Physics Letters A 341 (2005) 390–400
Is there any unique frequency operator for quantum-mechanical
, Francisco M. Fernández
Department of Physics, Jaypee Institute of Information Technology, A-10, Sector-62, Noida 201307, India
Cequinor (Conicet), Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Calle 47 y 115,
Casilla de Correo 962, 1900 La Plata, Argentina
Received 16 August 2004; accepted 23 January 2005
Available online 17 May 2005
Communicated by A.R. Bishop
We discuss some recently proposed methods that yield operator solutions free from secular terms for the quantum-mechanical
anharmonic oscillators. The frequency operators obtained in those methods are not same. In the present work we show that
the apparent differences between the frequency operators derived by different approaches are due to the different ordering
of noncommuting observables. Here we derive some relations between the existing frequency operators. In some cases we
generalize results from quartic to higher anharmonicities.
2005 Elsevier B.V. All rights reserved.
Straightforward application of perturbation theory to nonlinear equations of motion in classical mechanics
gives rise to secular terms that increase unboundedly with time even for periodic motion [1–3]. There are sev-
eral approaches that enable one to correct such unphysical behavior of the approximate solutions; among them
we mention the Lindstedt–Poincaré technique, the method of renormalization, and the method of multiple scales
[1–3]. The actual frequency of the motion, which appears explicitly as a natural consequence of the application of
those procedures, plays an important role in the perturbation calculation.
Unphysical secular terms also appear in the application of time-dependent perturbation theory to quantum-
mechanical systems. In order to remove them several authors have adapted the methods of classical mechanics
E-mail address: email@example.com (A. Pathak).
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