We present a semiclassical study of level widths for a class of one-dimensional potentials in the presence of an ohmic environment. Using a semiclassical approach for the dipole matrix element we obtain the level widths within the golden rule approximation. For potentials with an asymptotic power-law behavior, which may in addition be limited by an infinite wall, we find a universal result: The level widths are proportional to the corresponding quantum number.
REFERENCES
1.
See, e.g., Ulrich Weiss, Quantum Dissipative Systems (World Scientific, Singapore, 1993);
Thomas Dittrich, Peter Hänggi, Gert-Ludwig Ingold, Bernhard Kramer, Gerd Schön, and Wilhelm Zwerger, Quantum Transport and Dissipation (Wiley-VCH, Weinheim, 1998), Chap. 4.
2.
M.
Brune
, E.
Hagley
, J.
Dreyer
, X.
Maı̂tre
, A.
Maali
, C.
Wunderlich
, J. M.
Raimond
, and S.
Haroche
, “Observing the progressive decoherence of the ‘Meter’ in a quantum measurement
,” Phys. Rev. Lett.
77
, 4887
–4890
(1996
).3.
David P.
DiVincenzo
, “Quantum computation
,” Science
270
, 255
–261
(1995
).4.
See, e.g.,
Serge
Haroche
and Daniel
Kleppner
, “Cavity quantum electrodynamics
,” Phys. Today
42
, 24
–30
(January 1989
).5.
Martin C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, Berlin, 1990).
6.
Matthias Brack and Rajat K. Bhaduri, Semiclassical Physics, Frontiers in Physics (Addison-Wesley, Reading, 1997), Vol. 96.
7.
For recent work on accurate WKB wave functions for 1D potentials see:
H.
Friedrich
and J.
Trost
, Phys. Rev. Lett.
76
, 4869
–4873
(1996
);H.
Friedrich
and J.
Trost
, Phys. Rev. A
54
, 1136
–1145
(1996
).8.
Uday P.
Sukhatme
, “WKB energy levels for a class of one-dimensional potentials
,” Am. J. Phys.
41
, 1015
–1016
(1973
).9.
J. F.
Cariñena
, C.
Farina
, and Cássio
Sigaud
, “Scale invariance and the Bohr-Wilson-Sommerfeld (BWS) quantization for power law one-dimensional potential wells
,” Am. J. Phys.
61
, 712
–717
(1993
).10.
As an early reference we mention
V. B.
Magalinskiı̌
, “Dynamical model in the theory of the Brownian motion
,” Zh. Eksp. Teor. Fiz.
36
, 1942
–1944
(1959
)V. B.
Magalinskiı̌
, [Sov. Phys. JETP
9
, 1381
–1382
(1959
)]. Further references can be found for example in Ref. 1.11.
A. O.
Caldeira
and A. J.
Leggett
, “Quantum tunneling in a dissipative system
,” Ann. Phys. (N.Y.)
149
, 374
–456
(1983
).12.
C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, New York, 1978), Vol. 2, p. 1318.
13.
See, e.g., Claude Cohen-Tannoudji, Jacques Dupont-Roc, and Gilbert Grynberg, Atom-Photon Interactions (Wiley, New York, 1992).
14.
A.
Hanke
and W.
Zwerger
, “Density of states of a damped quantum oscillator
,” Phys. Rev. E
52
, 6875
–6878
(1995
).15.
See, e.g., L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Sec. 48 (Pergamon, New York, 1959).
16.
Robert
Karrlein
and Hermann
Grabert
, “Semiclassical theory of vibrational energy relaxation
,” J. Chem. Phys.
108
, 4972
–4983
(1998
).17.
S. M.
Susskind
and R. V.
Jensen
, “Numerical calculations of the ionization of one-dimensional hydrogen atoms using hydrogenic and Sturmian basis functions
,” Phys. Rev. A
38
, 711
–728
(1988
).18.
Hermann
Marxer
and Larry
Spruch
, “Semiclassical estimation of the radiative mean lifetimes of hydrogen like states
,” Phys. Rev. A
43
, 1268
–1274
(1991
).
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