Mean Field Approach to Quantum Duffing Oscillator

We propose a mean-field approach to the Duffing oscillator to construct perturbatively the bounded operators that generate the quantum states, and to define a non-Gaussian density operator. 1 The Duffing oscillator is a typical anharmonic oscillator whose classical aspect is relatively well-understood [1]. It has been studied in many fields of physics from classical mechanics to quantum mechanics, especially as a toy model for one-dimensional quantum field theory. In particular, more physical interests have been focused on the application of various perturbation techniques both as a classical and as a quantum model [2]. In this paper we study the quantum Duffing oscillator, a time-independent quantum anharmonic oscillator, ˆ H = ˆ p 2 2m + mω 2 ˆ q 2 2 + mλˆq 4 4. (1) Even though one can find in principle the exact quantum states by solving the time-independent Schrödinger equation, the results of most of attempts are minimal. In this Letters to Editor we shall propose a mean-field approach to the quantum Duffing oscillator, which has the following points different from other approaches: It takes into account a physically meaningful frequency that is quite close to the exact classical one. There appear no secular terms at least at the order of mλ for the coupling constant up to a critical value. A kind of renormalization can be used at the order of (mλ) 2. The motivation for this approach is based on two observations: both the extremization of the