We propose a high order numerical method for computing time dependent 4-D Wigner equation with unbounded potential and study a canonical quantum double-slit interference problem. To address the difficulties of 4-D phase space computations and higher derivatives from the Moyal expansion of nonlocal pseudo-differential operator for unbounded potentials, an operator splitting technique is adopted to decompose the 4-D Wigner equation into two sub-equations, which can be computed analytically or numerically with high efficiency. The first sub-equation contains only linear convection term in $(\bm x, t)$-space and can be solved with an advective method, while the second involves the pseudo-differential term and can be approximated by a plane wave expansion in $\bm k$-space. By exploiting properties of Fourier transformation, the expansion coefficients for the second sub-equation have explicit forms and the resulting scheme is shown to be unconditionally stable for any higher derivatives of the Moyal expansion, ensuring the feasibility of the 4-D Wigner numerical simulations for quantum double-slit interferences. Numerical experiments demonstrate the spectral convergence in $(\bm x, \bm k)$-space and provide highly accurate information on the number, position, and intensity of the interference fringes for different types of slits, quantum particle masses, and initial states (pure and mixed).
The global energy crisis and water pollution drive the researchers to develop highly effective and less energy intensive water purification technologies. In this study, a highly active WO3@TiO2–SiO2 nanocomposite was synthesized and used for photocatalytic degradation of persistent organic pollutants under simulated solar light. The optimum WO3@TiO2–SiO2 prepared with 2 wt% WO3 loading and calcination at 800 °C exhibited higher photocatalytic activity, as the rate constant (k1) for phenanthrene degradation was ∼7.1 times of that for the commercial TiO2 (P25). The extremely large specific surface area (>400 m2/g) of WO3@TiO2–SiO2 afforded it with enlarged pollutants adsorption performance and abundant active surface sites. The heterojunction of anatase with SiO2 as well as loading of WO3 decreased the band gap energy (Eg) of TiO2, which extended the utilization spectrum of TiO2 to visible region. Formation of Ti–O–Si band indicated the excess charges can cause Brønsted acidity due to the absorption of protons to compensate the charges. Moreover, the migration of photo-excited electrons from the conduction band of anatase to WO3 and holes in the opposite direction restrained the electron-hole recombination. The photocatalytic degradation mechanism and pathway for phenanthrene degradation were proposed based on experimental analysis and density functional theory (DFT) calculation, and the toxicities of the degradation intermediates were evaluated by quantitative structure–activity relationship (QSAR) analysis. WO3@TiO2–SiO2 also showed good separation (settling) performance and high stability. Our work is expected to offer new insight into the photocatalytic mechanism for WO3, TiO2 and SiO2 based heterojunctions, and rational design and synthesis of highly efficient photocatalysts for environmental application.