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arXiv:hep-th/9605181 · Physical Review Letters

Multiple-Scale Analysis of the Quantum Anharmonic Oscillator

Carl M. Bender, Luís M. A. Bettencourt

Published Nov 11, 1996• 85 citations
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Abstract

Conventional weak-coupling perturbation theory suffers from problems that arise from the resonant coupling of successive orders in the perturbation series. Multiple-scale perturbation theory avoids such problems by implicitly performing an infinite reordering and resummation of the conventional perturbation series. Multiple-scale analysis provides a good description of the classical anharmonic oscillator. Here, it is extended to study the Heisenberg operator equations of motion for the quantum anharmonic oscillator. The analysis yields a system of nonlinear operator differential equations, which is solved exactly. The solution provides an operator mass renormalization of the theory.

DOI
10.1103/physrevlett.77.4114

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Conventional weak-coupling perturbation theory suffers from problems that arise from the resonant coupling of successive orders in the perturbation series. Multiple-scale perturbation theory avoids such problems by implicitly performing an infinite reordering and resummation of the conventional perturbation series. Multiple-scale analysis provides a good description of the classical anharmonic oscillator. Here, it is extended to study the Heisenberg operator equations of motion for the quantum anharmonic oscillator. The analysis yields a system of nonlinear operator differential equations, which is solved exactly. The solution provides an operator mass renormalization of the theory.
The standard perturbation theory for time-dependent problems in quantum mechanics is reconsidered through renormalization-group methods. This approach justifies a posteriori the theory of quantum resonance given by the multiple-scale analysis of the perturbation series applied to the coupled equations of wave-function amplitudes. The resonance equations for the leading-order amplitude probabilties are then obtained up to second order and a general method for the algorithmic computation of higher-order terms of the perturbation series is given. The three-level model in a monochromatic wave with two resonant states is discussed in the light of the present results.
We develop a general formalism for calculating the large-order behavior of perturbation theory for quantized systems of unequal-mass coupled anharmonic oscillators. Our technique is based on a generalization of the semiclassical approximation which was used to study equal-mass oscillators in the first paper of this series. The unequal-mass problem is much more difficult because the path which minimizes the classical action is not a straight line. Assuming that this tunneling path is known, we derive a general expression for the physical-optics approximation to the wave function of a tunneling particle. This derivation rests on the construction of a WKB approximation in curved space. We thus completely reduce the general quantum problem to the much simpler classical one of determining the path. Then we present a perturbation scheme for finding the classical path for systems of oscillators whose masses only differ by a small amount. Finally, we illustrate our techniques by solving a two-mode unequal-mass oscillator and comparing these results with a computer calculation. Our theoretical predictions and numerical calculations agree.
We investigate the perturbative expansions of the ground-state wavefunctions of the quantum anharmonic oscillators. With an appropriate change of spatial scale, the weak-coupling Schrödinger equation is transformed to an equivalent strong-coupling one. The Friedberg–Lee–Zhao method is applied to obtain the improved perturbative expansions. These perturbative expansions give a correction to the WKB results for large spatial distances, and reproduce the conventional weak-coupling results for small spatial distances.
This paper considers the Heisenberg equations of motion $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{q}=\ensuremath{-}i[q, H]$, $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{p}=\ensuremath{-}i[p, H]$, for the quantum-mechanical Hamiltonian H(p,q)H(p, q)H(p,q) having one degree of freedom. It is a commonly held belief that such operator differential equations are intractable. However, a technique is presented here that allows one to obtain exact, closed-form solutions for huge classes of Hamiltonians. This technique, which is a generalization of the classical action-angle-variable methods, allows us to solve, albeit formally and implicitly, the operator differential equations of the anharmonic oscillator whose Hamiltonian is H=p22+q44H=\frac{{p}^{2}}{2}+\frac{{q}^{4}}{4}H=2p2​+4q4​.
We construct a generalised expression for the normal ordering of (a+a^{\dagger})^{m} for integral values of m and use the result to study the quantum anharmonic oscillator problem in the Heisenberg approach. In particular, we derive generalised expressions for energy eigen values and frequency shifts for the Hamiltonian H=\frac{x^{2}}{2}+\frac{\dot{x}^{2}}{2}+\fracλ{m}x^{m}. We also derive a closed form first order multi scale perturbation theoretic operator solution of this Hamiltonian with a view to generalise some recent results of Bender and Bettencourt .