The mechanism of energy quantization is studied for classical dynamics on a highly anharmonic potential, ranging from integrable, mixed, and chaotic motions. The quantum eigenstates (standing waves) are created by the phase factors (the action integrals and the Maslov index) irrespective of the integrability, when the amplitude factors are relatively slowly varying. Indeed we show numerically that the time Fourier transform of an approximate semiclassical correlation function in which the amplitude factors are totally removed reproduces the spectral positions (energy eigenvalues) accurately in chaotic regime. Quantization with the phase information alone brings about dramatic simplification to molecular science, since the amplitude factors in the lowest order semiclassical approximation diverge exponentially in a chaotic domain.

1.
L. E.
Reichl
,
The Transition to Chaos
(
Springer
,
New York
,
1992
);
(
University Press
,
Cambridge
,
1993
).
2.
Quantum Chaos Y2K Proceedings of the Nobel Symposium 116
, edited by
K.-F.
Berggren
and
S.
Åberg
, Physica Scripta Vol.
T90
,
The Royal Swedish Academy of Sciences
,
Stockholm, Sweden
,
2001
.
3.
K. G.
Kay
,
J. Chem. Phys.
101
,
2250
(
1994
).
4.
A.
Inoue-Ushiyama
and
K.
Takatsuka
,
Phys. Rev. E
64
,
056223
(
2001
).
5.
W. H.
Miller
,
J. Chem. Phys.
53
,
3578
(
1970
),
E. J.
Heller
,
J. Chem. Phys.
94
,
2723
(
1991
),
M. A.
Sepúlveda
and
E. J.
Heller
,
J. Chem. Phys.
101
,
8004
(
1994
),
G.
Campolieti
and
P.
Brumer
,
Phys. Rev. A
50
,
997
(
1994
),
[PubMed]
K. G.
Kay
,
J. Chem. Phys.
100
,
4377
(
1994
),
W. H.
Miller
,
J. Phys. Chem. A
105
,
2942
(
2001
).
6.
W. H.
Miller
,
Adv. Chem. Phys.
25
,
69
(
1974
).
7.
L. S.
Schulman
,
Techniques and Applications of Path Integration
(
Wiley
,
New York
,
1981
),
P.
Gaspard
,
D.
Alonso
, and
I.
Burghardt
,
Adv. Chem. Phys.
90
,
105
(
1995
).
8.
M. V.
Berry
and
N. L.
Balazs
,
J. Phys. A
12
,
625
(
1979
).
9.
S.
Tomsovic
and
E. J.
Heller
,
Phys. Rev. Lett.
67
,
664
(
1991
),
[PubMed]
S.
Tomsovic
and
E. J.
Heller
,
Phys. Rev. E
47
,
664
(
1993
).
10.
R. J.
Hinde
,
R. S.
Berry
, and
D. J.
Wales
,
J. Chem. Phys.
96
,
1376
(
1992
),
R. J.
Hinde
and
R. S.
Berry
,
J. Chem. Phys.
99
,
2942
(
1993
).
11.
K.
Hotta
and
K.
Takatsuka
,
J. Phys. A
36
,
4785
(
2003
).
12.
M. C.
Gutzwiller
,
J. Math. Phys.
11
,
1791
(
1970
),
M. C.
Gutzwiller
,
J. Math. Phys.
12
,
343
(
1971
);
M. C.
Gutzwiller
,
Chaos in Classical and Quantum Mechanics
(
Springer
,
Berlin
,
1990
).
14.
M. V.
Berry
and
J. P.
Keating
,
J. Phys. A
23
,
4839
(
1990
),
M. V.
Berry
and
J. P.
Keating
,
Proc. R. Soc. London, Ser. A
437
,
151
(
1992
).
15.
P.
Cvitanović
and
B.
Eckhardt
,
Phys. Rev. Lett.
63
,
823
(
1989
).
16.
F.
Christiansen
and
P.
Cvitanović
,
Chaos
2
,
61
(
1992
).
17.
M.
Sieber
and
F.
Steiner
,
Phys. Rev. Lett.
67
,
1941
(
1991
).
18.
R.
Aurich
and
F.
Steiner
,
Phys. Rev. A
45
,
583
(
1992
).
19.
T.
Szeredi
and
D. A.
Goodings
,
Phys. Rev. E
48
,
3529
(
1993
).
20.
G.
Vattay
and
P. E.
Rosenqvist
,
Phys. Rev. Lett.
76
,
335
(
1996
).
21.
H.
Ushiyama
and
K.
Takatsuka
,
J. Chem. Phys.
122
,
224112
(
2005
). Note that the discussion in this paper can be readily generalized to a periodic orbit.
22.
E. J.
Heller
,
J. Chem. Phys.
76
,
2923
(
1981
).
23.
C.
Jaffé
and
W. P.
Reinhardt
,
J. Chem. Phys.
77
,
5191
(
1982
).
24.
R. B.
Shirts
and
W. P.
Reinhardt
,
J. Chem. Phys.
77
,
5204
(
1982
).
25.
J. B.
Delos
and
R. T.
Swimm
,
Chem. Phys. Lett.
47
,
76
(
1977
),
R. T.
Swimm
and
J. B.
Delos
,
J. Chem. Phys.
71
,
1706
(
1979
).
26.
J.
Main
and
G.
Wunner
,
Phys. Rev. Lett.
82
,
3038
(
1999
).
27.
M. V.
Berry
and
M.
Tabor
,
Proc. R. Soc. London, Ser. A
349
,
101
(
1976
),
M. V.
Berry
and
M.
Tabor
J. Phys. A
10
,
371
(
1977
).
28.
S.
Takahashi
and
K.
Takatsuka
,
Phys. Rev. A
70
,
052103
(
2004
).
29.
S.
Takahashi
and
K.
Takatsuka
,
J. Chem. Phys.
124
,
144101
(
2006
).
30.
H.
Teramoto
and
K.
Takatsuka
,
J. Chem. Phys.
125
,
194301
(
2006
).
31.
K.
Hotta
and
K.
Takatsuka
,
J. Chem. Phys.
122
,
174108
(
2005
).
32.
M. F.
Herman
and
E.
Kluk
,
J. Chem. Phys.
91
,
27
(
1984
).
33.
K.
Takatsuka
and
A.
Inoue
,
Phys. Rev. Lett.
78
,
1404
(
1997
),
A.
Inoue-Ushiyama
and
K.
Takatsuka
,
Phys. Rev. A
59
,
3256
(
1999
),
A.
Inoue-Ushiyama
and
K.
Takatsuka
,
Phys. Rev. A
60
,
112
(
1999
).
34.
H.
Goldstein
,
Classical Mechanics
(
Addison-Wesley
,
New York
,
1980
).
35.
K.
Takatsuka
,
Phys. Rev. E
64
,
016224
(
2001
), which proposes AFC-I.
36.

Take the form dp0G(p0)C̃IIIp0(t), where G(p0) is a slowly varying function of p0, and the stationary phase condition in the p0-integral gives qt=q0. Combining this with ptp0 arising from the stationary phase condition in Eq. (3), the periodic orbit condition is reproduced. Hence, even the sharply peaked function F in Eq. (3) is not necessary. This aspect will be discussed elsewhere.

37.
38.

The advantages of the AFC-III (insensitive to the selection to the Gaussian width) and FG (of no need to calculate the Maslov index) can be used complementarily bringing about a combined method for the calculation of bound states.

39.

The phase destructive interference is also important in erasing the non-eigenstates as precisely studied in Ref. 21.

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