Numerical Methods for the Wigner Equation with Unbounded Potential
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Unbounded potentials are always utilized to strictly confine quantum dynamics and generate bound or stationary states due to the existence of quantum tunneling. However, the existed accurate Wigner solvers are often designed for either localized potentials or those of the polynomial type. This paper attempts to solve the time-dependent Wigner equation in the presence of a general class of unbounded potentials by exploiting two equivalent forms of the pseudo-differential operator: integral form and series form (i.e., the Moyal expansion). The unbounded parts at infinities are approximated or modeled by polynomials and then a remaining localized potential dominates the central area. The fact that the Moyal expansion reduces to a finite series for polynomial potentials is fully utilized. In order to accurately resolve both the pseudo-differential operator and the linear differential operator, a spectral collocation scheme for the phase space and an explicit fourth-order Runge–Kutta time discretization are adopted. We are able to prove that the resulting full discrete spectral scheme conserves both mass and energy. Several typical quantum systems are simulated with a high accuracy and reliable estimation of macroscopically measurable quantities is thus obtained.
KeywordsWigner equation Moyal expansion Spectral method Quantum dynamics Unbounded potential Uncertainty principle Double-well Pöschl–Teller potential Anharmonic oscillator
Mathematics Subject Classification81Q05 65M70 81S30 45K05 82C10
This research is supported by grants from the National Natural Science Foundation of China (Nos. 11471025, 11421101, 11822102). Z. C. is partially supported by Peking University Weng Hongwu original research fund (No. WHW201501). Y. X. is partially supported by The Elite Program of Computational and Applied Mathematics for PhD Candidates in Peking University. The authors are grateful to the useful discussions with Wei Cai, Jian Liu and Jing Shi.