 0⬍␪

⬍␲it is nonvanishing and assumes its largest negative value for␪=␲^{/ 2 whenr}ជandkជare orthogonal. Moreover, the com- binationrksin␪emerging in the polynomial can be be inter- preted as an angular momentum variable. This interpretation is rather remarkable since the elementaryswave is an eigen- state of the angular momentum operator corresponding to zero eigenvalue. Nevertheless, the Wigner function also in- volves nonvanishing values of the angular momentum in full accordance with the discussion in the Introduction.

B. Examples

In Figs. 1 and 2 we compare and contrast the Wigner functionsW_S⬅W_⬁^共D兲 andW_max⬅W_−2/D^共D兲 corresponding to the shell wave function and the maximally entangled state emerging from the elementaryswave Eq.共32兲in the limit of a→±⬁anda= −2 /D, respectively. Since the phase space of the Wigner functionWof answave is three dimensional, we depictW_a^共D兲 in its dependence onr andk in a single three- dimensional figure and then study the dependence of this figure on the angle ␪. Moreover, we also discuss the influ- ence of the numberDof dimensions.

For this purpose each row of Figs. 1 and 2 shows the Wigner functionW_a^共D兲共r,k,␪兲for three different angles␪ ^but fixed numberDof dimensions. Likewise for a fixed angle␪ each column displays the dependence onD. In order to bring out the characteristic features we have chosen the angles

␪1=␲^{/ 6,}␪2=␲^{/ 3, and}␪3=␲^{/ 2.}

IV. FREE PARTICLE

We now turn to the second application of our expressions, Eqs. 共11兲 and 共17兲, for the Wigner function of an s wave.

Here we derive and discuss the Wigner functionWE corre- sponding to the free nonrelativistic particle of massM in an eigenstate of energy with E⬅共បk0兲²/共2M兲 with vanishing angular momentum. We first recall the essential features of

the corresponding wave function␺E in position space. Then we start from the expression, Eq.共17兲, for the Wigner func- tion in terms of the wave functions␺^˜E in wave-vector space and calculateWE. This approach is most convenient since␺^˜E

is given by a␦ function which allows us to perform imme- diately the two integrations over q and␽ ^{defining W}E. For the relevant calculations we refer to Appendix E.

Since we dedicate this section to a discussion ofWE, we also present a heuristic derivation starting from the differen- tial equations in phase space determiningWE. This approach brings to light the origin of an intimate connection between the wave function in position space and the corresponding Wigner function. Indeed, we show that the Wigner function of a free particle inD dimensions is determined by the ap- propriately scaled energy wave function inD− 1 dimensions.

This hierarchy originates from a transversality condition of the Wigner function, which is very much analogous to the Coulomb gauge in electrodynamics. We conclude by analyz- ing the dependence of the Wigner functionWE on the num- berDof dimensions.

A. Position wave function

We start our discussion of the Wigner function of a free particle by first recalling the corresponding wave function.

Here we do not solve the appropriate time-independent

Schrödinger equation in D dimensions but rather construct the wave function by interfering plane waves. In this way we lay the formulation for understanding the rather unusual hi- erarchical connection between the wave function in D− 1 dimensions and the Wigner function inDdimensions.

The superposition

␺E共rជ兲 ⬅N

冕

^{dDqe−iqជ·rជ␦共q−k0兲 共39兲}

of plane waves satisfies the time-independent Schrödinger equation

共⌬^共D兲+k₀²兲␺E共rជ兲= 0, 共40兲 corresponding to the energyE⬅共បk₀兲²/共2M兲. Here⌬^共D兲 de- notes the Laplacian inD-dimensional position space.

The normalization constantN of␺Efollows from the or- thonormality condition

冕

^{dDr␺E*共rជ兲␺E⬘共rជ兲=␦共E−E}

^⬘

^{兲 共41兲}

of energy eigenstates corresponding to a continuous spec- trum.

In the definition, Eq.共39兲, of ␺E we integrate the wave vector qជ over all space directions with equal weight which according to Appendix E yields the expression

FIG. 1. Wigner functionW_Sof the shell wave function corresponding to the limita→±⬁in Eq.共32兲in its dependence on the angle␪ between rជ and kជ 共horizontal兲 and the number D of dimensions 共vertical兲. Here we do not show the Wigner function W_S but W_S共rk兲^D−1共sin^D−2␪兲S^共D兲S^共D−1兲. In the bottom of each figure we display contour lines ofW_S. The thick line marks the curve where the Wigner function vanishes, separating positive domains from negative domains. The horizontal axesrandkare identical in all figures. However, the vertical axis changes with increasing angles—that is, going from one row to the next—but is identical in each column.

␺E共rជ兲=N^共D兲f_k

0

共D兲共r兲 共42兲

for the energy wave function in a D-dimensional position space with the wave

f_k

0

共D兲共r兲 ⬅J_{共D−2兲/2}共k0r兲

r^{共D−2兲/2} =

冑

rJ_{共D−2兲/2}共k0r兲

r^{共D−1兲/2} 共43兲 and the normalization constant

N^共D兲⬅

冑

M ប

冑

S1^共D兲, 共44兲 whereS^共D兲, Eq. 共A10兲, denotes the surface of a sphere in D dimensions. Since ␺E only depends on the hyperradius, ␺E

corresponds to a state of angular momentum zero.

B. Wigner function The form, Eq.共43兲, of the wave f_k

0

共D兲 makes it difficult to perform the necessary integrations in the definition, Eq.共11兲, of the Wigner function WE corresponding to ␺E. However, the wave vector representation of␺^˜Ediscussed in Appendix E is rather elementary and contains a␦ function in k. This feature allows us to obtain the Wigner function W_E in a straightforward way starting from Eq.共17兲. In Appendix E we perform the necessary integrations and arrive at

W_E共rជ^,kជ_兲=共N^共D兲兲²g_k

0 共D兲共k兲f_q

0

共D−1兲共rsin␪兲, 共45兲

with

g_k

0

共D兲共k兲 ⬅ 1

S^共D−1兲 2

␲⌫

冉

^{D2− 1}

冊

^{共k02−kk0D−22兲共kD−3兲/4 共46兲}

and

q₀= 2共k₀²−k²兲^1/2. 共47兲 This expression describesWEfork艋k₀, only. In the domain k₀⬍kthe Wigner functionWE vanishes exactly, which is a rather unusual behavior.

The expression, Eq. 共45兲, for WE consists of three basic elements:共i兲the square of the normalization constantN^共D兲of the wave function␺E,共ii兲the functiong_k

0

共D兲which is indepen- dent of position and depends solely on the wave vector and the numberDof dimensions, and共iii兲the wave f_q

0

共D−1兲which also describes the position dependence of the energy wave function␺E.

We now discuss these constituents of W_E in more detail and start our analysis with the square ofN^共D兲 and the func- tiong_k

0

共D兲. The familiar definition, Eq.共4兲, of the Wigner func- tion brings out most clearly that the marginal distribution

P共rជ兲=

冕

^{dDkW共rជ,kជ兲=兩␺共rជ兲兩2, 共48兲}

that is, the Wigner function integrated over wave-vector space, must yield the probability densityP共rជ兲=兩␺共rជ兲兩²in po- sition space. As a consequence the normalization constant FIG. 2. Wigner functionW_maxof the maximally entangled state corresponding to the choicea= −2 /Din Eq.共32兲. The arrangement of figures is as in Fig.1.

N^共D兲has to appear quadratically inW_E. Moreover, the func- tiong_k

0

共D兲is needed in order to obtainP共rជ兲. In Appendix E we perform this integration over kជ and verify that indeed we arrive at the position density.

However, the most important contribution to WE origi- nates from the wave f_q

0

共D−1兲. Here it is remarkable that the position dependence ofW_Eenters through the same function which also determines the wave function ␺E in position space. However, there are three subtleties: 共i兲 Whereas the wave function ␺E in D space dimensions follows from the wavef_q

0

共D兲 inDdimensions, the corresponding Wigner func- tion WE follows from f_q

0

共D−1兲—that is, from the wave in D− 1 dimensions—that is, a space whose dimensions are re- duced by 1—共ii兲the wave vectork₀is replaced byq₀defined by Eq.共47兲, and共iii兲ris replaced byr₂⬅rsin␪^.

C. Discussion

In order to compare and contrast the properties of a quan- tum free particle with its classical counterpart we choose the absolute valuekof the wave vector and the angle␪^between kជandrជas the variables of the system and analyze the behav- ior ofWEas a function of the numberDof dimensions. The absolute valuerof the position vector and the energy eigen- value E⬅共បk0兲²/共2M兲 are the characteristic parameters of

the system. We emphasize that this choice of variables is different from the one used in the discussion of the elemen- taryswave illustrated in Figs.1and2.

In Fig.3we show the Wigner functionWE of a free par- ticle for a fixed energyE determined byk₀= 1 for three val- ues ofrandD. Moreover, the Wigner function is multiplied by the factor V^共D兲⬅共rk兲^D−1共sin^D−2␪兲S^共D兲S^共D−1兲 due to the D-dimensional volume element.

According to Eqs. 共45兲–共47兲 the Wigner function in two dimensions, shown in first row in Fig.3, reads

V^共2兲WE=2rcos共2

冑

k₀²−k²rsin␪兲

冑

k₀²−k² 共49兲 and becomes singular fork→k0. This feature is reminiscent of the classical phase-space representation of a free particle with zero angular momentum. Here we find a singularity at the classical energy—that is, k=k₀. Furthermore, WE as- sumes a maximum for␪^{= 0 and}␪⁼␲ which corresponds to the fact that the wave vectorkជ of the particle is parallel or antiparallel to the position vectorrជ^.

In contrast the Wigner functionW_E changes dramatically in higher dimensions. Already forD= 3 the Wigner function FIG. 3. Wigner functionW_Ecorresponding to the free nonrelativistic particle in an eigenstate of energyE⬅共បk₀兲²/共2M兲withk₀= 1 in its dependence on the absolute valuerof the position vector共horizontal兲and the numberDof dimensions共vertical兲. Here we do not show the Wigner functionW_E but V^共D兲W_E=共rk兲^D−1共sin^D−2␪兲S^共D兲S^共D−1兲W_E. Along the thick line the Wigner function vanishes, separating positive domains from negative domains. The horizontal axis␪andkare identical in all figures. However, the vertical axis changes with increasing angles—that is, going from one row to the next—but is identical in each column.

V^共3兲WE= 2

k₀␲^kr2共sin␪兲J0共2

冑

k₀²−k²rsin␪兲 共50兲 vanishes for␪= 0 and␪=␲. Furthermore, there is no cusp at k=k₀anymore, but a maximum.

For even larger dimension—that isD艌4—the deviation between the classical phase-space representation and the Wigner function of a free particle is more pronounced. In D= 5 dimensions the Wigner function

V^共5兲WE= 2

k₀³␲^k3r3共sin2␪兲

冑

k₀²−k²J₁共2

冑

k₀²−k²rsin␪兲 共51兲 vanishes fork=k₀and for␪= 0 or␪=␲. It reaches its maxi- mal value for␪⁼␲/ 2, which corresponds to the classically forbidden movement along a circle.

In summary the borderlinek=k₀of phase space is rather special and crucially depends on the number of dimensions.

InD= 2 it is a cusp with an inverse square-root dependence.

ForD= 3 the singularity is softened but still a maximum. For D艋4 there is an exact zero. Moreover, the values of the Wigner function at ␪^{= 0,} ␪⁼␲^{/ 2, and} ␪⁼␲ show a strong dependence on the number of dimensions.

D. Phase-space equations

All features—that is, the hierarchy of wave functions and Wigner functions with respect to dimensions and the special form in which position and wave vector enter into the ex- pression ofW_E throughq₀andrsin␪—follow from the dif- ferential equations

关4共k₀²−k²兲+⌬^共D兲兴WE共rជ,kជ兲= 0 共52兲 and

kជ· ⳵

⳵^rជ^WE共rជ^,kជ_兲= 0 共53兲 in phase space derived in Appendix E. The first equation corresponds to the Schrödinger equation whereas the second one is the remnant of the Liouville equation for a stationary state. We now rederive the expression, Eq. 共45兲, for the Wigner functionWEby solving these two equations using a Fourier transform and by making use of the symmetry rela- tion of the Fourier transform of the Wigner function of an swave discussed in Appendix E 5.

1. Solution by Fourier transform

In order to solve the set of equations, Eqs.共52兲and共53兲, we make a Fourier ansatz

WE共rជ,kជ_兲=

冕

^{dDqe−iqជ·rជW˜E共qជ,kជ兲, 共54兲}

which when substituted into Eq.共52兲immediately yields 关q₀²−q²兴W˜_E共qជ,kជ兲= 0, 共55兲 that is,

W˜_E共qជ,kជ兲=␦共q₀²−q²兲w˜_E共qជ,kជ兲. 共56兲 Consequently it is the Schrödinger-type equation Eq.共52兲for the Wigner functionWE which determines the length of the wave vectorqជto beq₀as defined by Eq.共47兲.

The functionW˜_E共qជ,kជ兲still depends on the direction ofqជ andkជ. However, Eq.共53兲puts a constraint on the direction of qជ with respect to kជ which specifies W_E and explains the above-mentioned hierarchy. Indeed, when we substitute the ansatz, Eq.共54兲, into Eq.共53兲we find the constraint

共kជ·qជ兲W˜

E共qជ,kជ_兲= 0, 共57兲 which translates into the condition that the wave vectorqជhas to be always orthogonal on the wave vectorkជ. This require- ment is reminiscent of the transversality condition enforced on the vector potentialAជ of electrodynamics by the Coulomb gauge

ⵜជ _·Aជ_{= 0. 共58兲} We are now in the position to perform the integration over qជ ^{in D} dimensions. We first note that the Schrödinger-type equation 共52兲 for WE determines the length of qជ to be q₀, leaving us withD− 1 integrations over angles. However, due to the transversality condition, Eq.共57兲, these angle integra- tions are restricted to theD− 1 dimensions orthogonal to the wave vectorkជ.

In order to bring out this geometry most clearly we sum- marize the situation in Fig.4for the case ofD= 3. We align the coordinate system ofqជintegration such that theeជ1axis is along the wave vector. Theeជ2axis is orthogonal tokជand the plane defined by rជ ^{and k}ជ. The other D− 2 dimensions are orthogonal to these axes. With this choice of the coordinate system the vectorsqជ andrជtake the form

FIG. 4. Integration over the wave vectorqជin the Fourier ansatz, Eq.共54兲, for the Wigner functionW_Eof an energy eigenstate. Since the Schrödinger equation共52兲forW_Edetermines the lengthq₀ofqជ, we are left with an integration over a sphere inDdimensions. Due to the transversality condition, Eq.共57兲, this integration takes place in the space orthogonal tokជ, leading to a reduction of the number of dimensions. For the case ofD= 3 depicted here we find a circle.

qជ=共0,q2,q₃, . . . ,qD兲=共0,q0cos␽^,q3, . . . ,qD兲 共59兲 and

rជ=共r1,r₂,0, . . . ,0兲=r共cos␪,sin␪,0, . . . ,0兲 共60兲 and the phase in the Fourier representation, Eq.共54兲, reads

qជ^·rជ^=q2r₂=q₀cos␽^rsin␪^. 共61兲 As a result we find the representation

W_E=

冕

^{dD−1qe−iqជ·rជ2␦共q2−q02兲w˜E共qជ,kជ兲 共62兲}

of the Wigner function whererជ2⬅r₂eជ2 and the functionw˜_E depends only on the direction ofqជ and the wave vectorkជ.

In contrast to the original Fourier ansatz, Eq. 共54兲, the integration over the directions of qជ is now restricted to the unit sphere in D− 1 dimensions. When we recall the well- known identity

␦共x²−y²兲= 1

2兩x兩关␦共x−y兲+␦共x+y兲兴 共63兲 and note that 0艋qand 0艋q₀we arrive at

WE= 1

2q₀

冕

^{dD−1qe−iqជ·rជ2␦共q−q0兲w˜E共qជ,kជ兲. 共64兲}

So far this calculation is valid for an energy eigenstate of a free particle of arbitrary angular momentum. We now specify this expression for answave.

2. Symmetry property of s wave

In Appendix E 4 we show that the Fourier transformW˜ of the Wigner functionWof anyswave depends only on three variables: namely, on the q⬅兩qជ兩, k⬅兩kជ_兩 and on the angle

␹⬅arccos关共qជ·kជ兲/rq兴 between them—that is,

W˜共qជ,kជ_兲=W˜共q,k,␹兲. 共65兲 When we apply this property to the Fourier transformW˜

Eof the energy eigenstate defined in Eq.共56兲we find

W˜_E共qជ,kជ兲=␦共q₀²−q²兲w˜_E共q,k,␹兲.

Due to the transversality condition, Eq. 共57兲, the scalar product betweenqជ ^andkជ vanishes—that is,␹= 0. As a con- sequence, the expression, Eq. 共64兲, of the Wigner function WE reduces to

WE= 1

2q₀w˜_E共q0,k,0兲

冕

^{dD−1qe−iqជ·rជ2␦共q−q0兲. 共66兲}

When we recall from Appendix E 1 the relation f_k

0

共D兲共r兲=共2␲^k0兲^−D/2

冕

^{dDqe−iqជ·rជ␦共q−k0兲, 共67兲}

we arrive at WE= 1

2q₀w˜_E共q0,k,0兲共2␲^q0兲^{共D−1兲/2}f_q

0

共D−1兲共r2兲. 共68兲

This expression is identical to the one obtained by direct integration, Eq.共45兲, when we identify

共N^共D兲兲²g_k

0

共D兲共k兲 ⬅ 1

2q₀w˜_E共q0,k,0兲共2␲^q0兲^{共D−1兲/2}. 共69兲 The right-hand side still depends on k through q₀ and w˜_E. The corresponding functional dependence can only be deter- mined from arguments which go beyond the scope of the present paper.

V. SUMMARY

The Wigner function of an s wave in D dimensions de- pends only on three variables: the modulus of the position vector, the modulus of the wave vector, and the angle be- tween them. We have derived integral representations of the s-wave Wigner function in terms of the wave functions in position representation or wave-vector representation. In each case we have expressed the Wigner function as a double integral with respect to a radial variable and a polar angle in the D-dimensional hyperspace. We have illustrated our for- malism using the two examples of the elementaryswave and the energy eigenstate of a free particle with vanishing angu- lar momentum. Here we have discussed the dependence of the results on the number of dimensions.

ACKNOWLEDGMENTS

We thank I. Białynicki-Birula and D. Kobe for many fruit- ful discussions. One of us 共S.V.兲 is grateful to the DAAD Grant Nos. A/01/19250 and A/04/21056 and to the Hungar- ian National Science Foundation 共OTKA兲 Project No.

T048324. Moreover, we appreciate the support of the Minis- try of Science, Research and the Arts of Baden-Württemberg and the Landesstiftung Baden-Württemberg in the frame- work of the Quantum Information Highway A8 and the Cen- ter of Quantum Engineering.

APPENDIX A: HYPERSPHERICAL COORDINATES In this appendix we briefly summerize the essential ingre- dients of the concept of hyperspherical coordinates. Here we follow very closely the treatment of Ref.关12兴.

We consider a vector bជ _{with D components} 共b1,b₂, . . . ,bD兲 in a Cartesian coordinate system defined by mutually orthogonal unit vectors eជ1,eជ, . . . ,eជD. In hyper- spherical coordinatesb⬅兩bជ_兩,␽^{, and}␸1, . . . ,␸D−2 the Carte- sian componentsb₁, . . . ,bDtake the form

b₁=bcos␽^, b₂=bsin␽^cos␸1, b₃=bsin␽^sin␸1cos␸2,

]

b_D−1=bsin␽^sin␸1sin␸2¯cos␸D−2,