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Approximation of the Quantum Anharmonic Oscillator: Investigating the Accuracy of Effective Models for the Double Well Potential

Linda Nordmark, Jacob Pirinen

Published Jan 1, 2026• 0 citations
OpenAlex

Abstract

The report investigates how the quantum mechanical model of a particle in one dimension underthe influence of the double well potential V (x) = λ(x^2 − x_0^2)^2 can be approximated using aneffective model, where the system is approximated with a system of truncated dimension. Theinfinite dimensional space of energy eigenfunctions to the double well potential was estimatedby two finite dimensional spaces, ones which were spanned by the ground- and first excitedeigenstates of the harmonic oscillators corresponding to the two minima of the potential. Theenergy eigenvalues given by these models were then studied. Additionally, the effects on thepartition function, the internal energy and the heat capacity of the system were checked tofurther study the accuracy of the models. To determine the accuracy of the approximation ahigh order, highly resolved finite element method was used for comparison as stand in for anexact, analytical solution. All investigated quantities were found to be better approximated forlarger λ and lower temperatures, with the effective model of larger dimension achieving a betterapproximation. The result with respect to λ likely results from that, for low energy, the doublewell potential resembles two harmonic oscillator potentials more the higher the energy barrier,i.e. the higher the λ. The effective models being better approximations for lower temperaturescan be explained by the fact that at low temperature, the lowest energies have the strongestimpact on the behaviour of the system.

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The report investigates how the quantum mechanical model of a particle in one dimension underthe influence of the double well potential V (x) = λ(x^2 − x_0^2)^2 can be approximated using aneffective model, where the system is approximated with a system of truncated dimension. Theinfinite dimensional space of energy eigenfunctions to the double well potential was estimatedby two finite dimensional spaces, ones which were spanned by the ground- and first excitedeigenstates of the harmonic oscillators corresponding to the two minima of the potential. Theenergy eigenvalues given by these models were then studied. Additionally, the effects on thepartition function, the internal energy and the heat capacity of the system were checked tofurther study the accuracy of the models. To determine the accuracy of the approximation ahigh order, highly resolved finite element method was used for comparison as stand in for anexact, analytical solution. All investigated quantities were found to be better approximated forlarger λ and lower temperatures, with the effective model of larger dimension achieving a betterapproximation. The result with respect to λ likely results from that, for low energy, the doublewell potential resembles two harmonic oscillator potentials more the higher the energy barrier,i.e. the higher the λ. The effective models being better approximations for lower temperaturescan be explained by the fact that at low temperature, the lowest energies have the strongestimpact on the behaviour of the system.
Both single-well and multiwell one-dimensional anharmonic oscillators have provided an extremely fruitful testing ground for various modern techniques of microscopic quantum many-body theory and quantum field theory. They have served both as (0+1)-dimensional field-theory models of their important counterparts in a higher number of dimensions, and as algebraically simple yet highly nontrivial problems on which various approximation or truncation schemes could be tested to very high orders. Such techniques have included perturbation theory and its many possible resummation schemes, and the coupled-cluster method. In the present paper the spotlight is focused on variational methods, and we concentrate on their application to the particularly demanding double-well quartic anharmonic oscillator. We examine in detail both single-state and multistate variational calculations based on either orthogonal (uncorrelated) or nonorthogonal (correlated) basis functions. Particular attention is paid to the question of the sensitivity and accuracy of the results with respect to any nonlinear parameters that characterize the basis states. Practical schemes are suggested for their choice and are shown to be capable of very high accuracy. Use of this simple model also allows detailed comparisons to be made with other techniques of a perturbative or perturbative-variational hybrid character.
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.
A self-consistent, nonperturbative approximation scheme is proposed which is potentially applicable to arbitrary interacting quantum systems. For the case of self-interaction, the scheme consists in approximating the original interaction HI(ϕ) by a suitable "potential" V(ϕ) which satisfies the following two basic requirements, (i) exact solvability (ES): the "effective" Hamiltonian H0 generated by V(ϕ) is exactly solvable i.e., the spectrum of states |n〉 and the eigenvalues En are known and (ii) equality of quantum averages (EQA): 〈n|HI(ϕ)|n〉 = 〈n|V(ϕ)|n〉 for arbitrary n. The leading order (LO) results for |n〉 and En are thus readily obtained and are found to be accurate to within a few percent of the "exact" results. These LO-results are systematically improvable by the construction of an improved perturbation theory (IPT) with the choice of H0 as the unperturbed Hamiltonian and the modified interaction, λH′(ϕ)≡λ(HI(ϕ) - V(ϕ)), as the perturbation where λ is the coupling strength. The condition of convergence of the IPT for arbitrary λ is satisfied due to the EQA requirement which ensures that 〈n|λH′(ϕ)|n〉 = 0for arbitrary λ and n. This is in contrast to the divergence (which occurs even for infinitesimal λ!) in the naive perturbation theory where the original interaction λHI(ϕ) is chosen as the perturbation. We apply the method to the different cases of the anharmonic and the double well potentials, e.g. quartic-, sextic- and octic-anharmonic oscillators and quartic-, sextic-double well oscillators. Uniformly accurate results for the energy levels over the full allowed range of λ and n are obtained. The results compare well with the exact results predicted by supersymmetry for the case of the sextic anharmonic potential and the double well partner potential. Further improvement in the accuracy of the results by the use of IPT, is demonstrated. We also discuss the vacuum structure and stability of the resulting theory in the above approximation scheme.
Specific heats of quantum systems with symmetric and asymmetric double-well potentials have been calculated. In numerical calculations of their specific heats, we have adopted the combined method which takes into account not only eigenvalues of ϵn\epsilon_nϵn​ for 0≤n≤Nm0 \leq n \leq N_m0≤n≤Nm​ obtained by the energy-matrix diagonalization but also their extrapolated ones for Nm+1≤n<∞N_m+1 \leq n < \inftyNm​+1≤n<∞ (Nm=20N_m=20Nm​=20 or 30). Calculated specific heats are shown to be rather different from counterparts of a harmonic oscillator. In particular, specific heats of symmetric double-well systems at very low temperatures have the Schottky-type anomaly, which is rooted to a small energy gap in low-lying two-level eigenstates induced by a tunneling through the potential barrier. The Schottky-type anomaly is removed when an asymmetry is introduced into the double-well potential. It has been pointed out that the specific-heat calculation of a double-well system reported by Feranchuk, Ulyanenkov and Kuz'min [Chem. Phys. 157, 61 (1991)] is misleading because the zeroth-order operator method they adopted neglects crucially important off-diagonal contributions.
This is an investigation of an anharmonic oscillator characterized by the potential (ω02/ 2) q2 + λ′ q4. By using the exact relations obtained by evaluating ≪i|[H, O]∓|j > where O is an arbitrary operator, H is our total Hamiltonian and |i > and |j > are exact eigenstates of H, we derive an exact recurrence formula. This formula allows us to express τ-functions with a higher power of the variables through τ-functions with a lower power of the variables and energy eigenvalues. In this way we derive several exact relations, which are, in a sense, generalizations of the virial theorem and sum rules (Eqs. (4·5), (4·6) and (5·4)). These exact relations are the central equaions of this paper. On the basis of these exact relations we propose our “nearest neighbour level” (N. N. L) approximation, which seems to provide a good approximation scheme. We can also use our exact relations to test the validity of various approximation methods, and as an example, we discuss the “New-Tamm-Dancoff” (N. T. D)-type of approximation in detail.
An expression for the Green function G(E;x_1,x_2) of the Schroedinger equation is obtained through the approximations of the path integral by n-fold multiple integrals. The approximations to Re{G(E;x,x)} on the real E-axis have peaks near the values of the energy levels E_{j}. The analytic and numerical examples for one-dimensional and multi-dimensional harmonic and anharmonic oscillators, and Poeschl-Teller potential wells, show that median values of these peaks for approximate G(E;0,0) corresponds with accuracy of order 10% to the exact values of even levels already in the lowest orders of approximation n=1 and n=2, i.e. when the path integral is replaced by a line or double integral. The weights of the peaks approximate the values of the squared modulus of the wave functions at x=0 with the same accuracy.