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Pure state condition for the semi-classical Wigner function
Physica 1lOA (1982) 501-517 North-Holland
PURE STATE CONDITION
Publishing Co.
FOR THE SEMI-CLASSICAL
WIGNER FUNCTION A.M. OZORIO de ALMEIDA Institute de Fkica “Gleb W&&in”
Universidade Estadual de Campinas, Campinas, 13100 SP., Brasil
Received 30 June 1981
The Wigner function W(p, q) is a symmetrized Fourier transform of the density matrix p(q,, q2), representing quantum-mechanical states or their statistical mixture in phase space. Identification of these two alternatives in the case of density matrices depends on the projection identity p* = p; its Wigner correspondence is the pure state condition. This criterion is applied to the Wigner functions obtained from standard semiclassical wave functions, determining as pure states those whose classical invariant tori satisfy the generalized Bohr-Sommerfeld conditions. Superpositions of eigenstates are then examined and it is found that the Wigner function corresponding to Gaussian random wave functions are smoothed out in the manner of mixedstate Wigner functions. Attention is also given to the pure-state condition in the case where an angular coordinate is used.
1. Introduction
The Wigner function W(p, q) is a representation of pure quantummechanical states or their statistical mixture in phase space and reduces to a Liouville distribution in the classical limit h +O. In the original work’,2V3), W(p, q) was obtained as symmetrized Fourier transform of the density matrix p(q,, qJ while Takabayasi4) and Baker’) were the first to consider the condition for a given W to represent a pure state. More recently Berry6) and coworkers’) have obtained explicit functional forms for the semi-classical (SC) limit of Wigner functions representing pure bound states whose corresponding classical motion is integrable. For a system with f degrees of freedom (2f-dimensional phase space) the classical limits of W for these states are the f-dimensional tori which correspond to the invariant manifolds of the classical orbits*). Relaxing h from zero reveals a rich oscillatory structure dominated by the tori. Systems described by bounded (angle) variables must be treated with caution’.“), but they have the advantage that the corresponding classical invariant manifolds may have topologies simpler than tori. In particular, free rotations provide an example where the S.C. and even the classical approximations of W coincide with the exact quantumResearch supported by FAPESP, CNPq and FINEP.
0378-4371/82/0000-0000/$02.75
@ 1982 North-Holland
502
A.M. OZORIO
mechanical
evaluation.
also given
by Berry6).
The main purpose satisfy
the pure
the S.C. Wigner
state
(PS) condition
the
determine
which
for general
geometric the
restriction which results explicitly from satisfy the generalized Bohr-Sommerfeld
rotations
were
that the S.C. form of W’,‘) does
in the integral
in a cunning
chords
functions
of this work is to verify
This will be seen to occur Berry
de ALMEIDA
S.C.
form
derived
by Baker’).
way by the interlocking Wigner
the calculation conditions”).
function.
of
The
only
is that the state
must
The SC form of W for the eigenstates of an integrable Hamiltonian, which is thus confirmed by this analysis, displays many oscillations and diverges the Wigner caustic’). Coherent superalong (2f - 1)-dimensional surfaces: positions of eigenstates are even more complicated, since the Wigner integral is nonlinear. The temptation is hence felt to relegate the use of this phase space representation to statistical mixtures of states”). These will appear as linear superpositions of pure W’s whose individual oscillations will tend to cancell on the whole. Are there pure states whose Wigner function resembles a mixed state? This question is answered by means of the PS integral in the last part of this paper. There it is shown that Gaussian random wave functions have this property. It is interesting to note that Berry13) has put forward the hypothesis that this is exactly the form of semi-classical wave that will appear for non-integrable systems. Before getting on to these subjects we derive systems described by angle coordinates. Contrary an immediate extention of the usual PS condition.
the exact PS condition for to previous belief this is not
2. The exact pure state condition The Wigner function corresponding (p 1I+!J),where p and q are the continuous momentum
W(P,
and coordinate
operators
1
4) = (nh)Y’ dQ exp(-
where f is the number This quasi-probability
to the wave functions (q I$) and and unbounded eigenvalues of the
6 and 4, is given
zip-Q/h)(q + QI$XICIlq - Q>.
of degrees of freedom. distribution has the important
I dqW(p,9) = 1$cI)Iz, I dpW(p> lb )44’.
by
property
(2.1)
that
KP
9) =
c2.2)
PURE
STATE
WIGNER
FUNCTION
503
W is not a linear function of I$) so that a linear superposition of W’s does not correspond to a pure state. It is thus called a mi.xed-state Wigner function and also reduces in the classical limit to a Liouville distribution. A necessary and sufficient condition for W to represent a pure state is P=
I
d5, dS‘zW(&)W(&) cos(Alh) = (N4)‘WE).
(2.3)
--z Here 5 = (P, 4)
(2.4)
and A is twice the skew product (6 - 5) A ( t2 - 4):
(2.5) Geometrically, A is the sum of the areas of the projections on each conjugate (p,, 4.) plane of the triangle which has &, 52 and 5 as the midpoint of each side, as shown in fig. 1. This symplectic area8) is invariant under canonical transformations and hence also for metaplectic transformations: the quantum version of linear canonical transformations. Since W is also invariant under metaplectic transformations’8,7), it follows that this property also holds for the P.S. integral (2.3). If one of the coordinates $ has bounded eigenvalues: -7T
(2.6)
PI f
Fig. 1. 41, &Zand f determine
the midpoints
of the sides of the triangle A.
504
A.M. OZORIO
the eigenvalues each real
of the corresponding
component. angle
angular
We will consider
in coordinate
momentum
space
de ALMEIDA
fi will be discrete only
so that
the case
with a spacing
where
pn corresponds
of h in
q corresponds
of
and
--sc
to a
to a component
(2.7)
integrations
over
q to (2.6) and replacing
integrations
over
p by
summation
(2.X) Berry6) obtains
the Wigner
function
for one degree
of rotational
freedom
as
iii? W(P”,
4) =
I
&
dQ
ew-
2inQHq + QI$I>($I q - QL
(2.9)
with the arguments q ? Q lying, modulo 27r, in the range (2.6). It is interesting that in this case the range of integration does not include the whole domain of variation of Q as opposed to (2.1). This will affect the PS integral as follows: Introducing (2.9) into (2.3) we get
P = (47r)_? I
xc nl”2
dQldQz j dq,dq,
exp{-2ilhhQl+
x (ql+ QI 1+)(4 1qln 1
=4
pnzQ2+
p,,(q-q2)+pa(ql-q)+p,(q2-q1)]}
QJ(qz + Qz
1rcl)(+142 - Qd
TT
I dQ1 dQz 1 dql dq2c S(QI+ q ~
q2 -
jlr)6(Q2
+
ql - q - j2n)
Jlk
x
Zxpl-
2ip,t42
-
41)/R]
x (qi+ QI 1GHIcI ( 4, - Q,Xqz+
Q2 1 $)(d~ 1 a-
In the integration range of ql and q2 there so that with the transformation
(2.10)
Q2).
will be always
be two fi-functions
(QI>QJ-*(Q;, Qi) = (Q, + QT,Q‘, - Q2)
(2.11)
we obtain P=;
II
dQ; dQ%q + Q; I GH4
x {(ICI14 + QiXq + Qi 1+)I.
1q -
Q3 exp(- 2ip,Q;/h)J (2.12)
505
PURESTATEWIGNERFUNCTION
The range of integration is bounded by the full line in fig. 2. To avoid this nonseparated boundary we note that the integrand is a periodic function in the (Q;, Qi) plane, so that an integral over the region bounded by the dashed line (the second Brillouin zone) is exactly twice the original integral. The result is then rr
P=
I
dQXq
+
Qi
IW$I q+ Qi)
ew(- %Q;lfi)
-a
(2.13)
=s[w(P",q)+w(P",q+~)l,
from which we see that W(p,, q) and W(p,, q + 7~)are complementary Wigner functions. The simplest system with which we can exemplify the use of these formulae is the free rotator. The wave functions are simply (4 I VV = (2~)-“’
(2.14)
ewhdil~h
so that W(P”,
(2.15)
4) = &“I274
from which (2.2) and (2.13) immediately
3. Bound states and the Bohr-Sommerfeld
follow.
conditions
The semiclassical approximation of W is constructed from the f-dimensional torus which contains the classical orbits in phase space. For a start,
Fig. 2. The area inside the full line is the range to the dashed line and the result halved.
of integration
in (2.12). This can be extended
up
506
A.M. OZORIO
considering action
only
coordinates,
we can
specify
the tori
by their
f
variables
2rrl,, =
where
extended
de ALMEIDA
p - dq,
the RHS
is the symplectic
(3.1)
area for the nth
irreducible
circuit
of the
torus. For any reducible circuit the RHS of (3.1) is zero. Considering the full family of tori this integral defines an f-dimensional vector function Z(t) of the points in phase space (2.4) constant along each torus. Introducing standard semi-classical wave functions into (2.1) and evaluating the integrals by stationary phase it is shown’) that
(3.2)
j
where 6,: and 4; are the two tips of the jth Berry chord of the torus centered on 6, each { ,} is the Poisson Bracket of one of the functions I”((’ - 6;) with Z,,,(g’- &) evaluated at t’= 0, and gj is the signature (i.e. the excess of positive over negative eigenvalues) of this matrix. Aj is the symplectic area for the closed path which starts out at 6; on the torus and returns to 4: along the Berry
A, =
chord:
p . dq.
(3.3)
The configuration for f = 1 is shown in fig. 3. For f 3 2 the number of chords bisecting themselves
at a given
point
4 of
any region of phase space is always even’). Chords can coalesce at the boundary of each region - the (2f - I)-dimensional Wigner cuustic where (3. I) diverges. This divergence is a shortcoming of the SC approximation, since the exact W is everywhere finite if h > 0. Nonetheless, if one is not too close to the Wigner caustic, (3.1) is asymptotically correct, so that it is necessary to examine the change in the signature of {I’, Z .} when the caustic is touched. To do this we make the linear canonical transformation (p, q)+ (K, Z ) taking as coordinates and momenta the action and angle variables linearized at t-. The immediate result is that {I’, I_} = lYZ’(K, z-)/a&, i.e. {I’, Z-} can be interpreted as the Jacobean formation between I+ and 8-, with I- fixed.
(3.4) of the (locally linear) transThe zeroes of aZ’/aK will
PURE
STATE
Fig. 3. One-dimensional
WIGNER
example
FUNCTION
501
of torus and Berry chord.
coincide’) with those of the more familiar matrix K/X(or ap/aqwith the torus a multivalued function p(q)). But there are two important differences: Unlike K/a&, al’/aK is not necessarily symmetrical. Also {I+, I-} is always regular, so that the only possible changes of signature occur on the Wigner caustic. The diagonalization of {If, I-} can be interpreted physically. First we note that for a separated torus this matrix is necessarily diagonal. For a general matrix, diagonalization is obtained by transforming the K variables so that the new axes coincide with the f vectors Ve-I+. Thus by performing the transpose of the inverse transformation on the I- variables we obtain new canonical coordinates in which @/a& is diagonal. One should note that this is not generally equivalent to local separation, since aI’/aZ- need not be diagonal. It will prove necessary to consider the effect on a chord contribution to (3.2) of taking one of its endpoints through an irreducible circuit. If a given circuit is described by an integer vector n, where nl is the number of cycles around the elementary circuit with action I, and so on, then W?(S) = (27&\/h)-’
cos[(Aj + 2m * 1)/h - (~9 + 2n - p)7~/4)] 9 Idet{I+, I-}(“*
(3.5)
where ~1 are the Maslov indices for the f elementary circuits. The symplectic area Aj must fit into a single sheet. If it is possible to fit the entire circuit into a single sheet p(q) of the torus, then the Maslov index will coincide with the change of signature of the Poisson bracket matrix {I’, I-}. To see this, one can use as the q-plane the K-plane (tangent at t-). On completing an irreducible cycle, the directions of VB-I+ will be the same but the orientations of the axes may have altered, leading to a change of signature. An example is the deseparated torus’) with f = 2. Choosing the fixed endpoint of the chord as the
SO8
A.M. OZORIO
origin
(2n7r, 2sn)
whose
centres
lie on the Wigner
whose
crossing
alters
irreducible circuit ment
circuits
is then that
Sommerfeld
of angular
coordinates
leads
determine
For separated
sheet.
The Maslov
is no change
WY, be univalued
on the torus,
caustic,
the signature.
into a single
2, but there
de ALMEIDA
chords,
tori, it is not possible index
in signature
immediately
the singular
in fig. 4 a set of lines to fit
for an elementary
of {I’, I-}. The requireto the
generalized
Bohr-
conditions
I = (n + p/4)h.
(3.6)
The global validity of (3.2) as a PS Wigner function will now be established by directly working out the integral (2.3) by the method of stationary phase. The key integrand
point is that the three phase space vectors are the mid points of the sides of the triangle
which defined
determine the in (2.5) and of
the Berry chords. So, for any choice of 5, and & which results in A just touching the torus with its three corners, one of the chords of W, will coincide with a side of A and likewise for WZ. Furthermore, the total symplectic area bounded by the two chords plus A is exactly one of the chord areas of W(t). This situation for f = I is shown in fig. 5. Changes of 4, and & which only move the corner of A opposite 4 will not alter the total area and it
Fig. 4. A Berry chord with one of its ends fixed at the origin will be singular if its other endpoint touches one of the dashed lines for a separated torus or one of the full lines if the torus has been deseparated by a perturbation. In the latter case it becomes possible to have irreducible cycles in which no singular lines are crossed. The 2 signs refer to the eigenvalues of {I’, I-) near the origin. The Maslov indices for the cycles shown are wU = p,h = 2 and wc = IL,, = 0.
PURE
I
*9
Fig. 5. The symplectic areas bounded by the chords centred on 61 and 42 together with A add up to the symplectic area bounded by the chord centred on g if all the corners of A touch the torus.
will be shown below that even changes of e,, and & in general directions leave the total area stationary. Introducing the semiclassical Wigner function (3.2) into (2.3), we obtain exp{(i/h)[k A f (Ai, + hail&l) If:(Aiz + fMizv/4)11 Jdet{l+, I-}], det{I+, I-Jj~“* ’ (3.7) where we sum over all combinations of sign in the exponent. The phase of one of these terms will be stationary when (a A k Ai, k Ad is stationary, which will only occur in the circumstances already outlined for the combinations (+++) and(---), and only for the chords jr and j2 which coincide with sides of A. Thus neglecting all terms for which the phases are never stationary and relabling jr = 1 and j2 = 2, p=
(n*h)-’ 2
d5 d52 exp{i[(A + AI + AMI + (VI+ hd41) + c c I Idet{I+, I-}, det{I+, Z-}21”2 * *’
(3’8)
It is now necessary to change the integration variables from the midpoints of the sides of A to its corners as shown in fig. 5: 170=42+&L-5,
The stationary
(3.9)
1)1=52-51+5.
condition thus becomes simply
I(qo)=I(ql)=Z,
ql=5t
or
47,
(3.10)
510
A.M. OZORIO
where
47 is one of the Berry
chords
de ALMEIDA
centred
on 4. Finally,
with the canonical
transformations
where
the O’s are the angle variables
conjugate
to the Z’s, we obtain
exp[i(A 1 dHo{ I de’ 1 dzO dz’ 0 0 0
+ A, + A#I + (a, + o&r/4 Idet{Z,, Z,} det{Zo, Zz}l”’ )
+c.c..
(3.12)
The evaluation of the multiple integral in the curly brackets requires the knowledge of the determinant of the Hessian H of second derivatives of the phase of the integrand. This is derived in the appendix, equation (1.6) leading to cos[A#i P = (h/4)‘(7T V/h))’
UH
=
in the appendix d-
{h,
II}\
’ {h,
that the signature I,},’
with the siffix s indicating separated, that
the three
(3.13)
ldet{Zr, Z2}/“*
0 It is shown
+ (CM + ok + (~2)n/41
matrices
* {I3
of H is given
by (3.14)
b,),),
the symmetric
part
of the matrix.
in (3.14) are diagonal.
It is then
If the torus
is
easy to verify
(3.15)
(TH+c,+u2=uj,
as long as all three chords fit into a single sheet. When O. takes one of the chords through a caustic into a different sheet one eigenvalue of its Poisson bracket
matrix
changes
pn + oI $- or = But in this case
uj
f
sign and then (3.16)
nTT.
the Wigner
function
WI is no longer
the primitive
Wigner
function but one of the complementary ones defined by (3.5) with Maslov index ~2. This addition to the phase exactly cancels the r in (3.16) leading again to (3.15) if we use the quantization conditions (3.6). The validity of (3.15) for all 8” leads to the verification of the PS condition. For non-separable tori it is possible to use different Ai’s if they add up to an irreducible circuit which can be represented in a single sheet. The quantum conditions ensure that these lead to the same Wigner function (because of concomittant changes of the signatures). The pure-state condition will select between them because either choice may lead to (3.15) or (3.16). It is
PURE STATE WIGNER
FUNCTION
511
interesting that there is no difference between exchanging W, for its complementary or exchanging WZ. It thus arises that the quantization conditions, while they ensure that the SC Wigner function is univalued, are an essential part of the proof that one is dealing with the Weyl-Wigner representation of a pure state. The degree of self-consistency achieved by the ubiquitous use of the stationary phase approximation is remarkable; not only in the interlocking of A with the Berry chords, but in the calcellation of all the spurious singularities of the semiclassical Wigner function in exactly the right way. The selection of eigenstates of integrable systems considered in a phase space of extended coordinates thus turns out similar to that which holds for rotation eigenstates. Eigenstates of hindered rotations will have semiclassical Wigner function of the same form as (3.2), except that 5 can only take on discrete values of p. The evaluation of the PS integral is here entirely analogous to the treatment given for extended variables: only two cases need to be considered. For rotations the Maslov index is zero so that the quantized area under the periodic classical orbit is the same as in the exact quantization of the free rotator. In the case of librations the Maslov index is two and the quantization is the same as for a system with one degree of freedom described by extended coordinates.
4. Superpositions
of eigenstates-mixed
and pure
So far we have considered only single eigenstates of the Hamiltonian. Their linear combinations will have complicated Wigner functions because of the non-linearity of the defining integral (2.1). On the other hand it has been found that in the SC Wigner function for mixed states, corresponding to thermal equilibrium, the oscillations and even the spurious singularities particular to each eigenstate are averaged out’?. This result is promising as far as the future use of the Wigner function in statistical mechanics is concerned, but the objective here is different. Instead we investigate the possibility that pure combinations of eigenstates may also be smoothed in the manner of mixed states. We start by considering linear combinations of the Wigner functions corresponding to free rotators. The resulting Wigner function will have the form W(P,,
4) = wllh,
with the normalization
T
w”=
1.
(4.1) condition (4.2)
s12
A.M. OZORIO
The PS condition
de ALMEIDA
is in this case reduced
to
w=” = w n from which consider
(4.3)
it follows
the classical
that w, = 6,” is the only exact
normalized
solution.
Yet
limit in which
wn-h-4 MP),
(4.4)
where w(p) is here a smooth function of p -the quantized levels move closer together while normalization requires that the probability of individual levels diminishes. In this case the PS condition is also satisfied, and we may consider that even in the SC limit the mixed Wigner function is a good approximation of one that represents a pure state. We now examine the nature of this pure state. It is definitely not an eigenstate for some Hamiltonian describing a hindered rotation, since its Wigner function would then condense on a curve p(q) in the classical limit, instead of continuing to occupy a region of phase space. properties of the Wigner function (2.2) it follows that
From
the projection
I(4 I ICIV = (27T)F’,
(4.5)
I(P, I $)I = WM.
(4.6)
But the latter (pn
leading
projection
I $,) = (w,hP
exp(i W,
(4.7)
to
(4 I $) = Finally,
implies
T (w,/27-r)“2 exphqlfi
the comparison
of the square
+ &)I. of the modulus
(4.8) of (4.8) with (4.5) shows
that & must be a randomly distributed variable in the interval words (q 1 q!~)is a sample of a Gaussian random function’“.“).
(0,27~). In other It is interesting
that the corresponding (p ) $) d oes not have a smooth SC limit, as the allowed values of p are squeezed together, Instead its features resemble those of a Weierstrass-Mandelbrot fractal function14). We can now consider the linear combinations of Wigner functions corresponding to eigenstates of general integrable Hamiltonians:
W(5) =
cn
W”W”(i3,
(4.9)
where n is here the set of quantum numbers which select the allowed actions I, according to (3.6). Since each W”(t) satisfies individually the PS condition,
PURE STATE WIGNER FUNCTION
513
its application to (4.9) again reduces to (4.3). It also follows that in the SC limit (4.9) represents an asymptotically pure state. It appears more complicated than the one obtained from the free rotator eigenstates, but using the Poisson summation formula W(s) = $
exp(q
M -
f dZw(Z) 0
as phase phases Aj of
(4.4) is smooth of w.. Stationary of these integrals eliminates but the M = 0 term since (3.2) are fractions of 27rZ, so that
In the classical limit6 w’(5)
--,
NZ(5) - 1)
(2nTT)f’ leading to (4.13)
W(5) = w(Z(5))/(277)fY
This same result is achieved semiclassically for f = 1 by using the uniform approximation of W’(e) in terms of Airy functions6). Equation (4.13) shows that in the classical limit w(Z) reduces to the Liouville density. To what wave functions do these Wigner functions correspond? Clearly they are a superposition of SC wave functions with an amplitude whose square modulus is w(Z)/fi’:
(4 I $> = (27rfi)F”’ T
[w(L) g] “*ew{i[Si(q, LM
+ fM1,
where summation over the different branches of S(q, Z,,) is implied. Expansion of the exponent near any given position qo S(q, Z”)/h + 8. = pn(q0) * (4 - 40) + @iI,
(4.15)
puts (4.14) in the local form of a superposition of plane waves. The phases 8. must be such that the cross terms in the square modulus of (4.14) cancell out, so as to agree with the projection of the Wigner function m
0
w(P, dp
--oc
4)) =
(2?l)f
(4.16) 0
514
A.M. OZORIO
de ALMEIDA
I<
Fig. 6. Momentum
and coordinate
wave
intensities
91$>12 I
h- q for w of step function
form
thus, again the wave functions are samples of Gaussian random functions, but there are two important differences with respect to the superpositions of free rotations. The first is that the average intensity is now a function of q. None the less it is a smooth function. Consider the simple case with f = 1, where
W(I(c3) = 0(1(t) - r”Yr”7 proportional line contained
in the I(s)
= 10 curve
as show
to the length
of the q = constant
in fig. 6. More important
is the
fact that all the metaplectic projections of the Wigner function so constructed will be Gaussian random functions. The momentum wave intensity is also shown in fig. 6 for the step Wigner function. It has been conjectured by Berry”) that the quantum mechanical eigenfunctions of systems whose classical motion is may resemble Gaussian random functions. These differ markedly from the eigenfunctions of integrable systems, but not from their superpositions with random phases which have very simple Wigner representations. unintegrable motion the Wigner approach may yet prove fruitful.
PURESTATEWIGNERFUNCTION
515
Acknowledgement
I thank Prof. M.V. Berry for many helpful discussions.
Appendix
The evaluation calculate the determinant H
=
x
stationary phase requires that we 3f Hessian matrix
a’@ +A,+&)
(A. 1)
d2V”,11,w> .
One way of doing this is by determining the change in the total area brought about by an arbitrary change in the independent variables from a stationary configuration. This change in area can be reduced to the symplectic area of the three small triangles whose corners are respectively a corner of A and the tips of the Berry chords that have been displaced from it by the change in q. and q,. This is shown in fig. 7. So (61&H,, se&- * H * (610, SI,, se,) = so A s1+ so A 62 + a* A s,, showing that the change of symplectic
(A.3
area is indeed of second order in the
Fig. 7. A general displacement of the corners of A from the torus will change the symplectic area A + Ai + A2 by the area of the three triangles shown shaded in the figure.
A.M. OZORIO de ALMEIDA
516
independent invariant
variables.
The skew product on the right-hand
which can thus be calculated
done to arrive
at equation
side is a canonical
in the (fZ,, I,) coordinate
system,
as was
(3.4). Then
60 = {I,, I”} -’ * ({I,, 80) * 610 - SZ,). FZo,
6, = 1% I,} ’ * 68, -{&, O,} * SI,, {I,,. III * F8, - {lo, 0,) * sz, 62 = {I,, Z,} ’ * ({I?, Z,} * 68, -{Z2. 0,) * 61, + {Zz. O,,} * SZO),610.
(A.3)
We now make use of the identities {Ak, Bj} = - {Bj, A,,}‘, {A,, 41 * lo,, Ai} = {A,. Or1 * 14. A,}. {Oj, ZkI * {O,, Zj} = E + {Oj, f-Zk}* {a, Z,I,
(A.4)
{Ai> Bk} = {Ai, Z,} * {% Bk) - {Ai, 0,) * {I,, 6.1, {Ai, Bk}-’ * (Ai, c,} * {BL c,} ’ = -- {c,, &} ’ . {C,. A,} * {BL, Ai} ‘. where A, B and C can stand for either reduce H to the block form
Z or 8 and E is the unit S x f matrix.
to
- {Il. 10) ’ . {Ii. Zz) . {IO. 121 ’ - {rl. I,,) ’ + (I:. I,,) ’ . jrz. 0,) {ZL, I,,] ’ {IL. 1,) H = {Z,,,Id ’ + {HI. IT] . {Z,,.12)’ ie,. Z,l~{Z,. I,,) ‘{I!. Z?l{Z~hZz} ‘{Zh 0,) E+{&. rz}{z,,, 12)~‘{Zo. r,} I E + {Zl. I,,) . {I:. h) . 112.e,1 {II. Z,,}.{I?. I,,] ’ . {k. 1,) [ - {II. 121. {Z,,.Zz) (A.5) The determinant
of H can now be calculated matrices of the usual rule about multiplying constant’“), to be det H = 4’
detVl,
using a generalization to block a row of a determinant by a
1~1
(Ah)
detIZr, IO) det{Zo, Z?)’
The evaluation of the signature of H is more delicate. so we go back to the change in symplectic area (A.2) brought about by a change in the 3f independent variables, making the following transformation: 6”+s,=p,,
Sz+&,=&.
s~~s,=p?.
Since & is restricted to the nth tangent as appears in fig. 7. we obtain (%,
(A.7) plane,
with local coordinates
SI,, St),)’ - H - (610, SZ,, SO,) = (SO& SO;, se;)
(MA, 0).
. H’ . (Mh, SO;, SO;), (A.81
where 0 H’ =
-- {I,, I”} ! - 112.I”>
II”, II} 0 - (12, II}
{~“,I21 (11, 121 0
(A.% I
PURE STATE WIGNER FUNCTION
517
This is a much simpler matrix than H given in (AS), though it has the same signature, being as it is derived from H by a linear transformation. A matrix in which the blocks are on the diagonal can now be obtained as Wo, II>, 0
H”= i
0
0 - WC-I,III,
0
,
(A. 10)
hI
0
where the suffix s denotes the symmetric h = - 2&, 111s- (10, I,};’
0
* (12,
part and (A. 11)
The eigenvalues of h will be the eigenvalues of its block matrices, from which it is concluded that uH = q,.
References 1) 2) 3) 4) 5) 6) 7) 8)
E.P. Wigner, Phys. Rev. 40 (1932) 749. H.J. Groenenwold, Physica 12 (1946) 405. J.E. Moyal, Proc. Camb. Phyl. Sot. Math. Phys. Sci. 45 (1949) 99. T. Takabayasi, Prog. Theor. Phys. Jap. II (1954), 341. G.A. Baker Jr., Phys. Rev. 109 (1958) 2198. M.V. Berry, Phil. Trans. R. Sot. 287 (1977) 237. A.M. Ozorio de Almeida and J.H. Hannay, Ann. Phys. (N.Y.) 137 (1981) in press. V.I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1978). M.V. Berry, in Topics in Non-Linear Mechanics, S. Jorna, ed. (Am. Inst. Phys. Proc. 46 (1978) 16. 9) P. Carruthers and M.M. Nieto, Rev. Mod. Phys. 40 (1968) 411. 10) B. Leaf, J. Math. Phys. 9 (1968) 65, 769; 10 (1969) 1971, 1980. 11) A. Einstein, Verh. dt. Phys. Ges. 19 (1917) 82. M.L. Brillouin, J. de Physique (Ser. 6) 7 (1926) 353. J.B. Keller, Ann. Phys. (N.Y.) 4 (1958) 180. V. Maslov, Theorie des Perturbations (Dunod, Paris, 1972). 12) H.J. Korsch, J. Phys. A: Math. Gen. 12 (1979) 811. 13) M.V. Berry, J. Phys. A: Math. Gen. 10 (1977) 2083. 14) M.V. Berry and Z.V. Lewis, Proc. R. Sot. A370 (1980) 459. 15) V. Gantmacher, Matrix Theory, Vol. 1 (Chelsea, New York, 1977). 16) SO. Rice, Bell. Syst. Tech. J. 23 (1944) 282; 24 (1945) 433. 17) M.S. Longuet-Higgins, Phil. Trans. R. Sot. A249 (1956) 321. 18) A. Voros, Ann. Inst. Henri Poincare 26 (1977) 343.
PURE STATE CONDITION
Publishing Co.
FOR THE SEMI-CLASSICAL
WIGNER FUNCTION A.M. OZORIO de ALMEIDA Institute de Fkica “Gleb W&&in”
Universidade Estadual de Campinas, Campinas, 13100 SP., Brasil
Received 30 June 1981
The Wigner function W(p, q) is a symmetrized Fourier transform of the density matrix p(q,, q2), representing quantum-mechanical states or their statistical mixture in phase space. Identification of these two alternatives in the case of density matrices depends on the projection identity p* = p; its Wigner correspondence is the pure state condition. This criterion is applied to the Wigner functions obtained from standard semiclassical wave functions, determining as pure states those whose classical invariant tori satisfy the generalized Bohr-Sommerfeld conditions. Superpositions of eigenstates are then examined and it is found that the Wigner function corresponding to Gaussian random wave functions are smoothed out in the manner of mixedstate Wigner functions. Attention is also given to the pure-state condition in the case where an angular coordinate is used.
1. Introduction
The Wigner function W(p, q) is a representation of pure quantummechanical states or their statistical mixture in phase space and reduces to a Liouville distribution in the classical limit h +O. In the original work’,2V3), W(p, q) was obtained as symmetrized Fourier transform of the density matrix p(q,, qJ while Takabayasi4) and Baker’) were the first to consider the condition for a given W to represent a pure state. More recently Berry6) and coworkers’) have obtained explicit functional forms for the semi-classical (SC) limit of Wigner functions representing pure bound states whose corresponding classical motion is integrable. For a system with f degrees of freedom (2f-dimensional phase space) the classical limits of W for these states are the f-dimensional tori which correspond to the invariant manifolds of the classical orbits*). Relaxing h from zero reveals a rich oscillatory structure dominated by the tori. Systems described by bounded (angle) variables must be treated with caution’.“), but they have the advantage that the corresponding classical invariant manifolds may have topologies simpler than tori. In particular, free rotations provide an example where the S.C. and even the classical approximations of W coincide with the exact quantumResearch supported by FAPESP, CNPq and FINEP.
0378-4371/82/0000-0000/$02.75
@ 1982 North-Holland
502
A.M. OZORIO
mechanical
evaluation.
also given
by Berry6).
The main purpose satisfy
the pure
the S.C. Wigner
state
(PS) condition
the
determine
which
for general
geometric the
restriction which results explicitly from satisfy the generalized Bohr-Sommerfeld
rotations
were
that the S.C. form of W’,‘) does
in the integral
in a cunning
chords
functions
of this work is to verify
This will be seen to occur Berry
de ALMEIDA
S.C.
form
derived
by Baker’).
way by the interlocking Wigner
the calculation conditions”).
function.
of
The
only
is that the state
must
The SC form of W for the eigenstates of an integrable Hamiltonian, which is thus confirmed by this analysis, displays many oscillations and diverges the Wigner caustic’). Coherent superalong (2f - 1)-dimensional surfaces: positions of eigenstates are even more complicated, since the Wigner integral is nonlinear. The temptation is hence felt to relegate the use of this phase space representation to statistical mixtures of states”). These will appear as linear superpositions of pure W’s whose individual oscillations will tend to cancell on the whole. Are there pure states whose Wigner function resembles a mixed state? This question is answered by means of the PS integral in the last part of this paper. There it is shown that Gaussian random wave functions have this property. It is interesting to note that Berry13) has put forward the hypothesis that this is exactly the form of semi-classical wave that will appear for non-integrable systems. Before getting on to these subjects we derive systems described by angle coordinates. Contrary an immediate extention of the usual PS condition.
the exact PS condition for to previous belief this is not
2. The exact pure state condition The Wigner function corresponding (p 1I+!J),where p and q are the continuous momentum
W(P,
and coordinate
operators
1
4) = (nh)Y’ dQ exp(-
where f is the number This quasi-probability
to the wave functions (q I$) and and unbounded eigenvalues of the
6 and 4, is given
zip-Q/h)(q + QI$XICIlq - Q>.
of degrees of freedom. distribution has the important
I dqW(p,9) = 1$cI)Iz, I dpW(p> lb )44’.
by
property
(2.1)
that
KP
9) =
c2.2)
PURE
STATE
WIGNER
FUNCTION
503
W is not a linear function of I$) so that a linear superposition of W’s does not correspond to a pure state. It is thus called a mi.xed-state Wigner function and also reduces in the classical limit to a Liouville distribution. A necessary and sufficient condition for W to represent a pure state is P=
I
d5, dS‘zW(&)W(&) cos(Alh) = (N4)‘WE).
(2.3)
--z Here 5 = (P, 4)
(2.4)
and A is twice the skew product (6 - 5) A ( t2 - 4):
(2.5) Geometrically, A is the sum of the areas of the projections on each conjugate (p,, 4.) plane of the triangle which has &, 52 and 5 as the midpoint of each side, as shown in fig. 1. This symplectic area8) is invariant under canonical transformations and hence also for metaplectic transformations: the quantum version of linear canonical transformations. Since W is also invariant under metaplectic transformations’8,7), it follows that this property also holds for the P.S. integral (2.3). If one of the coordinates $ has bounded eigenvalues: -7T
(2.6)
PI f
Fig. 1. 41, &Zand f determine
the midpoints
of the sides of the triangle A.
504
A.M. OZORIO
the eigenvalues each real
of the corresponding
component. angle
angular
We will consider
in coordinate
momentum
space
de ALMEIDA
fi will be discrete only
so that
the case
with a spacing
where
pn corresponds
of h in
q corresponds
of
and
--sc
to a
to a component
(2.7)
integrations
over
q to (2.6) and replacing
integrations
over
p by
summation
(2.X) Berry6) obtains
the Wigner
function
for one degree
of rotational
freedom
as
iii? W(P”,
4) =
I
&
dQ
ew-
2inQHq + QI$I>($I q - QL
(2.9)
with the arguments q ? Q lying, modulo 27r, in the range (2.6). It is interesting that in this case the range of integration does not include the whole domain of variation of Q as opposed to (2.1). This will affect the PS integral as follows: Introducing (2.9) into (2.3) we get
P = (47r)_? I
xc nl”2
dQldQz j dq,dq,
exp{-2ilhhQl+
x (ql+ QI 1+)(4 1qln 1
=4
pnzQ2+
p,,(q-q2)+pa(ql-q)+p,(q2-q1)]}
QJ(qz + Qz
1rcl)(+142 - Qd
TT
I dQ1 dQz 1 dql dq2c S(QI+ q ~
q2 -
jlr)6(Q2
+
ql - q - j2n)
Jlk
x
Zxpl-
2ip,t42
-
41)/R]
x (qi+ QI 1GHIcI ( 4, - Q,Xqz+
Q2 1 $)(d~ 1 a-
In the integration range of ql and q2 there so that with the transformation
(2.10)
Q2).
will be always
be two fi-functions
(QI>QJ-*(Q;, Qi) = (Q, + QT,Q‘, - Q2)
(2.11)
we obtain P=;
II
dQ; dQ%q + Q; I GH4
x {(ICI14 + QiXq + Qi 1+)I.
1q -
Q3 exp(- 2ip,Q;/h)J (2.12)
505
PURESTATEWIGNERFUNCTION
The range of integration is bounded by the full line in fig. 2. To avoid this nonseparated boundary we note that the integrand is a periodic function in the (Q;, Qi) plane, so that an integral over the region bounded by the dashed line (the second Brillouin zone) is exactly twice the original integral. The result is then rr
P=
I
dQXq
+
Qi
IW$I q+ Qi)
ew(- %Q;lfi)
-a
(2.13)
=s[w(P",q)+w(P",q+~)l,
from which we see that W(p,, q) and W(p,, q + 7~)are complementary Wigner functions. The simplest system with which we can exemplify the use of these formulae is the free rotator. The wave functions are simply (4 I VV = (2~)-“’
(2.14)
ewhdil~h
so that W(P”,
(2.15)
4) = &“I274
from which (2.2) and (2.13) immediately
3. Bound states and the Bohr-Sommerfeld
follow.
conditions
The semiclassical approximation of W is constructed from the f-dimensional torus which contains the classical orbits in phase space. For a start,
Fig. 2. The area inside the full line is the range to the dashed line and the result halved.
of integration
in (2.12). This can be extended
up
506
A.M. OZORIO
considering action
only
coordinates,
we can
specify
the tori
by their
f
variables
2rrl,, =
where
extended
de ALMEIDA
p - dq,
the RHS
is the symplectic
(3.1)
area for the nth
irreducible
circuit
of the
torus. For any reducible circuit the RHS of (3.1) is zero. Considering the full family of tori this integral defines an f-dimensional vector function Z(t) of the points in phase space (2.4) constant along each torus. Introducing standard semi-classical wave functions into (2.1) and evaluating the integrals by stationary phase it is shown’) that
(3.2)
j
where 6,: and 4; are the two tips of the jth Berry chord of the torus centered on 6, each { ,} is the Poisson Bracket of one of the functions I”((’ - 6;) with Z,,,(g’- &) evaluated at t’= 0, and gj is the signature (i.e. the excess of positive over negative eigenvalues) of this matrix. Aj is the symplectic area for the closed path which starts out at 6; on the torus and returns to 4: along the Berry
A, =
chord:
p . dq.
(3.3)
The configuration for f = 1 is shown in fig. 3. For f 3 2 the number of chords bisecting themselves
at a given
point
4 of
any region of phase space is always even’). Chords can coalesce at the boundary of each region - the (2f - I)-dimensional Wigner cuustic where (3. I) diverges. This divergence is a shortcoming of the SC approximation, since the exact W is everywhere finite if h > 0. Nonetheless, if one is not too close to the Wigner caustic, (3.1) is asymptotically correct, so that it is necessary to examine the change in the signature of {I’, Z .} when the caustic is touched. To do this we make the linear canonical transformation (p, q)+ (K, Z ) taking as coordinates and momenta the action and angle variables linearized at t-. The immediate result is that {I’, I_} = lYZ’(K, z-)/a&, i.e. {I’, Z-} can be interpreted as the Jacobean formation between I+ and 8-, with I- fixed.
(3.4) of the (locally linear) transThe zeroes of aZ’/aK will
PURE
STATE
Fig. 3. One-dimensional
WIGNER
example
FUNCTION
501
of torus and Berry chord.
coincide’) with those of the more familiar matrix K/X(or ap/aqwith the torus a multivalued function p(q)). But there are two important differences: Unlike K/a&, al’/aK is not necessarily symmetrical. Also {I+, I-} is always regular, so that the only possible changes of signature occur on the Wigner caustic. The diagonalization of {If, I-} can be interpreted physically. First we note that for a separated torus this matrix is necessarily diagonal. For a general matrix, diagonalization is obtained by transforming the K variables so that the new axes coincide with the f vectors Ve-I+. Thus by performing the transpose of the inverse transformation on the I- variables we obtain new canonical coordinates in which @/a& is diagonal. One should note that this is not generally equivalent to local separation, since aI’/aZ- need not be diagonal. It will prove necessary to consider the effect on a chord contribution to (3.2) of taking one of its endpoints through an irreducible circuit. If a given circuit is described by an integer vector n, where nl is the number of cycles around the elementary circuit with action I, and so on, then W?(S) = (27&\/h)-’
cos[(Aj + 2m * 1)/h - (~9 + 2n - p)7~/4)] 9 Idet{I+, I-}(“*
(3.5)
where ~1 are the Maslov indices for the f elementary circuits. The symplectic area Aj must fit into a single sheet. If it is possible to fit the entire circuit into a single sheet p(q) of the torus, then the Maslov index will coincide with the change of signature of the Poisson bracket matrix {I’, I-}. To see this, one can use as the q-plane the K-plane (tangent at t-). On completing an irreducible cycle, the directions of VB-I+ will be the same but the orientations of the axes may have altered, leading to a change of signature. An example is the deseparated torus’) with f = 2. Choosing the fixed endpoint of the chord as the
SO8
A.M. OZORIO
origin
(2n7r, 2sn)
whose
centres
lie on the Wigner
whose
crossing
alters
irreducible circuit ment
circuits
is then that
Sommerfeld
of angular
coordinates
leads
determine
For separated
sheet.
The Maslov
is no change
WY, be univalued
on the torus,
caustic,
the signature.
into a single
2, but there
de ALMEIDA
chords,
tori, it is not possible index
in signature
immediately
the singular
in fig. 4 a set of lines to fit
for an elementary
of {I’, I-}. The requireto the
generalized
Bohr-
conditions
I = (n + p/4)h.
(3.6)
The global validity of (3.2) as a PS Wigner function will now be established by directly working out the integral (2.3) by the method of stationary phase. The key integrand
point is that the three phase space vectors are the mid points of the sides of the triangle
which defined
determine the in (2.5) and of
the Berry chords. So, for any choice of 5, and & which results in A just touching the torus with its three corners, one of the chords of W, will coincide with a side of A and likewise for WZ. Furthermore, the total symplectic area bounded by the two chords plus A is exactly one of the chord areas of W(t). This situation for f = I is shown in fig. 5. Changes of 4, and & which only move the corner of A opposite 4 will not alter the total area and it
Fig. 4. A Berry chord with one of its ends fixed at the origin will be singular if its other endpoint touches one of the dashed lines for a separated torus or one of the full lines if the torus has been deseparated by a perturbation. In the latter case it becomes possible to have irreducible cycles in which no singular lines are crossed. The 2 signs refer to the eigenvalues of {I’, I-) near the origin. The Maslov indices for the cycles shown are wU = p,h = 2 and wc = IL,, = 0.
PURE
I
*9
Fig. 5. The symplectic areas bounded by the chords centred on 61 and 42 together with A add up to the symplectic area bounded by the chord centred on g if all the corners of A touch the torus.
will be shown below that even changes of e,, and & in general directions leave the total area stationary. Introducing the semiclassical Wigner function (3.2) into (2.3), we obtain exp{(i/h)[k A f (Ai, + hail&l) If:(Aiz + fMizv/4)11 Jdet{l+, I-}], det{I+, I-Jj~“* ’ (3.7) where we sum over all combinations of sign in the exponent. The phase of one of these terms will be stationary when (a A k Ai, k Ad is stationary, which will only occur in the circumstances already outlined for the combinations (+++) and(---), and only for the chords jr and j2 which coincide with sides of A. Thus neglecting all terms for which the phases are never stationary and relabling jr = 1 and j2 = 2, p=
(n*h)-’ 2
d5 d52 exp{i[(A + AI + AMI + (VI+ hd41) + c c I Idet{I+, I-}, det{I+, Z-}21”2 * *’
(3’8)
It is now necessary to change the integration variables from the midpoints of the sides of A to its corners as shown in fig. 5: 170=42+&L-5,
The stationary
(3.9)
1)1=52-51+5.
condition thus becomes simply
I(qo)=I(ql)=Z,
ql=5t
or
47,
(3.10)
510
A.M. OZORIO
where
47 is one of the Berry
chords
de ALMEIDA
centred
on 4. Finally,
with the canonical
transformations
where
the O’s are the angle variables
conjugate
to the Z’s, we obtain
exp[i(A 1 dHo{ I de’ 1 dzO dz’ 0 0 0
+ A, + A#I + (a, + o&r/4 Idet{Z,, Z,} det{Zo, Zz}l”’ )
+c.c..
(3.12)
The evaluation of the multiple integral in the curly brackets requires the knowledge of the determinant of the Hessian H of second derivatives of the phase of the integrand. This is derived in the appendix, equation (1.6) leading to cos[A#i P = (h/4)‘(7T V/h))’
UH
=
in the appendix d-
{h,
II}\
’ {h,
that the signature I,},’
with the siffix s indicating separated, that
the three
(3.13)
ldet{Zr, Z2}/“*
0 It is shown
+ (CM + ok + (~2)n/41
matrices
* {I3
of H is given
by (3.14)
b,),),
the symmetric
part
of the matrix.
in (3.14) are diagonal.
It is then
If the torus
is
easy to verify
(3.15)
(TH+c,+u2=uj,
as long as all three chords fit into a single sheet. When O. takes one of the chords through a caustic into a different sheet one eigenvalue of its Poisson bracket
matrix
changes
pn + oI $- or = But in this case
uj
f
sign and then (3.16)
nTT.
the Wigner
function
WI is no longer
the primitive
Wigner
function but one of the complementary ones defined by (3.5) with Maslov index ~2. This addition to the phase exactly cancels the r in (3.16) leading again to (3.15) if we use the quantization conditions (3.6). The validity of (3.15) for all 8” leads to the verification of the PS condition. For non-separable tori it is possible to use different Ai’s if they add up to an irreducible circuit which can be represented in a single sheet. The quantum conditions ensure that these lead to the same Wigner function (because of concomittant changes of the signatures). The pure-state condition will select between them because either choice may lead to (3.15) or (3.16). It is
PURE STATE WIGNER
FUNCTION
511
interesting that there is no difference between exchanging W, for its complementary or exchanging WZ. It thus arises that the quantization conditions, while they ensure that the SC Wigner function is univalued, are an essential part of the proof that one is dealing with the Weyl-Wigner representation of a pure state. The degree of self-consistency achieved by the ubiquitous use of the stationary phase approximation is remarkable; not only in the interlocking of A with the Berry chords, but in the calcellation of all the spurious singularities of the semiclassical Wigner function in exactly the right way. The selection of eigenstates of integrable systems considered in a phase space of extended coordinates thus turns out similar to that which holds for rotation eigenstates. Eigenstates of hindered rotations will have semiclassical Wigner function of the same form as (3.2), except that 5 can only take on discrete values of p. The evaluation of the PS integral is here entirely analogous to the treatment given for extended variables: only two cases need to be considered. For rotations the Maslov index is zero so that the quantized area under the periodic classical orbit is the same as in the exact quantization of the free rotator. In the case of librations the Maslov index is two and the quantization is the same as for a system with one degree of freedom described by extended coordinates.
4. Superpositions
of eigenstates-mixed
and pure
So far we have considered only single eigenstates of the Hamiltonian. Their linear combinations will have complicated Wigner functions because of the non-linearity of the defining integral (2.1). On the other hand it has been found that in the SC Wigner function for mixed states, corresponding to thermal equilibrium, the oscillations and even the spurious singularities particular to each eigenstate are averaged out’?. This result is promising as far as the future use of the Wigner function in statistical mechanics is concerned, but the objective here is different. Instead we investigate the possibility that pure combinations of eigenstates may also be smoothed in the manner of mixed states. We start by considering linear combinations of the Wigner functions corresponding to free rotators. The resulting Wigner function will have the form W(P,,
4) = wllh,
with the normalization
T
w”=
1.
(4.1) condition (4.2)
s12
A.M. OZORIO
The PS condition
de ALMEIDA
is in this case reduced
to
w=” = w n from which consider
(4.3)
it follows
the classical
that w, = 6,” is the only exact
normalized
solution.
Yet
limit in which
wn-h-4 MP),
(4.4)
where w(p) is here a smooth function of p -the quantized levels move closer together while normalization requires that the probability of individual levels diminishes. In this case the PS condition is also satisfied, and we may consider that even in the SC limit the mixed Wigner function is a good approximation of one that represents a pure state. We now examine the nature of this pure state. It is definitely not an eigenstate for some Hamiltonian describing a hindered rotation, since its Wigner function would then condense on a curve p(q) in the classical limit, instead of continuing to occupy a region of phase space. properties of the Wigner function (2.2) it follows that
From
the projection
I(4 I ICIV = (27T)F’,
(4.5)
I(P, I $)I = WM.
(4.6)
But the latter (pn
leading
projection
I $,) = (w,hP
exp(i W,
(4.7)
to
(4 I $) = Finally,
implies
T (w,/27-r)“2 exphqlfi
the comparison
of the square
+ &)I. of the modulus
(4.8) of (4.8) with (4.5) shows
that & must be a randomly distributed variable in the interval words (q 1 q!~)is a sample of a Gaussian random function’“.“).
(0,27~). In other It is interesting
that the corresponding (p ) $) d oes not have a smooth SC limit, as the allowed values of p are squeezed together, Instead its features resemble those of a Weierstrass-Mandelbrot fractal function14). We can now consider the linear combinations of Wigner functions corresponding to eigenstates of general integrable Hamiltonians:
W(5) =
cn
W”W”(i3,
(4.9)
where n is here the set of quantum numbers which select the allowed actions I, according to (3.6). Since each W”(t) satisfies individually the PS condition,
PURE STATE WIGNER FUNCTION
513
its application to (4.9) again reduces to (4.3). It also follows that in the SC limit (4.9) represents an asymptotically pure state. It appears more complicated than the one obtained from the free rotator eigenstates, but using the Poisson summation formula W(s) = $
exp(q
M -
f dZw(Z) 0
as phase phases Aj of
(4.4) is smooth of w.. Stationary of these integrals eliminates but the M = 0 term since (3.2) are fractions of 27rZ, so that
In the classical limit6 w’(5)
--,
NZ(5) - 1)
(2nTT)f’ leading to (4.13)
W(5) = w(Z(5))/(277)fY
This same result is achieved semiclassically for f = 1 by using the uniform approximation of W’(e) in terms of Airy functions6). Equation (4.13) shows that in the classical limit w(Z) reduces to the Liouville density. To what wave functions do these Wigner functions correspond? Clearly they are a superposition of SC wave functions with an amplitude whose square modulus is w(Z)/fi’:
(4 I $> = (27rfi)F”’ T
[w(L) g] “*ew{i[Si(q, LM
+ fM1,
where summation over the different branches of S(q, Z,,) is implied. Expansion of the exponent near any given position qo S(q, Z”)/h + 8. = pn(q0) * (4 - 40) + @iI,
(4.15)
puts (4.14) in the local form of a superposition of plane waves. The phases 8. must be such that the cross terms in the square modulus of (4.14) cancell out, so as to agree with the projection of the Wigner function m
0
w(P, dp
--oc
4)) =
(2?l)f
(4.16) 0
514
A.M. OZORIO
de ALMEIDA
I<
Fig. 6. Momentum
and coordinate
wave
intensities
91$>12 I
h- q for w of step function
form
thus, again the wave functions are samples of Gaussian random functions, but there are two important differences with respect to the superpositions of free rotations. The first is that the average intensity is now a function of q. None the less it is a smooth function. Consider the simple case with f = 1, where
W(I(c3) = 0(1(t) - r”Yr”7 proportional line contained
in the I(s)
= 10 curve
as show
to the length
of the q = constant
in fig. 6. More important
is the
fact that all the metaplectic projections of the Wigner function so constructed will be Gaussian random functions. The momentum wave intensity is also shown in fig. 6 for the step Wigner function. It has been conjectured by Berry”) that the quantum mechanical eigenfunctions of systems whose classical motion is may resemble Gaussian random functions. These differ markedly from the eigenfunctions of integrable systems, but not from their superpositions with random phases which have very simple Wigner representations. unintegrable motion the Wigner approach may yet prove fruitful.
PURESTATEWIGNERFUNCTION
515
Acknowledgement
I thank Prof. M.V. Berry for many helpful discussions.
Appendix
The evaluation calculate the determinant H
=
x
stationary phase requires that we 3f Hessian matrix
a’@ +A,+&)
(A. 1)
d2V”,11,w> .
One way of doing this is by determining the change in the total area brought about by an arbitrary change in the independent variables from a stationary configuration. This change in area can be reduced to the symplectic area of the three small triangles whose corners are respectively a corner of A and the tips of the Berry chords that have been displaced from it by the change in q. and q,. This is shown in fig. 7. So (61&H,, se&- * H * (610, SI,, se,) = so A s1+ so A 62 + a* A s,, showing that the change of symplectic
(A.3
area is indeed of second order in the
Fig. 7. A general displacement of the corners of A from the torus will change the symplectic area A + Ai + A2 by the area of the three triangles shown shaded in the figure.
A.M. OZORIO de ALMEIDA
516
independent invariant
variables.
The skew product on the right-hand
which can thus be calculated
done to arrive
at equation
side is a canonical
in the (fZ,, I,) coordinate
system,
as was
(3.4). Then
60 = {I,, I”} -’ * ({I,, 80) * 610 - SZ,). FZo,
6, = 1% I,} ’ * 68, -{&, O,} * SI,, {I,,. III * F8, - {lo, 0,) * sz, 62 = {I,, Z,} ’ * ({I?, Z,} * 68, -{Z2. 0,) * 61, + {Zz. O,,} * SZO),610.
(A.3)
We now make use of the identities {Ak, Bj} = - {Bj, A,,}‘, {A,, 41 * lo,, Ai} = {A,. Or1 * 14. A,}. {Oj, ZkI * {O,, Zj} = E + {Oj, f-Zk}* {a, Z,I,
(A.4)
{Ai> Bk} = {Ai, Z,} * {% Bk) - {Ai, 0,) * {I,, 6.1, {Ai, Bk}-’ * (Ai, c,} * {BL c,} ’ = -- {c,, &} ’ . {C,. A,} * {BL, Ai} ‘. where A, B and C can stand for either reduce H to the block form
Z or 8 and E is the unit S x f matrix.
to
- {Il. 10) ’ . {Ii. Zz) . {IO. 121 ’ - {rl. I,,) ’ + (I:. I,,) ’ . jrz. 0,) {ZL, I,,] ’ {IL. 1,) H = {Z,,,Id ’ + {HI. IT] . {Z,,.12)’ ie,. Z,l~{Z,. I,,) ‘{I!. Z?l{Z~hZz} ‘{Zh 0,) E+{&. rz}{z,,, 12)~‘{Zo. r,} I E + {Zl. I,,) . {I:. h) . 112.e,1 {II. Z,,}.{I?. I,,] ’ . {k. 1,) [ - {II. 121. {Z,,.Zz) (A.5) The determinant
of H can now be calculated matrices of the usual rule about multiplying constant’“), to be det H = 4’
detVl,
using a generalization to block a row of a determinant by a
1~1
(Ah)
detIZr, IO) det{Zo, Z?)’
The evaluation of the signature of H is more delicate. so we go back to the change in symplectic area (A.2) brought about by a change in the 3f independent variables, making the following transformation: 6”+s,=p,,
Sz+&,=&.
s~~s,=p?.
Since & is restricted to the nth tangent as appears in fig. 7. we obtain (%,
(A.7) plane,
with local coordinates
SI,, St),)’ - H - (610, SZ,, SO,) = (SO& SO;, se;)
(MA, 0).
. H’ . (Mh, SO;, SO;), (A.81
where 0 H’ =
-- {I,, I”} ! - 112.I”>
II”, II} 0 - (12, II}
{~“,I21 (11, 121 0
(A.% I
PURE STATE WIGNER FUNCTION
517
This is a much simpler matrix than H given in (AS), though it has the same signature, being as it is derived from H by a linear transformation. A matrix in which the blocks are on the diagonal can now be obtained as Wo, II>, 0
H”= i
0
0 - WC-I,III,
0
,
(A. 10)
hI
0
where the suffix s denotes the symmetric h = - 2&, 111s- (10, I,};’
0
* (12,
part and (A. 11)
The eigenvalues of h will be the eigenvalues of its block matrices, from which it is concluded that uH = q,.
References 1) 2) 3) 4) 5) 6) 7) 8)
E.P. Wigner, Phys. Rev. 40 (1932) 749. H.J. Groenenwold, Physica 12 (1946) 405. J.E. Moyal, Proc. Camb. Phyl. Sot. Math. Phys. Sci. 45 (1949) 99. T. Takabayasi, Prog. Theor. Phys. Jap. II (1954), 341. G.A. Baker Jr., Phys. Rev. 109 (1958) 2198. M.V. Berry, Phil. Trans. R. Sot. 287 (1977) 237. A.M. Ozorio de Almeida and J.H. Hannay, Ann. Phys. (N.Y.) 137 (1981) in press. V.I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1978). M.V. Berry, in Topics in Non-Linear Mechanics, S. Jorna, ed. (Am. Inst. Phys. Proc. 46 (1978) 16. 9) P. Carruthers and M.M. Nieto, Rev. Mod. Phys. 40 (1968) 411. 10) B. Leaf, J. Math. Phys. 9 (1968) 65, 769; 10 (1969) 1971, 1980. 11) A. Einstein, Verh. dt. Phys. Ges. 19 (1917) 82. M.L. Brillouin, J. de Physique (Ser. 6) 7 (1926) 353. J.B. Keller, Ann. Phys. (N.Y.) 4 (1958) 180. V. Maslov, Theorie des Perturbations (Dunod, Paris, 1972). 12) H.J. Korsch, J. Phys. A: Math. Gen. 12 (1979) 811. 13) M.V. Berry, J. Phys. A: Math. Gen. 10 (1977) 2083. 14) M.V. Berry and Z.V. Lewis, Proc. R. Sot. A370 (1980) 459. 15) V. Gantmacher, Matrix Theory, Vol. 1 (Chelsea, New York, 1977). 16) SO. Rice, Bell. Syst. Tech. J. 23 (1944) 282; 24 (1945) 433. 17) M.S. Longuet-Higgins, Phil. Trans. R. Sot. A249 (1956) 321. 18) A. Voros, Ann. Inst. Henri Poincare 26 (1977) 343.