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Journal of Physics A Mathematical and General

Frequency operator for anharmonic oscillators

Francisco Marcelo Fernández

Published May 31, 2001• 6 citations
Publisher

Abstract

We develop a simple approach based on time-independent perturbation theory that leads to a frequency operator for quantum mechanical models. This method is suitable for the calculation of time-dependent physical properties free from secular terms. In particular we obtain the first-order perturbation correction to the frequency operator for a class of anharmonic oscillators.

DOI
10.1088/0305-4470/34/23/305

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We develop a simple approach based on time-independent perturbation theory that leads to a frequency operator for quantum mechanical models. This method is suitable for the calculation of time-dependent physical properties free from secular terms. In particular we obtain the first-order perturbation correction to the frequency operator for a class of anharmonic oscillators.
Abstract Within the framework of the method of variation of constants, time-dependent perturbation theory is presented in a form that naturally eliminates the undesirable ‘secular ’ terms for a time-independent perturbation, permits an order-by-order calculation of the exact wavefunction for a ‘resonant’ harmonic perturbation and offers a simple route to establish the adiabatic hypothesis in quantum mechanics. The formulation rests on the introduction of a flexibility in the choice of the phase factors associated with the varying amplitudes, together with development of a perturbation procedure that closely resembles the Brillouin-Wigner formalism in the static case, and which exploits the flexibility of the phase factors at each order of the theory.
In this article the anharmonic oscillator in quantum mechanics is discussed. First propagators of less complex systems are found by using the path integral method. With perturbation theory an expression, up to any order, has been found for the vacuum propagator of the anharmonic oscillator. In this article only the first and second order were calculated and by looking at the n-pointfunctions the Feynman diagrams of these orders were found. It appears that the anharmonic term up to first order causes a change in the frequency of the system. Looking at second order perturbation theory gives five diagrams. The first four of these diagrams are simple diagrams and again these diagrams alter the frequency of the system. The change in the frequency looks a lot like a series and it is possible to ’guess’ a general expression up to any order for all simple diagrams. This expression seems to fit qualitatively for higher orders however the prefactors were not checked nor is there a proof formulated. The fifth diagrams is a complex diagram and this diagram has not been solved in a satisfying way. Finally some recommendations for additional research are given.
The paper introduces a simple quantum model to calculate in a general way allowed frequencies and energy levels of the anharmonic oscillator. The theoretical basis of the approach has been introduced in two early papers aimed to infer the properties of quantum systems exploiting the uncertainty principle only. For clarity the anharmonic oscillator is described having in mind the lattice oscillations of atoms/ions, yet quantum formalism of the model and approach have general character and can be extended to any oscillating system. The results show that the harmonic energy levels split into a complex system of energy levels dependent upon the number of anharmonic terms that characterize the oscillator.
This is an investigation of the energy levels and wavefunctions of an anharmonic oscillator characterised by the potential 1/2 (omega 2 q 2 + lambda q 4 ). As a lowest-order approximation an extremely simple formula for energy levels, E i (0) = (i + 1/2)1/4(3/alpha i + alpha i ), is derived (i being the quantum number of the energy level), which covers any (lambda, i). alpha i is the real positive root of a cubic equation: y i alpha i 3 + alpha i 2 - 1 = 0, with y i = 6lambda(2i 2 + 2i + 1)/(2i+1). This formula reproduces the exact energy levels within an error of about 1% for any ( lambda ,i) (the worst case is 2% for i = 0, lambda to infinity). Systematically higher orders of this perturbation theory are developed, which contains the 'usual' perturbation theory for the limiting case of small lambda , but this perturbation theory is valid for any (i, lambda). The second-order perturbation theory reduces the errors of our lowest-order results by a factor of about 1/5 in general. Various ranges (large, intermediate, small) of (i, lambda ) are investigated and compared with the exact values obtained by the Montroll group (1975, 1978). For i = 0, 1, even the fourth-order perturbation calculation can be elaborated explicitly, which reduces the error to about 0.01% for any lambda . For small lambda it gives correct numerical coefficients up to lambda 4 terms, as it should.
Abstract Quantum perturbation is an important tool for developing approximate solutions to quantum problems that cannot be solved exactly. This chapter provides a brief, systematic development of both time-independent and time-dependent quantum perturbation theory and the physical interpretation of its results, including both first- and second-order corrections. A quartic perturbation of the simple harmonic oscillator is used to illustrate time-independent perturbation theory. A spin-1/2 system subject to an oscillating field is used in time-dependent perturbation theory to describe the absorption and emission of energy from the quantum system.