Wigner’s Function and Tunneling.pdf

7/27/2019 Wigners Function and Tunneling.pdf

1/18

ANNALS OF PHYSICS 1% 123-140 (1990)

Wigners Function and TunnelingN.L. BALAZS* AND A. VOROS+

Institute .for Theo retical Phy sics, U nioersily of California,Santa Barbara, C alifornia 93106

Received September 13, 1989

We construc t Wign ers function s of a system with the potential-iw2q2 to exhibit andresolve some paradoxical features present in this desc ription. Among others, we show thattunneling at negative energies arises through real trajectories asso ciated with positive energies,in contradistinction of the usual WKB picture where this effect comes about by using complextrajectories of the correct energy. We resolve this puzzle by showing (a) that the initia l dataof this quanta1 des cription already contain wrong energy region s, (b) the interferencephenomenon of WK B wave function s which provides the main contribution to Wigne rs func-tion in the sem iclassic al domain becomes strongly non-local in the presence of a separatrix,which no t only allows , but nec essitate s the presence of these wrong cla ss ica l trajectories inthe present descr iption. c: 1990 Academic Press, Inc.

Special examples often illuminate general problems. In this note we will study thequantum mechanical scattering on a concave parabolic potential, often called theinverted harmonic oscillator. This problem has a long history, and has been com-pletely solved, as far as the wave mechanical problem is concerned [ 11. However,we will concentrate on studying the problem from the point of view of Wignersfunction. We do this to augment our understanding how the quanta1 phenomena oftunneling and scattering are described within this formalism. As we shall see, thisdescription yields some unexpected, and seemingly paradoxical features due in partto (a) the scattering condition expressed in this formalism; (b) the role of Plancksconstant in the selection of initial data for Wigners function, leading to a utilizationof real orbits in tunneling; (c) the presence of separatrices in the classical descrip-tion. Since one can obtain an exact analytical expression for Wigners function(Eq. (lo)), these points can be explicitly studied.

I. THE SETTING OF THE PROBLEMLet the classical Hamiltonian be H = P2/2rn - mo*Q*/2, where w is not the fre-quency, but the Liapunov exponent of the classical motion. We rescale this classical* Permanent address: Department of Phy sics, State University of New York, Stony Brook, New York

11794. Permanent address: Service de Physique Theorique, CEA-CEN Saclay. F-91 191 Gif-sur-Yvette

CEDE X, France-Member of CNRS.123 0003-4916/90 $7.50

Copyright ci 1990 by Academic Press. Inc.All rights of reproduction m any form reserved


2/18

124 BALAZS AND VOROSHamiltonian so as to make it dimensionless, in a manner immediately adaptableto quantum theory, by introducing the dimensionless variables p = P/+,q = Q/e, and h = H/iiw. This yields

h = (p- q2)/2. (1)The classical solution curves h = E are hyperbola branches in the p, q spaces asshown in Fig. 1. The diagonals are the separatrices, and are the phase space trajec-tories for E = 0, corresponding to paths which connect infinity with the top of thebarrier. The E > 0, E < 0 families of hyperbola are disjoint, and arranged as shownin the figure. The negative E trajectories correspond to classical paths whichrebound from the potential hill, without being able to pass through it; the positive

E solutions correspond to classical paths which go over the barrier without beingreflected by it. The arrows on the solution curves show the direction of motion ona phase trajectory. The scattering convention usually introduced requires that forq > 0 only transmitted particles should exist, thus no solution curve reachingq + +oo should be used which has an arrow pointing to the left. In other words,all solution curves below the p = q diagonal should be omitted. In order to incor-porate conveniently this condition into the quantum description we introduceclassically a coordinatization of the phase space in which this diagonal is a coor-dinate curve. Put

u=(P-4)/J% u=(P+4)/$. (2)Then the Hamiltonian is

h = uv. (3)In this parametrization the scattering condition implies that no solution should beutilized for which u is negative.If we now quantize the problem u and v become operators. Since the commutatorof ti and 6 is equal to the commutator of 8 and 4, we can consider u a momentum

FIG. 1. Cla ssic al phas e plane with representative orbits.


3/18

WIGNERS FUNCTION AND TUNNELING 125and u a coordinate. Hence choosing a p representation we consider the operatorassociation

zi+ u, 13 id/du (4)fi -+ i(zZ + fiti) + (uid/du + i(d/du)u)/2, (5)

this being the symmetrized dimensionless Hamiltonian operator,The eigenvalue equation for the eigenfunction f(u) reads as

with the singular solutionf= U-w+ 112) (u I=-01, (7)

and a similar solution for negative u values. The two solutions are separated by thesingularity. The scattering condition requires that we select (7) as the solution foru > 0, and zero otherwise,f =u-E+112)@(U). (8)

(In the rest of the article E will always denote the energy of the eigenstate, and hor E will refer to the energy parameter in Wigners function.) In Appendix A weshow that this solution is indeed the correct one in the q representation. We wouldlike to stress that the wave functions obtained using the u representation are asgood as the ones using the q representation, since the 45 rotation described by(2) is a classical canonical transformation to which an exact unitary transformationof the wave function corresponds, with the transformation function (U 1q),(u 1 q) = lcm/&4 aq1 12 eiS-yh

with S(u, q)/h = q2/2 + J&q + u2/2 using our dimensionless variables. (While thisassociation is true in general only in the semiclassical limit, for linear canonicaltransformations it is an exact relation. Moreover, it is a consequence of the detini-tion of Wigners function that it transforms as a scalar under linear unimodulartransformations [2], and thus Wigners function Wk(u, o) gives immediatelyWE(P, 4) = K(U(P? 4)? 4P, q)).)

II. hZZLES

Our aim is to construct Wigners function associated with a stationary quantummechanical state of energy E satisfying the scattering condition.What are our expectations?Consider E


4/18

126 BALAZS AND VOROSclassical

quanta1expectations

FIG. 2. Top part: On the left, potential hill with incoming and reflected beam of particles ; on theright, the asso ciated phase portrait. Bottom part: On the left, potential hill with incom ing, reflected, andtransmitted beam of particles ; on the right, expected (but incorrect) Wign ers function.

corresponding to the incoming and reflected particle is the left hand branch of thehyperbola h = E, the classical distribution function will be proportional to a Diracdelta function, 6(h - E). The right hand branch cannot be utilized since it violatesthe scattering condition. In the quanta1 case, using a semiclassical approximation,we expect something similar to the sketch shown in Fig. 2. On the left, 6(h - E)should be replaced by an Airy function. (The immediate vicinity of its peak isrepresented by the shaded ribbon.) Similarly, we might expect that the particles,after being transmitted by tunneling should be described by an Airy functionexponentially damped in E; its peak is represented by the shading over the upperhalf of the hyperbola on the right. (The lower half of the hyperbola cannot be usedbecause it describes particles coming from the right.) The two structures arepresumably connected in some manner, depicting tunneling in this description.However, this picture must be wrong. Wigners function gives a full description ofthe quanta1 situation, including tunneling and scattering. At the same time it is awellknown fact that for Hamiltonians at most quadratic in the variables theintegral-differential equation describing the time evolution of Wigners function isidentical to the classical Liouville equation, a first order partial differential equa-tion. Its characteristics are the classical phase space trajectories, h = constant, and

on each characteristic the solution has the same value, the initial value. Conse-quently, if Wigners function does not vanish on the p > 0 part of the right branchof the hyperbola, it cannot vanish on the p < 0 part either. Since the scatteringcondition requires this vanishing (for p < 0) the function must vanish on the whole


5/18

WIGNERS FUNCTION AND TUNNELING 127right hand branch obliterating (in the quanta1 case) the transmission altogether.Thus, the right hand branch of the E < 0 hyperbola cannot be used at all to constructWigners function for the transmitted particles. How do they appear then in thispicture? This is puzzle 1.The only way to circumvent this predicament is to use the h > 0 hyperbolae aswell, even though the energy of our quanta1 state is negative. These, however, comeinto play only if Wigners function for E < 0 is compelled to spill over the h = 0separatrix. Why should this spillover be compulsory? This is puzzle 2.Thus, puzzles 1 and 2 arise from the fact that the equivalence of Wigners integraldifferential equation with Liouvilles equation and the scattering condition imposedprecludes tunneling, unless characteristics of the wrong energy must also be utilizedin the time evolution of a quantum state.This fact leads to subsidiary questions.

(1) How can transmitted particles be described by Wigners function in theclassically allowed region if not by an Airy type approximation over the classicallyallowed phase space where h - E?(2) How can a negative energy eigenfunction give rise to a descriptionutilizing classical trajectories with h positive? In particular: (a) will this notgenerate an inconsistency if we use negative energy, WKB wave functions todescribe the state; (b) how can the particles transmitted by positive h trajectories

still possess negative energies, as they must?III. THE CONSTRUCTION OF WIGNERS FUNCTION

Given a wave function f(u) (U being a momentum like coordinate), Wignersfunction w(u, u) is obtained asw(u, 11)=(1/2x) j dsf(u+s/2)f*(u-s/2)ePi, (9)

with !I = 1.According to Eq. (8) f(u) = u-(/~)-~, u>O; f(u)=O, U-CO, in the energyeigenstate with energy E, satisfying the scattering condition. Thenw,(u, u)= (l/271) jy2,, ds(u+s/2)-12--iE (a-s/2)--(12+iEe,

= 0,Putting s = 2ux we immediately find

w,(u, u) = W,(h; u)=(lIn) j, dx(1+x)~/2~(1-x)-/2)+~e2i~~,= 0,

u > 0,u < 0.

u>O (lOa)24< 0.

595/199/l-9


6/18

128 BALAZS AND VOROSPutting x = tanh y we obtain the alternative form

w,(h; u)= (I/X) I_, dy(l/cosh y)ep2ihtahY+iEy, a>0 (lob)(or its complex conjugate, W being real).Since the wave functions are not normalizable the integral of W over p and qdiverges. We fixed the multiplicative factor by requiring that j W,(h; u) dh = 1, orW,(O) = l/cash nE.The real integral in Eq. (1Oa) can also be expressed as a complex contour integralwhich identifies it with a Laguerre function

W,(h; u) = (cash nE)-l (l/274 $ dze4jhz(z+ 1/2))1/2)-iE (z - 1/2))(12)fiE= (cash TIE)- e2ihL- (,,*,-iE(-4ih)/r(1/2- W, (11)

the last line being obtained through the integral representation of the Laguerrefunctions. (See Appendix B.)We note that the parameters u, u appear in the combination UZ) h, as expected.It is less expected that the expression also depends explicitly on the sign of U, incor-porating the scattering condition. In the present case the energy constant surfaceconsists of several disjoint pieces, and the signs of u and u are additional constantsof the motion labeling the pieces. In the future we will omit the u label in W,(h; u)and agree that it describes Wigners function for u positive (it being zero otherwise).With Eq. (11) our task is in principle solved. However, the Laguerre functions forcomplex arguments and order are not tabulated and their asymptotic expansionsare not studied. For this reason, in the next section we study this function directly.

IV. THE ANA LYS IS OF WIGNER S FUNCTIONThree energy parameters appear in the problem, the energy of the eigenstate&BE; the classical energy fiwh(p, q) associated with the parameters p and q appear-ing in Wigners function; and ho. They are present in the two ratios E and h. Wenow analyze W,(h). We note that if both E and h change sign nothing is alteredin (lOa) (since w is real). Thus we can confine our interest to one sign of E.Let us take E < 0, the analysis of tunneling, and study W,(h) as a function of h,starting with h < E. According to Eq. (10) for a given E (and u positive), Wdepends on h only. Classically we would find 6(h - E), thus W would vanisheverywhere save on the hyperbola branch h = E, intersecting the negative q axis if

E is negative (or the positive p axis if E is positive). In the semiclassical region weexpect this delta function to be broadened, thus we consider h values near E. ThusE and h have the same sign.This broadening can be easily understood if we evaluate (lob) by the method of


7/18

WIGNER'S FUNCTION AND TUNNELING 129steepest descent. For large h and E and h near E the saddle points are at smallvalues of y; hence we can expand the integrand in Eq. (lob) and find

(l/n) Srn he WI ~ E).v -eihY3/3 (2/(2h)u3) ~i( _ 2(h _ ~)/(2h)u3),m c (2/(2E)3) Ai( -2(h - E)/(2E)3) (12)the familiar Airy type approximation. As the different values of h sweep around E,W exhibits ripples on the convex side of the h = E curve, (h < E), and an exponen-tial damping on the other side, in the region bounded by the h = E( ~0) curve, theseparatrix p = -4, and the boundary p = q corresponding to the u = 0 curve whichseparates the W= 0 region from the rest. The Airy approximation ceases to existoutside this wedge. This can be understood easily, if we reinterpret the argument ofthe Airy function geometrically as follows [3].Take a point yE(pE, qE) on the h = E curve and draw a straight line through thispoint along the afIine normal N, to the curve at this point. All points x : (p, q) onthis normal curve are given by x(d) = yENEd, where d is a scalar parameter andmeasures in this sense the separation of the two points along the affine normal.Then the Airy function dependence at x(d) is simply Ai( -2,4/fi213), (where werestored all the physical dimensions of p and q). Consequently, only at those pointsx is this approximation possible which lies on an affine normal direction of somepoint of the h = E curve. In the present case N, at the point yE is along the radiusvector yE (see Appendix C), thus one can only apply this approximation at thosepoints x which lie on a straight half line connecting the h = E curve to the origin,i.e., at the points in the quadrant between the asymptotes of the h = E curve.Near the separatrix Jhl is small, and a different approximation is to be used,(Appendix C). This gives, for 0 > h B E

1WE(h) = (cash nE) &,(4 m)

a monotonic function of h.If h is small but positive and E large negative we find1

WE(h) = (cash nE) Jcl(4 $-El)an oscillatory function of h. (A saddle point analysis indicates that the oscillatorybehavior starts at h= -1/16E.)A general view of W,(h) is given in Fig. 3 for several different energies.

V. THE RESOLUTION OF THE PUZZLESIn Section II we raised several puzzles implied in the present problem.Wigners integral-different equation is equivalent to Liouvilles equation; henceall solutions of the initial value problem are obtained by displacing the initial data


8/18

130 BALAZS AND VOROSALAZS AND VOROS

0.6-

0.4-

0.2-

B.B-

-9.2-

-8.4-

FIG. 3. Graphs of Wigne rs function, W,(h), for different values of E. (For E=O, W,(h) is propor-tional to J,(h). For E >O the oscilla tions of W,(h) for h >O are no longer visible on the sca le of thegraph. )


9/18

WIGNERS FUNCTION AND TUNNELING

cI.0-

8.8-

0.6-

0.4-4

0.2-

0.0-

-B.Z-

-0.61 hI , , , .I, , ,-9.5 -7.5 -5.5 -3.5 -1.5 8.5 2.5 4.5

0.81

I-8.6 I I h-13.0 I I I I I I I.-Il.0 -9.0 -7.0 -5.0 -3.0 -1.0 I.0 3.6

131

FIGURE 3-Continued

59WYY/I- IO


10/18

132 BALAZS AND VOROSalong the characteristics, the h = constant curves. However, the characteristics forwhich E < 0, and which originate at q = -co, p > 0 do not lead to the q > 0 region,hence, seemingly, no tunneling is possible!This puzzle is resolved by observing that a Wigner function for a stationary stateof negative energy, E < 0 cannot be confined to the negative h(p, q) region allowedby the scattering conditions. This region is the wedge around the negative q axisbounded by the q = +p separatrices.The scattering condition absolutely confines W to the u > 0 region; the additionalrequirement of a vanishing W for h > 0 would imply the further condition W = 0 foru > 0. We now express these conditions on the wave function in the o and urepresentation. Accordingly we seek a wave function $(u) which vanishes for u > 0,and whose Fourier transform d(u) vanishes for u < 0. However, such a functioncannot exist by Painlevts theorem [4]. Indeed, Ii/(u) = 1; du d(u)e, thus it isanalytical in U. Hence, if it vanishes on part of the real axis it will vanisheverywhere, contrary to the requirement that it be finite for negative u values.Consequently, the wave functions of physically realizable states must produce suchassociated Wigner functions which must spill across the u = 0 separatrix (since thespillover across the u = 0 separatrix is forbidden by the scattering condition). Thus,the correct choice of the initial form of a general W function is influenced by quanta1conditions and will in general contain fi. (It must correspond to the projector of aphysically realizable state for pure states; or it must correspond to a positive densityoperator in general.)Hence the transfer of the initial data corresponding to tunneling is accomplishedby the real characteristics for which h(q, p) > 0 although the quantum state hasE


11/18

WIGNER S FUNCTION AND TUNNELING 133These are found by a chord construction: the two stationary points s1 s2 contribut-ing to W&J, q) are the endpoints of that chord of the h = E curve for which thep, q point is the bisector. At the same time the h = E curve is that curve which wasutilized (a la Maslov) to construct the WKB wave function used. As the argumentof W,, the (p, q) point, approaches the h = E curve the s,, s2 points conflue withthe (p, q) points, leading to the Airy function behavior. (see Fig. 4.)In the present case the WKB wave functions utilized are constructed over the twobranches of the $J -q2) = E < 0 hyperbola with the p negative segment of theright branch omitted (since there are no incoming particles from q = +co). Now aconstructive interference can occur in two different ways. As before, a self-inter-ference of one branch. For example, at the point (p, q) the standard chordconstruction results in the stationary points S, , s2 and the possibility of the s,, s2confluence yields the Airy approximation. However, a point p, q can now be thebisector of a chord ending on two different branches of the h = E < 0 hyperbola inthe points s;, s; which are again stationary points. Given E, these (p, q) pointslie on a h = E > 0 hyperbola.Thus, the mutual interference of WKB associated waves with the differentbranches of the h = E < 0 hyperbola leads to an interference at points with hpositive. These saddle points cannot coalesce with each other and with (p, q); thusthey will not lead to the Airy form. We stress that the effect is due to the separatrixsince it produced the two branches of the hyperbola, and it prohibited theconfluence of the saddle points on the different branches.This also answers point(b). We see that the mutual interference from differentbranches will yield contributions where h has the opposite sign to E. At the sametime the resulting Wigner function is oscillatory which allows the average value(h ) to be negative in regions where the h > 0 trajectories hold sway. The averageenergy per particle associated with the region for which h > 0 and p > 0 is given by

ce> =.f dpdqhW&).f d 4 W,(h)

a

q>q

\ \FIG. 4. (a) Conventional chord construction where the h(p, q) = E curve has a single branch. Thepoints sl, s2 are the locatio ns of the sadd le points in the integrand of kV; the point (p, q) is the bisector

of the s, , s2 chord. (b) The two different chord constru ctions if the h(p, q) = E curve has two branches.The chord s,. s2 has (p, q) as its bisector; sir s2 lie on the same branch. The chord s; , 3; has (p, q)as its bisector; s;, s; lie on different branches.


12/18

134 BALAZS AND VOROSintegrated over the wedge shaped re ion between the top part of the two diagonals.Putting h = E, p = fi cash a, q = t 2s sinh a, this becomes

(e) = jam d& C-W&)/$ d& WE(&).According to Eq. (lob) WE(s) satisfies a differential equation derivable from theone defining L,(z). This gives

f (E dW,/d&) +4(&-E) W,= 0;integrating this differential equation from E = 0 to E = cc we immediately find that

(e) = E.Thus, for E < 0 the oscillations of WE(&) indeed yield a negative (E). Theseexplanations resolve the original puzzle; nevertheless a certain unease remains, sincethe WKB description in a classically permitted region has the same form irrespec-tive of whether this region is actually reached by trajectories in the classical limitor not. Yet this property does not carry over to the Wigner description, eventhough we use WKB wave functions. This comes about since the interferencemechanism which forms the basis of the semiclassical evaluation of Wignersfunction yields profoundly different results depending on whether a separatrix ispresent or not.

VI. THE REFLECTION AND TRANSMISSION COEFFICIENTSWe compute now the reflection and transmission coefficients using Wignersfunction. At a point q the current density is given by j dpp W,(h), where the rangeof integration depends on the particular current we seek. Consider Fig. 5 and

q = q < 0; the current on the left is defined as

the first term (which will be negative) is the current j, due to reflection. The secondterm is that part of the incoming current, ji;, for which h < 0; the last term, j& isthe incoming current for which h > 0. The sum of the last two terms gives theincoming current jinc. On the right hand side q= q >O and the current is

I

ccIright = pWdp=&,

4

where j, is the current in the transmitted beam. Current conservation requires thatjleft = j+&f. This is satisfied because the second term of jleft is equal and opposite to


13/18

WIGNER'S FUNCTION AND TUNNELING 135

FIG. 5. Visua lization of currents in scattering. The incom ing currents j& , j,;, are produced byparticles with positive, esp. negative energies; j, and j, are the transmitted and reflected currents. Thecurrents are proportional to the number of stream lines intercepted by the 9 = 9 and the 9 = 9 lines .

the first, hence cancels the first, and the last term of, jleft, in turn, is equal to jright.These observations follow immediately from the fact that W is constant along theh = constant trajectories, and the same number of trajectories go through thecorresponding p segments of integration. The transmission and reflection coef-ficients C, and CR are defined asCT = j,(q + + cc )/i,c(q + -a 1= cli,:;i,, =*+a

1CR=--1 +!Xwith CI= lj,l/lj,l.As q + - 00, I ,l + j:. ,mW,(h) dh, j, + j; W,(h) dh thus

a= amW,(h) h; W,(h) dh.~ ,nA straightforward calculation in Appendix E yields

and

the standard answer.

a = ,+ZEn,2nE

CT = e +2nEl+e

VI CONCLUSIONSThe present study shows that the presence of separatrices in the classical phasespace alters our intuitive understanding of the picture conjured .up by Wigners


14/18

136 BALAZS AND VOROSfunctions. The usually innocuous broadening of the classical distribution functionby quantum effects will be sent by the separatrix into widely separated regions,leading to a spreading of Wigners function into regions which are associated withthe wrong sign of the classical energy. This, then, allows tunneling to be producedby classical orbits of the wrong sign. Nevertheless, this does not conflict with theusual semiclassical explanation of Wigners function as being the result of an inter-ference by WKB waves because the presence of the separatrix produces an inter-ference which has strong nonlocal consequences in the (p, q) space. These results,then, suggest that while in the usual WKB description tunneling can be conceivedas mediated classically by complex classical orbits, in the Wigner picture the realclassical orbits can also play a significant role. The role of the complex classicalorbits in the Wigner description requires further clarification.

APPENDIX A: THE WAVE FUNCTION IN THE q REPRESENTATIONAccording to the notation of Abramowitz, Stegun [6] we seek that solution of

y+(ax*-a)y=O, (AlI(their Eq. (19.21.1)), which has only a transmitted wave for x$ Ial. This isy = E(a, x)x U(ia, xe- in4), (their Eq. (19.17.9)) where

U(a, 2) N ez214s ers-s=/2Sn- l/2 ds , (AZ)1with c1 being a counterclockwise contour around a branch cut connecting -cowith 0.Since the Schrodinger equation is

d2$/$q2 + q2t+b 2EIl/ = 0 (A3)we must put a = -E and x = $q.Inserting this in (A2) and changing the integration variable s to u by s = ei3n4u,we obtain

i(q2/2 + fiqu + ~12 +, - ( 112 ) - iE du 7 (A4)where we have chosen the branch cut to lie on the negative real axis.

Since the generating function of the canonical transformation (p, q) + (u, u) isq2/2 + ,,bqu + u*/2, the unitary transformation (q 1 u) is proportional toei(q2/2 + J&u + U2/2), th us the wave function U- (1/2)-iE does indeed correspond to aneigenfunction with the asymptotic properties required by the scattering condition.


15/18

WIGNER'S FUNCTION AND TUNNELING 137APPENDIX B: LAGLJERRE FUNCTIONS

The Laguerre polynomials are defined asL,(x) = e $ (xe-~-)

if they are normalized so that the coefficient of .Y be unity.Then by Cauchys theorem1111L,(x)e- = -!- 4 dy(ye-?)2ni (y-xy+

where the closed contour of integration surrounds the points y = 0, x.This definition remains valid if II ceases to be an integer, and thus the pointsy = 0, x become branch points. Putting y = ~(4 + Z) we obtain the symmetricexpression

where the contour surrounds the branch points z = f 4 and the branch cut con-necting them.

APPENDIX C: THE AFFINE INTERPRETATION OF W,(h)Let x: (p, q) be the argument of W(p, q) and let yE(A) : (pE(;l), q,(l)) be anypoint on the curve H(p, q) - E = 0, where A is an arbitrary parameter along thecurve. Then

defines the affine arclength along the H= E curve. The afline normal N(s), normalto the H = E curve at the point y(s) is given byN(s) = d2y/ds2, ((3)

thusx = y + A d 2y/ds2 (C3)

defines A.In the present case the curve is $(p - q2) = E, E < 0, and the relevant branch is4E(P) = - &=z (C4)


16/18

138 BALAZSAND VOROSchoosing p as the parameter. That is,

Y,(P) : (P, 4E(P)),dy,l@ : (4 p/q&d2y,Mp2 : (0, - Wk:),

ds/dp = 1 Pl4E 13= ( -2E)130 -2EJq; qE 1

NE = d2y,lds2 : ( _ 2EJ2,3(P, qE) = ( _ &3 YETand

x=y,+N,A=y, A( -2q2:3 *Squaring and subtracting each component we find

or

from whichh-En=(2E)Ll? if h near E.

Hence Ai( -24) = Ai( -2(h - E)/(2E)j3). The same as the Airy-factor in Eq. (12).

APPENDIX D: ASYMPTOTES OF W,(h) FOR SMALL h, LARGE EAccording to (lob)

e+2ihW(h) = ___cash nE Lp,,,,,wiE(-4ih)/f (Dl)Put c2jhW(h) into Laguerres differential equation. Then W satisfies the followingdifferential equation

hW+ W+4(h-E) W=O. P2)


17/18

WIGNER'S FUNCTION AND TUNNELING 139If h is small and E is large, neglect h next to E in the last term, giving

(D3)The solution finite at h = 0 is

J( 4i Jz) = Zo(4 Jz,,giving W up to an E dependent factor. Since W,(O) can be easily calculated fromthe integral representation (or from the normalization of L,(z) at z = 0) we obtain

(04)

APPENDIX E: THE COMPUTATION OF THE TRANSMISSION COEFFICIENTC, = LX/( + a), with CI= A/B. Here

A= lam W,(h) & = i j dy e+2i,2ni+sinh yB= so-?c W,(h) dh = &. j & ezfEJ.4-A .

The integrals are to be taken on the contours which avoid y = 0 as indicated, (tomake the integration over h convergent at the limits + co, -co).NowA = & Principal value integral + k Residue( y = 0)B = - & Principal value integral + t Residue(y = 0).

HenceA + B = Residue( y = 0) = 1

A - B = k. Principal value integraldyB,l/m -ni X sinh v (e2iEt -e-2 )d/r, dy =& tanh nE


18/18

140whence

BALAZSAND VOROS

A = enE/(enE+ eCnE)B=e-E/(&=E+e-E)M = e2nE.

ACKNOWLEDGMENTSThe authors owe their thanks to many: the National Scie nce Foundation for spons oring their visit to

the Institute for Theo retical Phy sics at Santa Barbara (NSF Grant No. PHY82-17853, supplem ented byfunds from NASA) and the Institute itself for their hosp itality. NLB a lso acknowledg es the support of(NSF Grant No. PHY88-16234) and the repeated and kind ho spitality of the Service de PhysiqueTheorique, CEN-Saclay (France), where much of the work ha s been initiated. Finally, NLB thanksProfessor E. Wigner for an illuminating discus sion on these matters some years ago.

REFERENCES1. E. C. KEMBLE, Phys. Rev. 48 (1935), 560; L. D. LANDAU AND E. M. LIFSHITS, Quantum Mechanics,

p. 177, Pergamon, London, 1958.2. N. L. BALAZS AND B. K. JENNINGS, Phys. Rep. 104 (1984), 347.3. N. L. BALAZS, Physica A 102 (1980), 236.4. Oeuvres de Pau l Pa inleve, Ed., CNRS 1974, Vol. II, p. 55; see R. F. STREAT EZR AND A. S. WIGHTMAN,

PCT, Spin & Statis tics, and All That, p. 74, Benjam in, New York, 1964.5. M. V. BERRY, Philos. Trans. R. Sot. London A 281 (1977), 237.6. M. ABRAMO WITZ AND I. A. STEGUN, Handbook of Mathematical Functions, pp. 686720, U.S.

National Bureau of Standards, Was hington, D.C., 1964.

Wigner’s Function and Tunneling.pdf

Documents

Transcript of Wigner’s Function and Tunneling.pdf