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Journal of Physics A Mathematical and General

Quantum theory of anharmonic oscillators-a variational and systematic general approximation method

K Yamazaki

Published Feb 1, 1984• 26 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/17/2/021

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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.
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.
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.
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.
This paper discusses some practical problems in quantum mechanics, particularly finding solutions of the Schrödinger equation for some cases, as it is a pivotal part of this area. The article focuses on using perturbation theory to calculate the approximate ground state energy for both helium atom and quartic anharmonic oscillator. For the helium atom, conventional perturbation theory is used to solve for the approximate energy, whilst for the anharmonic oscillator, a slightly different approach, that is perturbation theory incorporating a parameter, is chosen to improve the precision of the approximation. The final relative errors for the ground state energy of helium atom model are, for the first order perturbation, , and for the second order, , and that for the anharmonic oscillator is . The study of perturbation theory gives insights into understandings of how it may be modified further to pursue a better level of precision. This facilitates development of new approximation techniques for solving quantum-mechanical issues.
A new approach to the eigenvalue problem in quantum mechanics is proposed. This approach is based on three propositions: 1) a perturbation theory which does not require knowledge of the entire eigenvalue spectrum of the unperturbed problem and which uses a nonlinearization procedure (leaving some latitude in the choice of a zeroth order approximation); 2) a relationship between the perturbation theory and a variational principle, namely that any variational calculation is none other than the first two terms of some nontrivial perturbation theory which, when developed further, can reveal the accuracy of the variational calculations and can refine them by an iterative procedure; 3) Dyson's argument, which serves as a criterion for the reasonableness of the choice of a zeroth order approximation (the unperturbed problem). The realization of this perturbation theory in a k-dimensional space is equivalent to the solution of a k-dimensional electrostatic problem with a variable dielectric permittivity. In the one-dimensional case and in cases which reduce to the one-dimensional case, all the corrections are written in quadratures. It is shown that the construction of an ordinary perturbation theory (in which the zeroth order approximation is an exactly solvable problem) within the framework of this perturbation theory is a purely algebraic procedure, which reduces to the solution of some simple recurrence relations. An approximation analogous to the leading logarithmic approximation of quantum field theory is constructed. Some standard problems of quantum mechanics—the anharmonic oscillator, the Zeeman effect, and the Stark effect—are treated as examples. It is shown that this new approach makes it possible to develop systematically a theory for strong coupling and large perturbations.
The variational approximation to the thermal properties of a quantum-mechanical system proposed by Giachetti and Tognetti [Phys. Rev. Lett. 55, 912 (1985); Phys. Rev. B 33, 7647 (1986)], Feynman and Kleinert [Phys. Rev. A 34, 5080 (1986)], and others is taken as the starting point, and a systematic perturbation series is developed for both the free energy and the particle density. Generalizations of the usual Feynman propagator appear naturally in the formalism. The anharmonic oscillator is treated as an example; the second-order correction to the free energy and the first-order correction to the particle density are evaluated explicitly and shown to lead to significant improvements over the variational solution.