Mapping Wigner distribution functions into semiclassical distribution functions

Mapping Wigner distribution functions into semiclassical distribution functions

G. W. Bund1,*and M. C. Tijero2,† 1

Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona, 145, 01405-900 Sa˜o Paulo, SP, Brazil 2_{Pontifı´cia Universidade Cato´lica de Sa˜o Paulo, Rua Marqueˆs de Paranagua´, 111, 01303-000 Sa˜o Paulo, SP, Brazil}

共Received 1 April 1999; published 19 April 2000兲

A mapping that relates the Wigner phase-space distribution function of a given stationary quantum mechani-cal wave function, a solution of the Schro¨dinger equation, to a specific solution of the Liouville equation, both subject to the same potential, is studied. By making this mapping, bound states are described by semiclassical distribution functions still depending on Planck’s constant, whereas for elastic scattering of a particle by a potential they do not depend on it, the classical limit being reached in this case. Following this method, the mapped distributions of a particle bound in the Po¨schl-Teller potential and also in a modified oscillator potential are obtained.

PACS number共s兲: 03.65.Sq I. INTRODUCTION

The phase-space formulation of quantum mechanics be-gan with a paper of Wigner 关1兴. Because the phase-space formulation offers a framework in which quantum problems can be treated using the classical language as much as it is allowed, it has been applied to many areas of physics. Among these areas we mention statistical physics关2兴, quan-tum optics关3兴, collision theory 关4,5兴, nuclear physics 关6兴, and nonlinear physics 关7,8兴.

The phase-space distribution functions are the main tool of the phase-space formulation of quantum mechanics and among them the Wigner distribution function 共WDF兲 has been used with success, for instance, in the description of atom-molecule collision processes 关9,10兴, in nuclear physics 关6,11兴, and also for studying the classical limit of quantum mechanics关12,13兴. It is with this last topic that this work is concerned. In Sec. II we summarize Ref. 关12兴, where it is shown how to perform the mapping that relates the WDF of a given stationary quantum mechanical wave function, a so-lution of the Schro¨dinger equation, to a soso-lution of the Liou-ville equation, both subject to the same potential. Two ex-amples have already been considered in Ref. 关12兴: the infinite square well and the potential step. In Sec. III of this work we apply this same mapping to a particle bound in the Po¨schl-Teller potential关14兴 and in Sec. IV to a particle in the ground state of a modified oscillator potential. The potentials used in Sec. III and Sec. IV are smooth and more appropriate for classical or semiclassical approximations than those of Ref. 关12兴.

II. MAPPING THE WDF TO A SEMICLASSICAL DISTRIBUTION FUNCTION

The classical limit of quantum mechanics may be accom-plished in two steps. First the mapping of the WDF to a semiclassical phase-space distribution function共SDF兲 is ob-tained, and in a second step the limitប⫽0 is taken since the

SDF may still depend on ប. We consider only the first of those two steps 共Ref. 关12兴; the abbreviation SDF here corre-sponds to CDF in that reference兲.

The WDF␳(q, p,t) satisfies the equation关5兴

⳵␳ ⳵t ⫹ p m ⳵␳ ⳵q⫹

冕

K共q,p⫺p

⬘

兲␳共q,p

⬘

,t兲dp

⬘

⫽0, 共1兲

where (q, p) is a point in phase space. The kernel K is given by K共q,p⫺p

⬘

兲⫽ i ប

冕

dv 2␲បe (i/ប)(p⫺p⬘)v

冋

_V

冉

_q⫺v 2

冊

⫺V

冉

q⫹v 2

冊册

, 共2兲

V(q) being the potential. Equation 共2兲 becomes in the

clas-sical limit关5兴 Kc共q,p⫺p

⬘

兲⫽⫺ i ប

冕

dv 2␲បe (i/ប)(p⫺p⬘)v_v⳵V ⳵q ⫽⫺⳵_⳵V q ⳵ ⳵p␦共p⫺p

⬘

兲, 共3兲

and Eq.共1兲 goes over into the Liouville equation

⳵␳c ⳵t ⫹ p m ⳵␳c ⳵q ⫺ ⳵V ⳵q ⳵␳c ⳵p ⫽0. 共4兲

We are looking for a particular solution ␳c of Eq. 共4兲 which may be considered the correct semiclassical approxi-mation of a given␳, a solution of Eq.共1兲. In order to obtain it we write the integral equation

␳共q,p,t兲⫽␳c共q,p,t兲⫺

冕

⫺⬁ ⬁ dt

⬘

冕

dq

⬘

d p

⬘

⫻Gc共q,p;q

⬘

, p

⬘

,t⫺t

⬘

兲

冕

d p

⬙

关K共q

⬘

, p

⬘

⫺p

⬙

兲 ⫺Kc共q

⬘

, p

⬘

⫺p

⬙

兲兴␳共q

⬘

, p

⬙

,t

⬘

兲, 共5兲 *Electronic address: [email protected]

relating Eqs. 共1兲 and 共4兲, where Gc(q, p;q

⬘

, p

⬘

,t⫺t

⬘

) is the retarded Green’s function, which satisfies the inhomoge-neous Liouville equation

⳵Gc ⳵t ⫹ p m ⳵Gc ⳵q ⫺ ⳵V ⳵q ⳵Gc ⳵p ⫽␦共q⫺q

⬘

兲␦共p⫺p

⬘

兲␦共t⫺t

⬘

兲. 共6兲 A WDF that satisfies Eq. 共1兲 when inserted into Eq. 共5兲 gives a function␳cthat satisfies Eq.共4兲 as can be verified by inserting Eq.共5兲 directly into Eq. 共4兲. So Eq. 共5兲 generates a SDF from a known WDF. This is the starting point for ob-taining␳c. Using Eq.共1兲 and Eq. 共3兲, Eq. 共5兲 may be written

␳c共q,p,t兲⫽␳共q,p,t兲⫹

冕

⫺⬁ ⬁ dt

⬘

冕

dq

⬘

d p

⬘

⫻Gc共q,p;q

⬘

, p

⬘

,t⫺t

⬘

兲

冉

⫺ ⳵ ⳵t

⬘

⫺ p

⬘

m ⳵ ⳵q

⬘

⫹⳵V ⳵q

⬘

⳵ ⳵p

⬘

冊

␳共q

⬘

, p

⬘

,t

⬘

兲. 共7兲

We stress that Eq.共7兲 is equivalent to Eq. 共5兲 if ␳ satisfies Eq. 共1兲.

The WDF␳ generated by a given wave function⌿(q,t), a solution of the Schro¨dinger equation corresponding to the potential V(q), is given by关1兴

␳共q,p,t兲⫽

冕

₂dv_␲បei pv/ប⌿

冉

q⫺v

2,t

冊

⌿*

冉

q⫹

v

2,t

冊

. 共8兲 In Eq. 共8兲, when ⌿ is stationary,␳ does not depend on the time. In the particular case in which V(q) is quadratic in q, Eq. 共7兲 reduces to ␳⫽␳c, as in this case Eq. 共1兲 becomes identical to the Liouville equation.

The retarded Green’s function satisfying Eq.共6兲 is

G_c共q,p;q

⬘

, p

⬘

,␶,␧兲⫽e⫺␧␶␦„Q共q,p,⫺␶兲⫺q

⬘

…

⫻␦„P共q,p,⫺␶兲⫺p

⬘

…␩⫹共␶兲, 共9兲

where the convergence factor exp(⫺␧␶) (␧→0), was intro-duced. Here „Q(q,p,t),P(q,p,t)… represents the classical trajectory in phase space of a particle subject to the potential

V(q), which at time t⫽0 occupies the phase-space point

(q, p), and ␩_⫹ is the step function.

Using the classical equations of motion P/m

⫽⳵Q(q, p,t

⬘

⫺t)/⳵t

⬘

and⫺⳵V/⳵Q⫽⳵P(q, p,t

⬘

⫺t)/⳵t

⬘

we get

冋冉

⳵ ⳵t

⬘

⫹ p

⬘

m ⳵ ⳵q

⬘

⫺ ⳵V ⳵q

⬘

⳵ ⳵p

⬘

冊

⫻␳共q

⬘

, p

⬘

,t

⬘

兲

册

q⬘⫽Q(q,p,t⬘⫺t),p⬘⫽P(q,p,t⬘⫺t) ⫽ ⳵ ⳵t

⬘

␳„Q共q,p,t

⬘

⫺t兲,P共q,p,t

⬘

⫺t兲,t

⬘

… 共10兲

and Eq.共7兲 becomes 关12兴

␳c共q,p,t兲⫽␳共q,p,t兲⫺

冕

⫺⬁ t dt

⬘

e⫺␧(t⫺t⬘) ⫻ ⳵ ⳵t

⬘

␳„Q共q,p,t

⬘

⫺t兲,P共q,p,t

⬘

⫺t,t

⬘

兲…. 共11兲 When␧ may be set equal to zero from the beginning, we get from Eq.共11兲

␳c共q,p,t兲⫽ lim t⬘_→⫺⬁

␳„Q共q,p,t

⬘

兲,P共q,p,t

⬘

兲,t

⬘

…. 共12兲 Thus, if the limit in Eq.共12兲 does exist the value of␳cat the phase-space point (q, p) is the value of ␳ at the initial (t⫽ ⫺⬁) phase-space point of the classical trajectory that at time

t⫽0 reaches the point (q,p). Assuming that␳ does not de-pend on the time, Eq. 共11兲 gives, after integrating by parts, the mapping equation

␳c共q,p兲⫽ lim ␧→0⫹ ␧

冕

⫺⬁ 0 d␶e␧␶␳„Q共q,p,␶兲,P共q,p,␶兲…. 共13兲 For systems with one degree of freedom at phase-space points where the classical energy E(q, p) is negative, Eq. 共12兲 cannot be applied and then Eq. 共13兲 has to be used. Making the Fourier decomposition

␳„Q共q,p,␶兲,P共q,p,␶兲…⫽

兺

n⫽⫺⬁

⬁

Rn共q,p兲ein␻(q,p)␶, 共14兲 where the period associated with the trajectory is T(q, p) ⫽2␲/␻(q, p), and inserting Eq. 共14兲 into Eq. 共13兲, we get

␳c(q, p)⫽R0( p,q), which is equal to ␳c共q,p兲⫽ 1 T共p,q兲

冕

0 T(q, p) d␶␳„Q共q,p,␶兲,P共q,p,␶兲…, 共15兲 so the only nonvanishing term in the ␧⫽0_⫹ limit corre-sponds to n⫽0 in Eq. 共14兲.

For points (q, p) on the same closed classical path, Eq. 共15兲 gives the same value of ␳c, because this expression is invariant under time translations of the integrand, and the trajectories „Q(q,p,t),P(q,p,t)… associated with these points are related to each other by making time translations. This invariance shows that ␳c depends only on the energy

E(q, p) associated with the trajectory and it can be extended

to open classical paths by making use of Eq.共13兲.

When the WDF corresponds to a bound state in a poten-tial that vanishes at infinity, Eq.共12兲 may be applied directly for points with positive energy E(q, p), giving the result

␳c(q, p)⫽0. It can be shown also that the phase-space aver-aged classical energy for the SDF coincides with the quan-tum energy of the bound state. The mapped SDF in this case may be viewed as a stationary distribution function of clas-sical particles trapped by the potential.

In order to see how this theory works two examples were considered.

A. The infinite square well The ground state of the infinite square well

V共q兲⫽

再

0, 兩q兩⭐d ⬁, 兩q兩⬎d 共16兲 is given by ⌿共q兲⫽

_冑

1 dcos ␲q 2d, 兩q兩⭐d. 共17兲

The WDF corresponding to this bound state is关4兴

␳共Q,P兲⫽_2␲1_បd

再

_{2 P}ប

冋

sin

冉

2 P ប 共d⫺Q兲⫹ ␲ dQ

冊

⫹sin

冉

2 P_{ប 共}d⫺Q兲⫺␲ dQ

冊册

⫹

冉

2 P_{ប ⫹}␲_d

冊

⫺1sin

冋冉

2 P ប ⫹ ␲ d

冊

共d⫺Q兲

册

⫹

冉

2 P_{ប ⫺}␲ d

冊

⫺1 sin

冋冉

2 P ប ⫺ ␲ d

冊

共d⫺Q兲

册冎

共18兲 for 0⬍Q⬍d and ␳共Q,P兲⫽␳共⫺Q,P兲 共19兲 for ⫺d⬍Q⬍0.

Making a change of variables in Eq. 共15兲, we obtain for the SDF ␳c共q,p兲⫽ 1 2d

冕

_⫺d d ␳共q

⬘

, p兲dq

⬘

⫽ ␲ 4ប cos2共pd/ប兲 关共␲/2兲2⫺共dp/ប兲2兴2. 共20兲 We see that as the SDF ␳c depends only on the classical energy, which in this case is p2/2m, it cannot depend on the coordinate q. In the limit ប→0 Eq. 共20兲 gives ␳c(q, p)

→(1/2d)⫺1_{␦( p), which describes a uniform distribution of}

particles at rest inside a box.

B. The potential step The wave function for the potential step

V共q兲⫽

再

0, q⬍0

V0⬎0, q⭓0 共21兲

for incident energies E⫽k2/2m⬍V0 is given by

␺共q兲⫽

再

2A cos共kq/ប⫺␣/2兲, q⬍0

2A cos共␣/2兲e⫺␬q/ប, q⬎0, 共22兲

where ␬⫽(2mV0⫺k2)1/2 and ei␣⫽(ik⫹␬)/(ik⫺␬) gives the phase ␣. The WDF is given by 关4兴

␳共Q,P兲⫽4兩A兩 2 ␲

冋

C(⫺)共P,k兲cos

冉

2共P⫺k兲 Q ប

冊

⫹C(⫹)_共P,k兲cos

冉

_2共P⫹k兲Q ប

冊

⫹S(_{⫺)共P,k兲sin}

冉

_2共P⫺k兲Q ប

冊

⫹S(⫹)_共P,k兲sin

冉

_2共P⫹k兲Q ប

冊册

, 共23兲 for Q⬍0, where C⫿共P,k兲⫽␬k关共⫿2P⫹k兲2⫹␬2兴⫺1共⫾2P兲⫺1, S⫿共P,k兲⫽k共⫾2Pk⫺k2⫹␬2兲/关共⫿2P⫹k兲2⫹␬2兴 ⫻4P共k⫿P兲. 共24兲 For Q⬎0 the WDF is ␳共Q,P兲⫽4兩A兩 2 ␲ e⫺2␬Q

冋

C0共P,k兲cos

冉

2 PQ ប

冊

⫹S0共P,k兲sin

冉

2 PQ_ប

冊册

, 共25兲 where C0共P,k兲⫽4␬k2关共2P⫹k兲2⫹␬2兴⫺1关共2P⫺k兲2⫹␬2兴⫺1, S0共P,k兲⫽ k 2_共k2_⫹␬2_⫺4P2_兲 关共2P⫹k兲2_⫹␬2_{兴关共2P⫺k兲}2_⫹␬2_兴P. 共26兲 The SDF that corresponds to the WDF given in Eqs.共23兲 and共25兲 is 关12兴

␳c共q,p兲⫽

再

兩A兩2_{关␦共p⫺k兲⫹␦共p⫹k兲兴, q⬍0}

0, q⬎0. 共27兲

Equation 共27兲 describes the reflection of a classical par-ticle subject to the potential given by Eq. 共21兲 for energies below the height of the barrier. Equation共27兲 can be written

␳c共q,p兲⫽

再

k兩A兩2 m ␦

冉

p2 2m⫺ k2 2m

冊

, q⬍0 0, q⬎0, 共28兲 which explicitly shows the energy dependence of the SDF.

We remark that, in order to treat the potential step with a normalized wave function, the time dependent wave packet approach for scattering can be used 关15兴. The corresponding time dependent WDF and SDF are calculated afterwards. In the limit when the incident packet reduces to a single plane wave one expects the result of Eq. 共27兲.

III. THE PO¨ SCHL-TELLER POTENTIAL

Here we present a third example, in which the mapping of the WDF corresponding to the ground state of the potential

V共q兲⫽⫺ U0

cosh2共aq兲, U0⬎0, 共29兲 into the SDF is performed. The parameters U0 and a are chosen such that the binding energy of the ground state be-comes equal to U₀/2. This choice fixes the value U₀ ⫽ប2_a2_/m.

The normalized wave function, the solution of the Schro¨-dinger equation corresponding to the potential given by Eq. 共29兲, is

␺共q兲⫽

冉

a₂

冊

1/2

关cosh共aq兲兴⫺1_{. 共30兲}

The WDF, which we obtained following Eq. 共8兲, is

␳共q,p兲⫽ sin共2pqប

⫺1_兲

បsinh共2aq兲sinh关␲p共បa兲⫺1兴, 共31兲

which satisfies the condition兰␳(q, p)dq d p⫽1.

Making q⫽␲q

⬘

/(2a) and p⫽បap

⬘

the expression 共31兲 becomes symmetric in the dimensionless variables q

⬘

and p

⬘

共see Fig. 2兲. In Fig. 2 we draw the curves of constant density

␳ in the first quadrant; for the other quadrants the symmetry

q→⫺q, p→⫺p should be applied. We observe that regions

in which␳is positive alternate with regions where it is nega-tive. The points where ␳(q, p) vanishes are given by the parabolas 2 pq⫽n␲ប,n⫽⫾1,⫾2, . . . . The magnitude of␳ in the region of negative values is small and occurs on points of phase space in which a classical particle would have posi-tive energy, thus not being trapped by the potential.

The classical trajectories „Q(q,p,t),P(q,p,t)…, the solu-tions of Newton’s equasolu-tions, are obtained by performing the integral

冕

dt⫽

冕

dQ Q˙ ⫽

冕

dQ

冉

2E m ⫹ 2U0 mcosh2共aQ兲

冊

⫺1/2 . 共32兲 The constants of integration are determined by imposing the condition (Q, P)⫽(q,p) at t⫽0.

For E⬍0 the result is

sinh共aQ兲⫽a p cosh共aq兲sin共␻t兲

␻m ⫹sinh共aq兲cos共␻t兲,

共33兲

P cosh共aQ兲⫽p cosh共aq兲cos共␻t兲⫺␻m

a sinh共aq兲sin共␻t兲.

共34兲 The frequency ␻ is related to the energy E(q, p) of the tra-jectory by␻⫽a(2兩E兩/m)1/2. Equations共33兲 and 共34兲 may be written

sinh共aQ兲⫽

冉

U0 兩E兩 ⫺1

冊

1/2

sin⌽共q,p,t兲, 共35兲

FIG. 1. The Po¨schl-Teller potential as a function of the dimen-sionless variable q⬘⫽aq.

FIG. 2. Curves of constant density␳(q,p) 共in units of ប⫺1), the WDF corresponding to the ground state for the potential in Eq.共29兲. The variables q⬘and p⬘are dimensionless, q and p being the co-ordinate and momentum, respectively.

P cosh共aQ兲⫽关2m共U0⫺兩E兩兲兴1/2cos⌽共q,p,t兲, 共36兲 where

⌽共q,p,t兲⫽␻t⫹␸共q,p兲.

. Here

␸共q,p兲⫽arctan关␻m sinh共aq兲/ap cosh共aq兲兴.

The SDF ␳c is obtained by replacing (q, p) by „Q(q,p,t),P(q,p,t)… in Eq. 共31兲 for the WDF and integrat-ing it in t over one period. For this purpose, expressions共35兲 and共36兲 are solved explicitly in Q and P, the results being

Q⫽a⫺1ln关共1⫹F2兲1/2⫹F兴, P⫽共2mE兲1/2G共1⫹F2兲⫺1/2, 共37兲 where

F⫽共U0/兩E兩⫺1兲1/2sin⌽ and

G⫽共U0/兩E兩⫺1兲1/2cos⌽.

We mention here that␸(q, p) fixes the origin of the time and

␳c should not depend on it.

The final expression for ␳_c for E(q, p)⬍0 is

2␲ប␳_c共q,p兲⫽

冕

0

2␲

d␪sin兵2共aប兲

⫺1_{共2m兩E兩兲}1/2_{g共1⫹ f}2_兲⫺1/2_{ln关共1⫹ f}2_兲1/2_{⫹ f 兴其}

2 f共1⫹ f2兲1/2sinh兵␲共2m兩E兩兲1/2g关aប共1⫹ f2兲1/2兴⫺1其 , 共38兲

where f⫽(U0/兩E兩⫺1)1/2sin␪ and g⫽(U0/兩E兩⫺1)1/2cos␪. Equation共38兲 shows that␳cdepends on p and q only through

E(q, p).

For positive energies E(q, p) a similar calculation gives, as expected,␳c(q, p)⫽0, showing that the SDF corresponds indeed to a distribution of particles trapped inside the poten-tial.

In Fig. 3 we plot the curves along which ␳c is constant 共which coincide with the classical trajectories兲. The trajec-tory marked␳c⫽0 in Fig. 3 corresponds to E(q,p)⫽0. All

trajectories for which E(q, p)⬎0 are situated at the right of this trajectory in this figure. As in the previous examples, we find also that only non-negative values for the density sur-vive. Fig. 4 is a superposition of Fig. 2 and Fig. 3. In Fig. 4, when comparing the quantal curve for a given ␳ with the corresponding classical curve 共same value of the density兲, one notices that the classical curve tends to move toward the left, away from the region corresponding to negative values of ␳. Thus, the region in which the WDF assumes negative values is located in the domain where␳cvanishes.

FIG. 3. Same as Fig. 2 for␳c(q, p). In this case the curves are

also the classical trajectories. The curve for which␳c(q, p)⫽0

cor-responds to the energy E(q, p)⫽0. For points on the right of this curve E(q, p)⬎0 and␳c⫽0.

FIG. 4. Figures 2 and 3 are superposed. Dashed lines correspond to␳c and full lines to␳. Negative density curves are omitted.

IV. THE MODIFIED HARMONIC OSCILLATOR POTENTIAL

Now we present results corresponding to the modified harmonic oscillator 共MHO兲 potential

V共q兲⫽ប

2

2m关␣

2_q2_{⫺2␣␤qtanh共␤q兲兴. 共39兲}

The wave function that corresponds to the ground state of this potential is

␺⫽Ne(⫺␣q2/2)_{cosh共␤q兲, 共40兲} with normalization factor N2⫽2(␣/␲)1/2关exp(␤2/␣)⫹1兴⫺1 and energy eigenvalue

⑀共␣,␤兲⫽ ប 2

2m共␣⫺␤

2_{兲. 共41兲}

The corresponding WDF may be calculated in closed form,

␳共q,p兲⫽Me(⫺␣q2⫺p2/␣ប2)

冋

_e⫺␤2/␣_{cosh共2␤q兲⫹cos}

冉

2␤p

␣ប

冊册

, 共42兲 where M⫽兵␲ប关1⫹exp(⫺␤2/␣)兴其⫺1 is the normalization factor. We shall consider ␣⫽␤2, in which case we get

⑀(0,0)⫽0 and the potential, given by Fig. 5, has a hump in the middle of the oscillator.

Equation共15兲 corresponding to the SDF␳_cwas evaluated numerically making use of the expression

␳c共q,p兲⫽T⫺1

冖

␳共Q,P„Q,E共q,p兲…兲

mdQ P„Q,E共q,p兲…,

共43兲 where the integration is performed over a closed path with energy E(q, p). The period T was simply calculated through

T⫽

冖

mdQ

P„Q,E共q,p兲…, 共44兲

where P„Q,E(q,p)… is the momentum given as function of the coordinate Q,

P共Q,E兲⫽兵2m关E⫺V共Q兲兴其1/2. 共45兲

FIG. 5. The modified harmonic oscillator potential V⬘(q)

⫽2m(ប␤)⫺2_{V(q),␤q and V⬘(q) being dimensionless quantities.}

FIG. 6. Curves of constant density␳(q,p) 共in units of ប⫺1) for the WDF corresponding to the ground state of the modified har-monic oscillator potential. The variables q⬘and p⬘are dimension-less.

FIG. 7. Same as Fig. 6 for␳c. In this case the curves are also

the classical trajectories. The dashed lines correspond to negative energies.

Equation共43兲 is obtained from Eq. 共15兲 by replacing the time

␶ by the coordinate Q as variable of integration. A second convenient change of variables Q⫽Q(␪) is made such that the singularities associated with the zeros of P(Q,E) at the classical turning points are canceled by corresponding zeros in the numerator in Eqs.共43兲 and 共44兲. Alternatively, we use the subtraction method for the elimination of these singulari-ties in order to evaluate these integrals.

In Figs. 6 and 7 we plot, respectively, lines of constant␳ and␳c and in Fig. 8 we plot␳c versus E, corresponding to the MHO potential for the choice␣⫽␤2. From Fig. 6 we see that the domain of phase space where ␳ is negative has its coordinates restricted to the region of the hump of the poten-tial and it is periodic in the momentum variable. The value of

␳ decreases rapidly in magnitude as the momentum in-creases. As can be seen from Figs. 7 and 8, there may exist more than one trajectory for a given value of ␳c. In Fig. 9 we superpose parts of Figs. 6 and 7, plotting curves with fixed density ␳⫽␳c. The SDF we obtain by applying Eq. 共43兲 has similarities with the original WDF but again, as in the cases of Secs. II A, II B, and III, negative values are absent. One finds that for larger values of q the two curves with␳⫽␳_c⫽const tend to get closer, a result that is expected because of the dominance of the harmonic oscillator part of the potential.

We remark here that the point (q, p)⫽(0,0), at the top of the hump, is a point of unstable equilibrium for classical

motion. Thus the period associated with the trajectory that passes through the point (q, p)⫽(0,0) becomes infinite since the particle has to arrive at the point q⫽0 with zero velocity. For this trajectory, which corresponds to E⫽0, the mo-mentum P(Q,0) given by Eq.共45兲 is proportional to Q in the neighborhood of the origin so that there is a pole at Q⫽0 in the integrand of Eq.共43兲 leading to a logarithmic singularity in the integral. This singularity is cancelled by the analogous singularity of T given by Eq. 共44兲. In fact, for phase-space points (q, p) in the neighborhood of the trajectory E(q, p) ⫽0 the SDF is given by ប␳c⫽ 1 ␲ A⫹ln兩E共q,p兲兩 B⫹ln兩E共q,p兲兩, 共46兲

where A and B are slowly varying functions of E(q, p). Thus one gets the value ␳_c⫽1/␲ for points on the trajectory

E(q, p)⫽0. This value coincides with the maximum value of ␳, which occurs precisely at the point (q, p)⫽(0,0). The functions A and B have to be evaluated numerically after subtraction of the pole term in the integrals in Eqs.共43兲 and 共44兲. As can be seen in Fig. 8, Eq. 共46兲 leads to a very sharp maximum of␳_cfor the trajectory passing through the origin of phase space and for which E(q, p)⫽0 and ប␳c⫽1/␲. This path is compressed between the two curves for ប␳c ⫽0.2 in Figs. 7 and 9, the upper curve corresponding to ␧

⬘

(q, p)⫽2m(ប␤)⫺2E(q, p)⫽0.0010 and the lower one to

␧

⬘

(q, p)⫽⫺0.0010.

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FIG. 8. The density␳c共in units of ប⫺1) of the SDF as function

of the dimensionless classical energy⑀⬘(q, p)⫽E(q,p)2m(ប␤)⫺2.

FIG. 9. Figures 6 and 7 are superposed. Dashed lines correspond to␳ and full lines to ␳c.

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