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The anharmonic oscillator in quantum mechanics

Jordy de Vries

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

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.

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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.
In this article, for pedagogical purposes we have discussed the application of nondegenerate perturbation theory up to the third order to compute energy eigenvalues and wave functions for the quantum anharmonic oscillator. Energy levels of a single quartic oscillator for 𝜆 values in range of 0.1-1 are given. Perturbed and non-perturbed wave functions of the levels up to the fourth excited level are compared. Ground, first and second excited energy levels are also calculated by applying finite differences method and, results are compared with the ones obtained via perturbation theory. It is found that perturbation theory gives comparable results only for a small 𝜆 parameter and for the ground state. The quartic term in the Hamiltonian of the anharmonic oscillator leads to a more effective confinement of the particle which is deduced from the plots of wavefunctions and probability distributions. Meanwhile, the number of zero crossing nodes of the wavefunctions increases as the energy level increases, which is an expected result for both the harmonic and anharmonic oscillator.
In quantum mechanics, the propagator is basically the most important; It contains all physical properties, Feynman's path integration is widely accepted to be the formal method for finding the propagator. However, the method is limited to only some type of problems. One challenging and interesting problem which has not been able to be solved since the time of Feynman is potential with quadratic and cubic terms. The problem is analysed in detail by using Baker - Hausdorff lemma in order to find the possibility of the best approximation; It is found that the best practical one is the first-order semiclassical approximation by using condition the cubic potential term has value very smaller than the quadratic potential term. The approximated propagator has been obtained for quadratic and cubic potential terms when it is analogous to the propagator with quadratic potential term, we have increased prefactor and exponential terms due to the influence of cubic potential term which expresses the anharmonic part of the propagator. The approximated propagator is reduced to the well-known propagator for the simple harmonic oscillator when the anharmomc part is zero.
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.
A new formalism of the thermodynamic perturbation theory for the quantum-anharmonic oscillator is proposed. By this method, for each order of the perturbation series, the partition function and the free energy of the system can be obtained in a compact form which is expressed by the temperature Green's function of the harmonic quanta. Based upon a unified method, the various expressions of the perturbation series for the partition function and the free energy are also given. It is shown that the perturbation series for the free energy can be naturally formulated in terms of a cluster expansion.
This paper shows how to calculate the terms of a semiclassical (WKB) expansion of the quantum-mechanical propagator corresponding to the quartic anharmonic-oscillator potential, V=mω2q2/2 +λq4/4. This nonperturbative treatment expresses each term in the series as a path integral, which is then evaluated in the framework of a formalism, introduced by C. DeWitt-Morette, which does not entail the usual time-slicing operation followed by a limiting procedure. The Gaussian measure used absorbs all the quadratic terms in the expansion of the action functional about a classical path. The covariance of this Gaussian measure is Feynman’s Green function for the small-disturbance operator of the system. This function can be obtained by varying the constants of integration in the classical solution, and therefore the coefficients of the expansion depend only on this classical solution. If the latter is chosen to be the one which tends to its harmonic counterpart when λ→0, then it is seen that the propagator also tends to its harmonic counterpart when λ→0.