PHYSICAL REVIEW A VOLUME 46, NUMBER 2 15 JULY 1992

Quantum and semiclassical Husimi distributions for a one-dimensional resonant system

R. Egydio de Carvalho

M. A. M. de Aguiar
Instituto de Ftsica "Gleb 8'ataghin, "Universidade Estadual de Campinas, Campinas, 13081 Sicko Paulo, Brazil

(Received 13 November 1991)

We compare exact and semiclassical Husimi distributions for the single eigenstates of a one-
dimensional resonant Hamiltonian. We find that both distributions concentrate near the unstable fixed

points even when these points are made complex by suitably varying a parameter.

PACS number(s): 05.45.+b, 03.20.+i

The study of the semiclassical limit of quantum
mechanics is as old as quantum theory itself. Comparing
the features of both theories is, however, not a trivial
task, due to the very di8'erent languages that they em-
ploy. In order to overcome this difficulty, Wigner [1] in
1932 provided a rule to associate to every quantum state
g(q ) a phase-space function W&(q, p ) through

W&(q, p) = f dx f'(q+x /2)g(q —x /2)

/px /A

(1)

Although 8' is real valued and projects onto the
correct marginal distributions, it is necessarily negative in
some regions of phase space [2]. This implies that the
Wigner function cannot be interpreted as a probability
density. In practice, simple examples show that W(q, p)
oscillates violently in the semiclassical limit [3], suggest-
ing that these oscillations could be washed out be a suit-
able averaging. Indeed, Husimi [4] showed that a Gauss-
ian smoothing of (1) would result in a positive quantity.
The Husimi function for the eigenstate g is defined by

classical orbit with energy E, E =H(q, p ) and the over-
dot means time derivative. The two important features of
(4) are the Gaussian concentration on the energy shell
and the dependence on ~z ~, showing that the fixed points
play an important role in h (q,p).

The purpose of this paper is to compare exact and
semiclassical Husimi distributions for a one-dimensional
integrable Hamiltonian system exhibiting two resonances.
Although simple, this model presents a complex tori
structure where the fixed points determine that certain
regions of phase space are more frequently visited by the
orbits. Moreover, as a parameter is varied, one of the
resonances disappears. As we shall see, this model con-
stitutes a very interesting example to check the validity of
the semiclassical approximation for the Husimi distribu-
tions.

We start by considering an autonomous classical Ham-
iltonian system with two degrees of freedom in action-
angle variable (J,O) given by the truncated Birkhoff-
Gustavson normal form [7,8]:

K(J, O) =Ho(J)+aH&(J, O, ),

h (q,p)= f dq'dp'W (q', p')= 1

Xexp[ —(q —q') /2b —(p —p') b /2R I

where

Ho= J2+6M, —c(6J, } +a(6J, )

H, =(6J, ) (Jz —J, )'i cosO, ,

(Sb)

(Sc)

(2)

where ~z ) is the coherent state for the harmonic oscilla-
tor,

z =(1/&2)(q /b+ipb /fi), (3)

&2M
h (q,p)= exp

z r(E )

(E E)—
where E is the eigenenergy, r(E ) is the period of the

and b is a free parameter controlling the Gaussian width.
Semiclassical expressions for both Wigner [4,5] and

Husimi [6] functions have been obtained recently but not
yet tested numerically.

For one degree of freedom the semiclassical limit of (2}
is given by [6]

and A. , c, a, and a are parameters.
As J2 is conserved, we concentrate our analysis on the

J&-0& plane. It is easy to check that the origin J, =0 cor-
responds to a stable periodic orbit. The cubic depen-
dence of 00 on J& and the term cos8& in 0& produces two
separate resonances (two islands) of the same order, with
unstable periodic orbits at 0, =0 for one resonance and

0, =~ for the other. As the parameter a is varied, the
hyperbolic point at 0, =0 vanishes through a saddle-
center bifurcation and a single resonance remains. In
what follows we shall fix the parameters values as
X=1/32. 1, c=6 /4, a =6 /2, and JR=29.765.

By computational convenience we analyze the motion
in the Cartesian coordinates (p, q) connected with (J, , O, )

by the canonical transformation,

q =+2J,cosOi, p =+2JisinO, .

46 1128 1992 The American Physical Society



46 BRIEF REPORTS 1129

—5.80 -3.87 —1.93 0.00 1.93 3.87 5.8Q
5 80 '

s s ~ I I I I I s I I I I s I s s I s s I I s a.80
—5.80 —3 87 —1.93 0-00 1.93 3.87 5.8Q

I I I I I I I I I I I I I ~ I I I I I I I I I

3.87 - 3.87 3.87 - 3.87

1.93 - 1.93 1.93 - 1.93

P 000 I

- 0.00 P ooo- - 0.00

—1.93 - —1.93 —1.93 — —1.93

-3.87 - -3.87 -3.87 - -3.87

' -t.80 -3.87 -1.93 0.00 1.93 3.87 5.80 ' -t.80 -3.87 -1.93 0.00 1.93 3.87 5.80

FIG. 1. Separatrices of the two unstable fixed points in the

(q,p) plane for = 1.1 X 6
FIG. 2. Separatrix of the only remaining unstable fixed point

in the (q,p) plane for a=1.3X6 '.

a=&Iexp(i1p), a'=~Iexp( imp), — (7)

This transformation curves the space (Ji,8i) attaching
the point 8&=0 with 8, =2m. . Figures 1 and 2 show the
separatrices for the cases a =1.1 X 6 and 1.3 X6, re-
spectively.

To calculate the semiclassical and quantal Husimi dis-
tributions we need to know the eigenvalues and eigen-
functions of the corresponding Hamiltonian operator.
The quantization of the normal form [9] is better visual-
ized when described in terms of the complex variables
(a,a' ) defined by

creation operators, respectively, where [&i,&k]=A'5 k.
The operator J2 is also a constant of motion, so that a
base of eigenstates of J2 puts the Hamiltonian matrix in a
block form. To get these eigenstates we first define a
basis of harmonic oscillators for the operators
I =(ata + —,'), Ini, n2), so that IJIni, n2) =Pi(nj.
+—,

'
) I

n i, n2 ). Then, in analogy to Eqs. (8) we define the
operator J, and Jz which, acting on In, , ni), define the
quantum numbers (n, m) with ni=6n+np and
n2=m —n where n0 0, 1, 2, 3, 4, or 5 fixes one of the
sixfold symmetry blocks. In this paper we have chosen
n0=0. Therefore, we define the basis vector

where (I,y) are related to (J,e) by canonical transforma-
tion Ik )—:I6n+np, m n)— (9)

I) I]
Ji=, 8i =6yi q2, J2 —I2+, t92 —y2 (8)

The quantization of (a, a" ) defines the annihilation and

and for each fixed value of m we obtain a block of the
Hamiltonian matrix. The matrix elements are then given
by

(k'IPIk ) = [1!9[m+—',
7

)+RA'[6n+ —'] cubi [6n+ —,'] +—al [6n+ —] ]5„,„5

+—iri [(6n +4)(6n +5)(6n +6)(6n +7)(6n +8)(6n +9)(m —n) ]
' 5„.„+i5 ~ +H. c. (10)

where H.c. means Hermitian complex term.
The integrable resonant perturbation implies that the

Hamiltonian matrix is not only real and symmetric but
also separated by uncoupled blocks, where each block is
tridiagonal and finite. Therefore the system is reduced to
one degree of freedom and we interpret the classical
Hamiltonian Eq. (5) as a truly one-dimensional system
where J2 is just a parameter and every orbit is periodic.
The block which corresponds to the value used classically
for Jz has dimensions (179X179) and the eigenvalues
and eigenvectors of Hk. & are obtained by direct diagonal-
ization of the matrix.

Let Ig„), n =0, 1, . . . ,m be the eigenfunctions of the
block labeled by m. In terms of the basis vectors

I
k )

Iy„ ) = y c„Ik & .
k=0

Therefore, from the definition (2), the Husimi distribution
reads

m 2 m k —zz /2A
2

gc (zIk) = gc„
k=0 k=p iri k!

(12)
z =(1//2A')(q +ip) .

The Gaussian width b has been chosen as &3ii to fit the
harmonic-oscillator term of the Hamiltonian in terms of I
and y.



1130 BRIEF REPORTS

580580 387 -193
I I I I I I ~ I ~ ~

3.87-

1.93

P O.OO-

-1.93 .
W

-3.87-

0 00 1 93 3 87 5 8Q
~ I ~ ~ I I I 'I ~ I I ~ I

- 3.87

- 1.93

- 0.00

- —1.93

- -3.87

i
/I~

p ~m~ q

5 8-5.80 -3.87580 I I I ~

I

3.87

1.93
I

P ooo-

—1.93-

-3.87-

-1.93 0.00
~ ~ ~ ~ I I

1.93 3.87 5.8g 80

- 3.87

- 1.93

- O.oo

- —1.93

- -3.87

~ I ~ I I ~ ~ I I ~ ~ ~ I ~ I I I ~ I I
' -5.80 -3.87 -1.93 0.00 1.93 3.87 5.80

I I I I I I ~ ~ ~ ~ ~ ~ I ~ I ~ ~ I I I I ~

-5.80 -3.87 —1.93 0.00 1.93 3.87 5.80

FIG. 3. Husimi distribution for state 58, A'= 6, and a= l. 1 X 6 '. (a) shows contour plots of the semiclassical calculation [Eq. (4)],
with level curves from 0 to 0.096, in steps of 0.008. (b) shows the three-dimensional plot and (c) shows contour plots for the exact dis-
tribution [Eq. (12)],with level curves from 0 to 0.152, in steps of 0.008.

-5.80 -3.87 -1.93 0.00
I I I I I I I ~ I I I ~

1.93 3.87 5.8S (b) -s.eo -3.S7 -1.93
I ~ I 'I ~ I I

0.00 1.93 3.87 5.8(I

3.87 - 3.87 3.87
(c) - 387

1.93 - 1.93 1.93 - 1.93

P ooo- - 0.00 P ooo- - 0.00

-1.93- - -1.93 -1.93- - -1.93

I

-3.87- - -3.87 -3.87- - -3.87

~ ~ ~ I ~ ~ ~ ~ I ~ ~ ~ ~ I ~ I I ~ I I ~ ~

' -5.80 -3.87 -1.93 0.00 1.93 3.87 5.80

P

' -t.80 -3.87 -1.93 0.00 1 93 3.87 5 80

FIG. 4. The same as in Fig. 3 for state 72 and a = 1.1 X 6 . In (a) the curves go from 0 to 0.12, in steps of 0.02 and in (c) from 0 to
0.52, in steps of 0.02.

5.80 ' -3.87 -1.93 0.00 1.93 3.87
I I I I I ~ ~ I I I ~ I I I I I I ~ I . 5%.80

-s.eo -3.87 -1.93 0.00
~ I I I ~ ~ I ~ ~ I I I

1.93 3.87 5.+80

3.87 - 3.87 3.87 (c - 3.87

1.93- . 1.93 1.93 - 1.93

ooo- - 0.00 P ooo- - o.oo

-1.93- - —1.93 -1.93- - —1.93

-3,87- - —3.87 -3.87 - -3.87

' -t.80 -3.87 —1.93 0.00 1.93 3.87 5.80

I

~ ~ ~ ~ ~ ~ ~ I ~ ~ I I I ~ I ~ ~ I I I ~

—%.80 —3.87 -1.93 0.00 1.93 3.87 5.80

FIG. 5. The same as in Fig. 3 for state 51 and a= 1.3 X 6 '. In (a) the curves go from 0 to 0.056, in steps of 0.008 and in (c) from 0
to 0.096, in steps of 0.008.



46 BRIEF REPORTS 1131

5eo
—3.87 -1.93

I 1 I I I I ~ 1 I
0'00 1'93 3 87 5 8Q 0~ I ~ ~ I I I I I I I I ~ V

(b) S.8O
' -3.87 -1.93 0.00 1.93

I I I I I I I S I I 1 I ~ ~ I ~

3.87 5.89

3.87 - 3.87 3.87 - 3.87

1.93 - 1.93 1.93 - 1.93

ooo- - 0.00 o.oo- - 0.00

-1.93- - —1.93 —1.93 - -1.93

-3.87- - -3.87 -3.87- - -3.87

' %.80 -3.87 -1.93 0.00 1.93 3.87 5.80 .80 -3.87 -1.93 0.00 1.93 3.87 5.80

FIG. 6. The same as in Fig. 3 for state 74 and a=1.3X6 . In (a) the curves go from 0 to 0.22, in steps of 0.02 and in (c) from 0 to
0.18, in steps of 0.02.

Figures 3-6 show examples of the exact Husimi distri-
bution [Eq. (12)] and their semiclassical counterparts [Eq.
(4)]. In Figs. 3 and 4, a= l. 1 X6, corresponding to the
classical situation described in Fig. 1, while for Figs. 5
and 6, a=1.3X6, corresponding to Fig. 2. These
eigenstates were selected for having energies close to the
separatrices in each case. Both classical and quantal dis-
tributions have been normalized so as to have unit in-
tegral over q and p. In all contour plots the level curves
go from zero to the maximum value reached by h (q,p) at
constant step (see captions).

Besides the very good agreement between the distribu-
tions we notice that most semiclassical Husimi distribu-
tions are more spread when compared to the quantum
ones, an effect similar to the scarring phenomena of
chaotic systems. This is also verified in the behavior of

neighboring states: some are more uniformly distributed
along the energy shell and some exhibit concentrations
like the states shown in the figures. Another effect is,
however, more important in our results: although for
a=1.3X6 the fixed point corresponding to 8&=0 has
vanished from the real plane and has become complex, it
still has a very strong influence on the wave functions, as
can be seen by the peaked distribution of Fig. 6 around
this point. The scar of the complex point also appears in
several neighboring states. This fact suggests that com-
plex periodic orbits might play a very important role in
the theory of scars for higher-dimensional systems.

We thank A.M. Ozorio de Almeida for interesting dis-
cussions. This work was partly supported by FAPESP,
CNPq, and FINEP.

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[2] M. Hillery et al. , Phys. Rep. 106, 121 (1984).
[3] A. M. Ozorio de Almeida, Hamiltonian Systems: Chaos

and Quantization (Cambridge University Press, Cam-
bridge, 1989).

[4] K. Husimi, Proc. Phys. Math. Soc. Jpn. 22, 264 (1940); K.
Takahashi, Jpn. 55, 1443 (1986).

[5] M. V. Berry, Proc. R. Soc. London Ser. A 423, 219 (1989).
[6] M. Baranger and M. A. M. de Aguiar (unpublished); J.

Kurchan et al. , Phys. Rev. A 40, 6800 (1989).
[7] F. G. Gustavson, Astron. J. 71, (1966) 670.
[8] R. Egydio de Carvalho, Ph.D. thesis, UNICAMP, 1989;

R. Egydio de Carvalho and A. M. Ozorio de Almeida,
Phys. Lett. A 162, 457 (1992).

[9]M. Robnik, J. Phys. A 17, 109 (1984); R. Egydio de Car-
valho and A. M. Ozorio de Almeida, Rev. Bras. Fis. 18,
400 (1988).