6 November 2000
Physics Letters A 276 (2000) 225–232
www.elsevier.nl/locate/pla
Quantum-mechanical tunnelling and the renormalization group
A.S. Kapoyannis
a
,N.Tetradis
a,b,∗
a
Nuclear and Particle Physics Sector, University of Athens, GR-15771 Athens, Greece
b
Department of Physics, University of Crete, GR-71003 Heraklion, Greece
Received 22 August 2000; accepted 29 September 2000
Communicated by C.R. Doering
Abstract
We explore the applicability of the exact renormalization group to the study of tunnelling phenomena. We investigate
quantum-mechanical systems whose energy eigenstates are affected significantly by tunnelling through a barrier in the potential.
Within the approximation of the derivative expansion, we find that the exact renormalization group predicts the correct
qualitative behaviour for the lowest energy eigenvalues. However, quantitative accuracy is achieved only for potentials with
small barriers. For large barriers, the use of alternative methods, such as saddle-point expansions, can provide quantitative
accuracy.
2000 Elsevier Science B.V. All rights reserved.
1. Introduction
The exact renormalization group [1,2] is a power-
ful method with a wide range of applications in var-
ious fields. It provides a framework in which it is
possible to study non-perturbative aspects of physi-
cal phenomena. This is made possible by an exact
renormalization-group equation [2–4] that describes
the dependence of generating functionals for the cor-
relation functions of the theory on a coarse-graining
scale. In particular formulations, such as the effective
averageaction [5,6],an exact equation can be obtained
for the flow of the coarse-grained free energy of the
system, a quantity with intuitive physical interpreta-
tion.
The biggest difficulty one faces in this approach
concerns the approximations that must be made in
order to turn the exact renormalization-group equa-
*
Corresponding author.
E-mail address: nikolaos.tetradis@cern.ch (N. Tetradis).
tion, a functional differential equation, into evolution
equations for quantities such as the effective poten-
tial. A widely used approximation method employs
an expansion of the effective action in the number of
derivatives of the fields appearing in it. Its validity, at
the formal and practical level, has been studied and
tested extensively [6–9]. The absence of a small ex-
pansionparametermakesthe precision of a given trun-
cation in the number of derivatives hard to estimate,
and one often must rely on comparisons with alter-
native methods. However, one can obtain answers for
quantities that are difficult to compute, such as criti-
cal exponents, amplitudes and the critical equation of
state for second-order phase transitions, or the bubble-
nucleation rate beyondthe semiclassical level for first-
order phase transitions [6].
In this letter we are interested in the applicability
of the approach to tunnelling phenomena. We work
within the formulation of the effective average action
[5,6]. Recent work [10,11] has demonstrated how the
presence of a coarse-graining scale leads to a con-
sistent quantitative description of such phenomena,
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