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arXiv:physics/9905056

Post-Gaussian variational method for quantum anharmonic oscillator

Akihiro Ogura

Published May 28, 1999• 1 citation
arXivHTMLPDF

Abstract

Using the post-Gaussian trial functions, we calculate the variational solutions to the quantum-mechanical anharmonic oscillator. We evaluate not only the ground state but also some excited energies, and compare them with numerical results.
DOI
10.48550/arxiv.physics/9905056

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Three sets of trial wave functions were applied to calculate the ground state and first excited state energies of the x 4 and x 6 anharmonic oscillator potentials. We demonstrate that, even with a rather crude initial guess, results can be improved by proper adjustment of the trial wave functions after careful study of their general features. Meanwhile, we found some interesting and unexpected results when applying the second set of trial wave functions. Once again, we suggest using together with computer algebra software packages, these examples when teaching the variational method in a quantum mechanics course.
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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.
We present a systematic study of the Rayleigh--Ritz variational method for quantum oscillators in the Segal--Bargmann space. We rigorously derive the normalizability condition ∣α∣<12|α| < \tfrac{1}{2}∣α∣<21​ for generalized Gaussian trial functions ψ(z)=eαz2+βzψ(z) = e^{αz^2 + βz}ψ(z)=eαz2+βz through convergence analysis of Gaussian integrals in the complex plane. Applications to the harmonic oscillator demonstrate exact recovery of the ground state in Segal--Bargmann space when the trial family contains the true solution. For the quartic anharmonic oscillator (H^=−12∂x2+12x2+λx4\hat{H} = -\tfrac{1}{2}\partial_x^2 + \tfrac{1}{2}x^2 + λx^4H^=−21​∂x2​+21​x2+λx4), adaptive Gaussian ansätze in position space yield a cubic stationarity equation and perturbative energy expansions beyond first order, capturing anharmonic wavefunction narrowing. In contrast, monomial trial functions (ψn(z)=znψ_n(z) = z^nψn​(z)=zn) in the Segal--Bargmann space -- while providing rigorous upper bounds En=n+12+3λ4(2n2+2n+1)E_n = n + \tfrac{1}{2} + \tfrac{3λ}{4}(2n^2 + 2n + 1)En​=n+21​+43λ​(2n2+2n+1) for excited states -- lack width adaptability and are limited to first-order accuracy for ground-state calculations. We further analyze displaced Gaussians and displaced monomials for asymmetric potentials (e.g., x3+x4x^3 + x^4x3+x4), showing that displacement parameters are essential to capture parity breaking and stabilization effects (E0≈12+3μ4−9λ24+⋯E_0 \approx \tfrac{1}{2} + \tfrac{3μ}{4} - \tfrac{9λ^2}{4} + \cdotsE0​≈21​+43μ​−49λ2​+⋯).
A quantum anharmonic oscillator in the ground state has been considered under the conditions of loading with an external force. The wave functions have been calculated for different forces, and the eigenvalues of the energy of the system have been determined as a function of the load. It has been established that the zero-point energy of the oscillator varies linearly with a variation in the force (decreases under tension and increases under compression) and that the average kinetic and potential components of the energy are also characterized by linear variations.